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On sampling theories and discontinuous Dirac systems with eigenparameter in the boundary conditions

Mohammed M Tharwat

Author Affiliations

Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

Boundary Value Problems 2013, 2013:65  doi:10.1186/1687-2770-2013-65

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/65


Received:8 November 2012
Accepted:11 March 2013
Published:29 March 2013

© 2013 Tharwat; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The sampling theory says that a function may be determined by its sampled values at some certain points provided the function satisfies some certain conditions. In this paper we consider a Dirac system which contains an eigenparameter appearing linearly in one condition in addition to an internal point of discontinuity. We closely follow the analysis derived by Annaby and Tharwat (J. Appl. Math. Comput. 2010, doi:10.1007/s12190-010-0404-9) to establish the needed relations for the derivations of the sampling theorems including the construction of Green’s matrix as well as the eigen-vector-function expansion theorem. We derive sampling representations for transforms whose kernels are either solutions or Green’s matrix of the problem. In the special case, when our problem is continuous, the obtained results coincide with the corresponding results in Annaby and Tharwat (J. Appl. Math. Comput. 2010, doi:10.1007/s12190-010-0404-9).

MSC: 34L16, 94A20, 65L15.

Keywords:
Dirac systems; transmission conditions; eigenvalue parameter in the boundary conditions; discontinuous boundary value problems

1 Introduction

Sampling theory is one of the most powerful results in signal analysis. It is of great need in signal processing to reconstruct (recover) a signal (function) from its values at a discrete sequence of points (samples). If this aim is achieved, then an analog (continuous) signal can be transformed into a digital (discrete) one and then it can be recovered by the receiver. If the signal is band-limited, the sampling process can be done via the celebrated Whittaker, Shannon and Kotel’nikov (WSK) sampling theorem [1-3]. By a band-limited signal with band width σ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M1">View MathML</a>, i.e., the signal contains no frequencies higher than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M2">View MathML</a> cycles per second (cps), we mean a function in the Paley-Wiener space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M3">View MathML</a> of entire functions of exponential type at most σ which are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M4">View MathML</a>-functions when restricted to ℝ. In other words, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M5">View MathML</a> if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M6">View MathML</a> such that, cf.[4,5],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M7">View MathML</a>

(1.1)

Now WSK sampling theorem states [6,7]: If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M5">View MathML</a>, then it is completely determined from its values at the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M10">View MathML</a>, by means of the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M11">View MathML</a>

(1.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M12">View MathML</a>

(1.3)

The sampling series (1.2) is absolutely and uniformly convergent on compact subsets of ℂ.

The WSK sampling theorem has been generalized in many different ways. Here we are interested in two extensions. The first is concerned with replacing the equidistant sampling points by more general ones, which is very important from the practical point of view. The following theorem which is known in some literature as the Paley-Wiener theorem [5] gives a sampling theorem with a more general class of sampling points. Although the theorem in its final form may be attributed to Levinson [8] and Kadec [9], it could be named after Paley and Wiener who first derived the theorem in a more restrictive form; see [6,7,10] for more details.

The Paley-Wiener theorem states that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M10">View MathML</a>, is a sequence of real numbers such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M15">View MathML</a>

(1.4)

and G is the entire function defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M16">View MathML</a>

(1.5)

then, for any function of the form (1.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M17">View MathML</a>

(1.6)

The series (1.6) converges uniformly on compact subsets of ℂ.

The WSK sampling theorem is a special case of this theorem because if we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M18">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M19">View MathML</a>

The sampling series (1.6) can be regarded as an extension of the classical Lagrange interpolation formula to ℝ for functions of exponential type. Therefore, (1.6) is called a Lagrange-type interpolation expansion.

The second extension of WSK sampling theorem is the theorem of Kramer, [11] which states that if I is a finite closed interval, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M20">View MathML</a> is a function continuous in t such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M21">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M22">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M23">View MathML</a> be a sequence of real numbers such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M24">View MathML</a> is a complete orthogonal set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M25">View MathML</a>. Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M26">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M27">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M28">View MathML</a>

(1.7)

Series (1.7) converges uniformly wherever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M29">View MathML</a>, as a function of t, is bounded. In this theorem, sampling representations were given for integral transforms whose kernels are more general than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M30">View MathML</a>. Also Kramer’s theorem is a generalization of WSK theorem. If we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M33">View MathML</a>, then (1.7) is (1.2).

The relationship between both extensions of WSK sampling theorem has been investigated extensively. Starting from a function theory approach, cf.[12], it was proved in [13] that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M36">View MathML</a>, satisfies some analyticity conditions, then Kramer’s sampling formula (1.7) turns out to be a Lagrange interpolation one; see also [14-16]. In another direction, it was shown that Kramer’s expansion (1.7) could be written as a Lagrange-type interpolation formula if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M37">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M38">View MathML</a> were extracted from ordinary differential operators; see the survey [17] and the references cited therein. The present work is a continuation of the second direction mentioned above. We prove that integral transforms associated with Dirac systems with an internal point of discontinuity can also be reconstructed in a sampling form of Lagrange interpolation type. We would like to mention that works in direction of sampling associated with eigenproblems with an eigenparameter in the boundary conditions are few; see, e.g., [18-20]. Also, papers in sampling with discontinuous eigenproblems are few; see [21-24]. However, sampling theories associated with Dirac systems which contain eigenparameter in the boundary conditions and have at the same time discontinuity conditions, do not exist as far as we know. Our investigation is be the first in that direction, introducing a good example. To achieve our aim we briefly study the spectral analysis of the problem. Then we derive two sampling theorems using solutions and Green’s matrix respectively.

2 The eigenvalue problem

In this section, we define our boundary value problem and state some of its properties. We consider the Dirac system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M39">View MathML</a>

(2.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M40">View MathML</a>

(2.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M41">View MathML</a>

(2.3)

and transmission conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M42">View MathML</a>

(2.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M43">View MathML</a>

(2.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44">View MathML</a>; the real-valued functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M46">View MathML</a> are continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M48">View MathML</a> and have finite limits <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M49">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M50">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M52">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M54">View MathML</a>.

In [24] the authors discussed problem (2.1)-(2.5) but with the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M55">View MathML</a> instead of (2.3). To formulate a theoretic approach to problem (2.1)-(2.5), we define the Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M56">View MathML</a> with an inner product, see [19,20],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M57">View MathML</a>

(2.6)

where ⊤ denotes the matrix transpose,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M58">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M61">View MathML</a>. For convenience, we put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M62">View MathML</a>

(2.7)

Equation (2.1) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M63">View MathML</a>

(2.8)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M64">View MathML</a>

(2.9)

For functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M65">View MathML</a>, which are defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M66">View MathML</a> and have finite limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M67">View MathML</a>, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M69">View MathML</a>, we denote the functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M70">View MathML</a>

(2.10)

which are defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M71">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M72">View MathML</a>, respectively.

In the following lemma, we prove that the eigenvalues of problem (2.1)-(2.5) are real.

Lemma 2.1The eigenvalues of problem (2.1)-(2.5) are real.

Proof Assume the contrary that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M73">View MathML</a> is a nonreal eigenvalue of problem (2.1)-(2.5). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M74">View MathML</a> be a corresponding (non-trivial) eigenfunction. By (2.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M75">View MathML</a>

Integrating the above equation through <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M77">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M78">View MathML</a>

(2.11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M79">View MathML</a>

(2.12)

Then from (2.2), (2.3) and transmission conditions, we have, respectively,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M80">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M81">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M82">View MathML</a>, it follows from the last three equations and (2.11), (2.12) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M83">View MathML</a>

(2.13)

This contradicts the conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M84">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M85">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M73">View MathML</a> must be real. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M87">View MathML</a> be the set of all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M88">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M90">View MathML</a> are absolutely continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M60">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M60">View MathML</a>. Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M97">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M98">View MathML</a>

(2.14)

Thus, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99">View MathML</a> is symmetric in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M100">View MathML</a>. Indeed, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M101">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M102">View MathML</a>

The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M103">View MathML</a> and the eigenvalue problem (2.1)-(2.5) have the same eigenvalues. Therefore they are equivalent in terms of this aspect.

Lemma 2.2Letλandμbe two different eigenvalues of problem (2.1)-(2.5). Then the corresponding eigenfunctions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M104">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M105">View MathML</a>of this problem satisfy the following equality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M106">View MathML</a>

(2.15)

Proof Equation (2.15) follows immediately from the orthogonality of the corresponding eigenelements:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M107">View MathML</a>

 □

Now, we construct a special fundamental system of solutions of equation (2.1) for λ not being an eigenvalue. Let us consider the next initial value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M108">View MathML</a>

(2.16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M109">View MathML</a>

(2.17)

By virtue of Theorem 1.1 in [25], this problem has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M110">View MathML</a>, which is an entire function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44">View MathML</a> for each fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M112">View MathML</a>. Similarly, employing the same method as in the proof of Theorem 1.1 in [25], we see that the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M113">View MathML</a>

(2.18)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M114">View MathML</a>

(2.19)

has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M115">View MathML</a>, which is an entire function of parameter λ for each fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M116">View MathML</a>.

Now the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M117">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M118">View MathML</a> are defined in terms of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M119">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M60">View MathML</a>, respectively, as follows: The initial-value problem,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M122">View MathML</a>

(2.20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M123">View MathML</a>

(2.21)

has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M124">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M125">View MathML</a>.

Similarly, the following problem also has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M126">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M127">View MathML</a>

(2.22)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M128">View MathML</a>

(2.23)

Let us construct two basic solutions of equation (2.1) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M129">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M130">View MathML</a>

(2.24)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M131">View MathML</a>

(2.25)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M132">View MathML</a>

(2.26)

By virtue of equations (2.21) and (2.23), these solutions satisfy both transmission conditions (2.4) and (2.5). These functions are entire in λ for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M133">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M134">View MathML</a> denote the Wronskian of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M135">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M136">View MathML</a> defined in [[26], p.194], i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M137">View MathML</a>

It is obvious that the Wronskian

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M138">View MathML</a>

(2.27)

are independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M139">View MathML</a> and are entire functions. Taking into account (2.21) and (2.23), a short calculation gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M140">View MathML</a>

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44">View MathML</a>.

Corollary 2.3The zeros of the functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M142">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M143">View MathML</a>coincide.

Then we may take into consideration the characteristic function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M144">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M145">View MathML</a>

(2.28)

In the following lemma, we show that all eigenvalues of problem (2.1)-(2.5) are simple.

Lemma 2.4All eigenvalues of problem (2.1)-(2.5) are just zeros of the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M144">View MathML</a>. Moreover, every zero of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M144">View MathML</a>has multiplicity one.

Proof Since the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M148">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M149">View MathML</a> satisfy the boundary condition (2.2) and both transmission conditions (2.4) and (2.5), to find the eigenvalues of the (2.1)-(2.5), we have to insert the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M148">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M149">View MathML</a> in the boundary condition (2.3) and find the roots of this equation.

By (2.1) we obtain for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M153">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M154">View MathML</a>

Integrating the above equation through <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M77">View MathML</a>, and taking into account the initial conditions (2.17), (2.21) and (2.23), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M157">View MathML</a>

(2.29)

Dividing both sides of (2.29) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M158">View MathML</a> and by letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M159">View MathML</a>, we arrive to the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M160">View MathML</a>

(2.30)

We show that equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M161">View MathML</a>

(2.31)

has only simple roots. Assume the converse, i.e., equation (2.31) has a double root <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M162">View MathML</a>, say. Then the following two equations hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M163">View MathML</a>

(2.32)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M164">View MathML</a>

(2.33)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M165">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M162">View MathML</a> is real, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M167">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M168">View MathML</a>. From (2.32) and (2.33),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M169">View MathML</a>

(2.34)

Combining (2.34) and (2.30) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M170">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M171">View MathML</a>

(2.35)

contradicting the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M85">View MathML</a>. The other case, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M173">View MathML</a>, can be treated similarly and the proof is complete. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M174">View MathML</a> denote the sequence of zeros of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M144">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M176">View MathML</a>

(2.36)

are the corresponding eigenvectors of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99">View MathML</a> is symmetric, then it is easy to show that the following orthogonality relation holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M179">View MathML</a>

(2.37)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M180">View MathML</a> is a sequence of eigen-vector-functions of (2.1)-(2.5) corresponding to the eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M174">View MathML</a>. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M182">View MathML</a> the normalized eigenvectors of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M184">View MathML</a>

(2.38)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M185">View MathML</a> satisfies (2.3)-(2.5), then the eigenvalues are also determined via

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M186">View MathML</a>

(2.39)

Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M187">View MathML</a> is another set of eigen-vector-functions which is related by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M180">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M189">View MathML</a>

(2.40)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M190">View MathML</a> are non-zero constants, since all eigenvalues are simple. Since the eigenvalues are all real, we can take the eigen-vector-functions to be real-valued.

Now we derive the asymptotic formulae of the eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M174">View MathML</a> and the eigen-vector-functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M180">View MathML</a>. We transform equations (2.1), (2.17), (2.21) and (2.24) into the integral equations, see [26], as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M193">View MathML</a>

(2.41)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M194">View MathML</a>

(2.42)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M195">View MathML</a>

(2.43)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M196">View MathML</a>

(2.44)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M197">View MathML</a> the following estimates hold uniformly with respect to x, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M133">View MathML</a>, cf. [[25], p.55],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M199">View MathML</a>

(2.45)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M200">View MathML</a>

(2.46)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M201">View MathML</a>

(2.47)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M202">View MathML</a>

(2.48)

Now we find an asymptotic formula of the eigenvalues. Since the eigenvalues of the boundary value problem (2.1)-(2.5) coincide with the roots of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M203">View MathML</a>

(2.49)

then from the estimates (2.47), (2.48) and (2.49), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M204">View MathML</a>

which can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M205">View MathML</a>

(2.50)

Then, from (2.45) and (2.46), equation (2.50) has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M206">View MathML</a>

(2.51)

For large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M207">View MathML</a>, equation (2.51) obviously has solutions which, as is not hard to see, have the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M208">View MathML</a>

(2.52)

Inserting these values in (2.51), we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M209">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M210">View MathML</a>. Thus we obtain the following asymptotic formula for the eigenvalues:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M211">View MathML</a>

(2.53)

Using the formulae (2.53), we obtain the following asymptotic formulae for the eigen-vector-functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M212">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M213">View MathML</a>

(2.54)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M214">View MathML</a>

(2.55)

3 Green’s matrix and expansion theorem

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M215">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M216">View MathML</a>, be a continuous vector-valued function. To study the completeness of the eigenvectors of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99">View MathML</a>, and hence the completeness of the eigen-vector-functions of (2.1)-(2.5), we derive Green’s function of problem (2.1)-(2.5) as well as the resolvent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99">View MathML</a>. Indeed, let λ be not an eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99">View MathML</a> and consider the inhomogeneous problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M220">View MathML</a>

where I is the identity operator. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M221">View MathML</a>

then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M222">View MathML</a>

(3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M223">View MathML</a>

(3.2)

and the boundary conditions (2.2), (2.4) and (2.5) with λ is not an eigenvalue of problem (2.1)-(2.5).

Now, we can represent the general solution of (3.1) in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M224">View MathML</a>

(3.3)

We applied the standard method of variation of the constants to (3.3), and thus the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M225">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M226">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M227">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M228">View MathML</a> satisfy the linear system of equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M229">View MathML</a>

(3.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M230">View MathML</a>

(3.5)

Since λ is not an eigenvalue and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M231">View MathML</a>, each of the linear system in (3.4) and (3.5) has a unique solution which leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M232">View MathML</a>

(3.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M233">View MathML</a>

(3.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M234">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M235">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M236">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M237">View MathML</a> are arbitrary constants, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M238">View MathML</a>

Substituting equations (3.6) and (3.7) into (3.3), we obtain the solution of (3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M239">View MathML</a>

(3.8)

Then from (2.2), (3.2) and the transmission conditions (2.4) and (2.5), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M240">View MathML</a>

Then (3.8) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M241">View MathML</a>

(3.9)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M242">View MathML</a>

(3.10)

which can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M243">View MathML</a>

(3.11)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M244">View MathML</a>

(3.12)

Expanding (3.12) we obtain the concrete form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M245">View MathML</a>

(3.13)

The matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M246">View MathML</a> is called Green’s matrix of problem (2.1)-(2.5). Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M246">View MathML</a> is a meromorphic function of λ, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M248">View MathML</a>, which has simple poles only at the eigenvalues. Although Green’s matrix looks as simple as that of Dirac systems, cf., e.g., [25,26], it is rather complicated because of the transmission conditions (see the example at the end of this paper). Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M249">View MathML</a>

(3.14)

Lemma 3.1The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99">View MathML</a>is self-adjoint in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M100">View MathML</a>.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99">View MathML</a> is a symmetric densely defined operator, then it is sufficient to show that the deficiency spaces are the null spaces, and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M253">View MathML</a>. Indeed, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M254">View MathML</a> and λ is a non-real number, then taking

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M255">View MathML</a>

implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M256">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M246">View MathML</a> satisfies the conditions (2.2)-(2.5), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M258">View MathML</a>. Now we prove that the inverse of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M259">View MathML</a> exists. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M260">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M261">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M262">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M263">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M264">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M265">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M266">View MathML</a>, the resolvent operator of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99">View MathML</a>, exists. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M268">View MathML</a>

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M269">View MathML</a>. The domains of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M270">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M271">View MathML</a> are exactly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M100">View MathML</a>. Consequently, the ranges of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M273">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M274">View MathML</a> are also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M100">View MathML</a>. Hence the deficiency spaces of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99">View MathML</a> are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M277">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99">View MathML</a> is self-adjoint. □

The next theorem is an eigenfunction expansion theorem. The proof is exactly similar to that of Levitan and Sargsjan derived in [[25], pp.67-77]; see also [26-29].

Theorem 3.2

(i) For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M279">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M280">View MathML</a>

(3.15)

(ii) For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M281">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M282">View MathML</a>

(3.16)

the series being absolutely and uniformly convergent in the first component for on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M66">View MathML</a>, and absolutely convergent in the second component.

4 The sampling theorems

The first sampling theorem of this section associated with the boundary value problem (2.1)-(2.5) is the following theorem.

Theorem 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M284">View MathML</a>. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M286">View MathML</a>

(4.1)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M135">View MathML</a>is the solution defined above. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M288">View MathML</a>is an entire function of exponential type that can be reconstructed from its values at the points<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M174">View MathML</a>via the sampling formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M290">View MathML</a>

(4.2)

The series (4.2) converges absolutely onand uniformly on any compact subset of ℂ, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M144">View MathML</a>is the entire function defined in (2.28).

Proof The relation (4.1) can be rewritten in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M292">View MathML</a>

(4.3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M293">View MathML</a>

Since both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M294">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M295">View MathML</a> are in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M100">View MathML</a>, then they have the Fourier expansions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M297">View MathML</a>

(4.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M299">View MathML</a> are the Fourier coefficients

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M300">View MathML</a>

(4.5)

Applying Parseval’s identity to (4.3), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M301">View MathML</a>

(4.6)

Now we calculate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M302">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M303">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M305">View MathML</a>. To prove expansion (4.2), we need to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M306">View MathML</a>

(4.7)

Indeed, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M305">View MathML</a> be fixed. By the definition of the inner product of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M100">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M310">View MathML</a>

(4.8)

From Green’s identity, see [[25], p.51], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M311">View MathML</a>

(4.9)

Then (4.9) and the initial conditions (2.21) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M312">View MathML</a>

(4.10)

From (2.40), (2.19) and (2.7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M313">View MathML</a>

(4.11)

Also, from (2.40) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M314">View MathML</a>

(4.12)

Then from (2.26) and (4.12) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M315">View MathML</a>

(4.13)

Substituting from (4.10), (4.11) and (4.13) into (4.8), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M316">View MathML</a>

(4.14)

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M317">View MathML</a> in (4.14), since the zeros of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M144">View MathML</a> are simple, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M319">View MathML</a>

(4.15)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M305">View MathML</a> are arbitrary, then (4.14) and (4.15) hold for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M125">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M305">View MathML</a>. Therefore, from (4.14) and (4.15), we get (4.7). Hence (4.2) is proved with a pointwise convergence on ℂ. Now we investigate the convergence of (4.2). First we prove that it is absolutely convergent on ℂ. Using Cauchy-Schwarz’ inequality for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M325">View MathML</a>

(4.16)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M294">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M327">View MathML</a>, then the two series on the right-hand side of (4.16) converge. Thus series (4.2) converges absolutely on ℂ. As for uniform convergence, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M328">View MathML</a> be compact. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M329">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M330">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M331">View MathML</a> to be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M332">View MathML</a>

(4.17)

Using the same method developed above, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M333">View MathML</a>

(4.18)

Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M334">View MathML</a>

(4.19)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M335">View MathML</a> is compact, then we can find a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M336">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M337">View MathML</a>

(4.20)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M338">View MathML</a>

(4.21)

uniformly on M. In view of Parseval’s equality,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M339">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M340">View MathML</a> uniformly on M. Hence (4.2) converges uniformly on M. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M288">View MathML</a> is an entire function. From the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M342">View MathML</a>

and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M343">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M344">View MathML</a>, are entire functions of exponential type, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M288">View MathML</a> is of exponential type. □

Remark 4.2 To see that expansion (4.2) is a Lagrange-type interpolation, we may replace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M144">View MathML</a> by the canonical product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M347">View MathML</a>

(4.22)

From Hadamard’s factorization theorem, see [4], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M348">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M349">View MathML</a> is an entire function with no zeros. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M350">View MathML</a>

and (4.1), (4.2) remain valid for the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M351">View MathML</a>. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M352">View MathML</a>

(4.23)

We may redefine (4.1) by taking kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M353">View MathML</a> to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M354">View MathML</a>

(4.24)

The next theorem is devoted to giving vector-type interpolation sampling expansions associated with problem (2.1)-(2.5) for integral transforms whose kernels are defined in terms of Green’s matrix. As we see in (3.12), Green’s matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M355">View MathML</a> of problem (2.1)-(2.5) has simple poles at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M356">View MathML</a>. Define the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M357">View MathML</a> to be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M358">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M359">View MathML</a> is a fixed point and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M144">View MathML</a> is the function defined in (2.28) or it is the canonical product (4.22).

Theorem 4.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M361">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M362">View MathML</a>be the vector-valued transform

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M363">View MathML</a>

(4.25)

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M364">View MathML</a>is a vector-valued entire function of exponential type that admits the vector-valued sampling expansion

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M365">View MathML</a>

(4.26)

The vector-valued series (4.26) converges absolutely onand uniformly on compact subsets of ℂ. Here (4.26) means

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M366">View MathML</a>

(4.27)

where both series converge absolutely onand uniformly on compact sets of ℂ.

Proof The integral transform (4.25) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M367">View MathML</a>

(4.28)

Applying Parseval’s identity to (4.28) with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M368">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M369">View MathML</a>

(4.29)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M44">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M371">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M305">View MathML</a>. Since each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M373">View MathML</a> is an eigenvector of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M99">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M375">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M376">View MathML</a>

(4.30)

From (3.14) and (4.30) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M377">View MathML</a>

(4.31)

Then from (2.26) and (2.40) in (4.31), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M378">View MathML</a>

(4.32)

Hence equation (4.32) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M379">View MathML</a>

(4.33)

The definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M380">View MathML</a> implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M381">View MathML</a>

(4.34)

Moreover, from (3.12) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M382">View MathML</a>

(4.35)

Then from (4.35), (2.26) and (2.40) in (4.34), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M383">View MathML</a>

(4.36)

Combining (4.36) and (4.33), yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M384">View MathML</a>

(4.37)

Taking the limit when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M385">View MathML</a> in (4.28), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M386">View MathML</a>

(4.38)

Making use of (4.37), we may rewrite (4.38) as, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M359">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M388">View MathML</a>

(4.39)

The interchange of the limit and summation is justified by the asymptotic behavior of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M389">View MathML</a> and that of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M144">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M391">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M392">View MathML</a>, then (4.39) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M393">View MathML</a>

(4.40)

Combining (4.37), (4.40) and (4.29), we get (4.28) under the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M391">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M392">View MathML</a> for all n. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M396">View MathML</a>, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M397">View MathML</a> or 2, the same expansions hold with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M398">View MathML</a>. The convergence properties as well as the analytic and growth properties can be established as in Theorem 4.1 above. □

Now we derive an example illustrating the previous results.

Example 4.1

Consider the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M399">View MathML</a>

(4.41)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M400">View MathML</a>

(4.42)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M401">View MathML</a>

(4.43)

and transmission conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M402">View MathML</a>

(4.44)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M403">View MathML</a>

(4.45)

This problem is a special case of problem (2.1)-(2.5) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M404">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M405">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M406">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M407">View MathML</a>. For simplicity, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M408">View MathML</a>

In the notations of the above section, the solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M135">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M185">View MathML</a> are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M411">View MathML</a>

(4.46)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M412">View MathML</a>

(4.47)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M413">View MathML</a>

The eigenvalues are the solutions of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M414">View MathML</a>

which can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M415">View MathML</a>

(4.48)

Green’s function of problem (4.41)-(4.45) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M416">View MathML</a>

(4.49)

By Theorem 4.1, the transform

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M417">View MathML</a>

(4.50)

has the following expansion:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M418">View MathML</a>

(4.51)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M174">View MathML</a> are the zeros of (4.48). In the view of Theorem 4.3, the vector-valued transform

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M420">View MathML</a>

(4.52)

has the following vector-valued expansion:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M421">View MathML</a>

(4.53)

It should be noted that with any choices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M422">View MathML</a>, α, β, we cannot always compute the eigenvalue of problem (4.41)-(4.45). Hence the eigenvalues are the points of ℝ which satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M423">View MathML</a>

(4.54)

This is illustrated in Figures 1 and 2.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, therefore, acknowledges with thanks DSR technical and financial support.

References

  1. Kotel’nikov, V: On the carrying capacity of the “ether” and wire in telecommunications. Material for the first all union conference on questions of communications. Izd. Red. Upr. Svyazi RKKA. 55, 55–64 (in Russian) (1933)

  2. Shannon, C: Communication in the presence of noise. Proc. IRE. 37, 10–21 (1949)

  3. Whittaker, E: On the functions which are represented by the expansion of the interpolation theory. Proc. R. Soc. Edinb., Sect. A, Math.. 35, 181–194 (1915)

  4. Boas, R: Entire Functions, Academic Press, New York (1954)

  5. Paley, R, Wiener, N: Fourier Transforms in the Complex Domain, Amer. Math. Soc., Providence (1934)

  6. Higgins, JR: Sampling Theory in Fourier and Signal Analysis: Foundations, Oxford University Press, Oxford (1996)

  7. Zayed, AI: Advances in Shannon’s Sampling Theory, CRC Press, Boca Raton (1993).

  8. Levinson, N: Gap and Density Theorems, Amer. Math. Soc., Providence (1940)

  9. Kadec, MI: The exact value of Paley-Wiener constant. Sov. Math. Dokl.. 5, 559–561 (1964)

  10. Hinsen, G: Irregular sampling of bandlimited <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/65/mathml/M430">View MathML</a>-functions. J. Approx. Theory. 72, 346–364 (1993). Publisher Full Text OpenURL

  11. Kramer, HP: A generalized sampling theorem. J. Math. Phys.. 38, 68–72 (1959)

  12. Everitt, WN, Hayman, WK, Nasri-Roudsari, G: On the representation of holomorphic functions by integrals. Appl. Anal.. 65, 95–102 (1997). Publisher Full Text OpenURL

  13. Everitt, WN, Nasri-Roudsari, G, Rehberg, J: A note on the analytic form of the Kramer sampling theorem. Results Math.. 34, 310–319 (1988)

  14. Everitt, WN, Garcia, AG, Hernández-Medina, MA: On Lagrange-type interpolation series and analytic Kramer kernels. Results Math.. 51, 215–228 (2008). Publisher Full Text OpenURL

  15. Garcia, AG, Littlejohn, LL: On analytic sampling theory. J. Comput. Appl. Math.. 171, 235–246 (2004). Publisher Full Text OpenURL

  16. Higgins, JR: A sampling principle associated with Saitoh’s fundamental theory of linear transformations. In: Saitoh S, Hayashi N, Yamamoto M (eds.) Analytic Extension Formulas and Their Applications, Kluwer Academic, Norwell (2001)

  17. Everitt, WN, Nasri-Roudsari, G: Interpolation and sampling theories, and linear ordinary boundary value problems. In: Higgins JR, Stens LR (eds.) Sampling Theory in Fourier and Signal Analysis: Advanced Topics, Oxford University Press, Oxford (1999) (Chapter 5)

  18. Annaby, MH, Freiling, G: Sampling integrodifferential transforms arising from second order differential operators. Math. Nachr.. 216, 25–43 (2000). Publisher Full Text OpenURL

  19. Annaby, MH, Tharwat, MM: On sampling theory and eigenvalue problems with an eigenparameter in the boundary conditions. SUT J. Math.. 42, 157–176 (2006)

  20. Annaby, MH, Tharwat, MM: On sampling and Dirac systems with eigenparameter in the boundary conditions. J. Appl. Math. Comput. doi:10.1007/s12190-010-0404-9 (2010)

  21. Annaby, MH, Freiling, G: A sampling theorem for transformations with discontinuous kernels. Appl. Anal.. 83, 1053–1075 (2004). Publisher Full Text OpenURL

  22. Annaby, MH, Freiling, G, Zayed, AI: Discontinuous boundary-value problems: expansion and sampling theorems. J. Integral Equ. Appl.. 16, 1–23 (2004). PubMed Abstract | Publisher Full Text | PubMed Central Full Text OpenURL

  23. Tharwat, MM: Discontinuous Sturm-Liouville problems and associated sampling theories. Abstr. Appl. Anal. doi:10.1155/2011/610232 (2011)

  24. Tharwat, MM, Yildirim, A, Bhrawy, AH: Sampling of discontinuous Dirac systems. Numer. Funct. Anal. Optim.. 34(3), 323–348 (2013). Publisher Full Text OpenURL

  25. Levitan, BM, Sargsjan, IS: Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, American Mathematical Society, Providence (1975)

  26. Levitan, BM, Sargsjan, IS: Sturm-Liouville and Dirac Operators, Kluwer Academic, Dordrecht (1991)

  27. Fulton, CT: Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. R. Soc. Edinb.. 77(A), 293–308 (1977)

  28. Hinton, DB: An expansion theorem for an eigenvalue problem with eigenvalue parameters in the boundary conditions. Q. J. Math.. 30, 33–42 (1979). Publisher Full Text OpenURL

  29. Wray, SD: Absolutely convergent expansions associated with a boundary-value problem with the eigenvalue parameter contained in one boundary condition. Czechoslov. Math. J.. 32(4), 608–622 (1982)