Discontinuous Sturm-Liouville Problems and Associated Sampling Theories

and Applied Analysis 3 The series 1.6 converges uniformly on compact subsets of C. The WKS sampling theorem is a special case of this theorem because if we choose tk kπ/σ −t−k, then


Introduction
The recovery of entire functions from a discrete sequence of points is an important problem from mathematical and practical points of view.For instance, in signal processing it is needed to reconstruct recover a signal function from its values at a sequence of 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 WKS sampling theorem 1-3 .By a band-limited signal with band width σ, σ > 0, that is, the signal contains no frequencies higher than σ/2π cycles per second cps , we mean a function in the Paley-Wiener space PW 2  σ of entire functions of exponential type at most σ 2 Abstract and Applied Analysis which are L 2 R -functions when restricted to R. This space is characterized by the following relation which is due to Paley and Wiener 4, 5 : e ixt g x dx, for some function g • ∈ L 2 −σ, σ . 1.1 Now WKS 6, 7 sampling theorem states the following.

1.3
The sampling series 1.2 is absolutely and uniformly convergent on compact subsets of C, uniformly convergent on R and converges in the norm of L 2 R , see [6,8,9].
The WKS 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 important from practical and theoretical point of view.The following theorem which is known in some literature as 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 10 and Kadec 11 , it could be named after Paley and Wiener who first derive the theorem in a more restrictive form, see 6, 7 for more details.and let G t be the entire function defined by the canonical product Then, for any f ∈ PW 2 The series 1.6 converges uniformly on compact subsets of C.
The WKS sampling theorem is a special case of this theorem because if we choose t k kπ/σ −t −k , then Expansion 1.6 is of Lagrange-type interpolation.
The second extension of WKS sampling theorem is the theorem of Kramer 12 .In this theorem sampling representations were given for integral transforms whose kernels are more general than exp ixt .Theorem 1. 3 Kramer .Let I be a finite closed interval, K •, t : I × C → C a function continuous in t such that K •, t ∈ L 2 I for all t ∈ C, and let {t k } k∈Z be a sequence of real numbers such that {K •, t k } k∈Z is a complete orthogonal set in L 2 I .Suppose that 1.8 Then 1.9 Series 1.9 converges uniformly wherever K •, t L 2 I as a function of t is bounded.
Again Kramer's theorem is a generalization of WKS theorem.If we take K x, t e itx , I −σ, σ , t k kπ/σ, then 1.9 will be 1.2 .The relationship between both extensions of WKS sampling theorem has been investigated extensively.Starting from a function theory approach, cf. 13 , it is proved in 14 that if K x, t , x ∈ I, t ∈ C satisfies some analyticity conditions, then Kramer's sampling formula 1.9 turns out to be a Lagrange interpolation one, see also 15-17 .In another direction, it is shown that Kramer's expansion 1.9 could be written as a Lagrange-type interpolation formula if K •, t and t k are extracted from ordinary differential operators, see the survey 18 and the references cited therein.The present work is a continuation of the second direction mentioned above.We prove that integral transforms associated with secondorder eigenvalue problems with an eigenparameter appearing in the boundary conditions and also 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, for example, 19, 20 .Also papers in sampling with discontinuous eigenproblems are few, see 21-23 .However sampling theories associated with eigenproblems, which contain eigenparameter in the boundary conditions and have at the same time discontinuity conditions, do not exist as for as we know.Our investigation will be the first in that direction, introducing a good example.To achieve our aim we will briefly study the spectral analysis of

The Eigenvalue Problem
In this section we define our boundary value problem and state some of its properties.Consider the boundary value problem with boundary conditions and transmission conditions where λ is a complex spectral parameter; r x r 2 1 for x ∈ −1, 0 , r x r 2 2 for x ∈ 0, 1 ; r 1 > 0 and r 2 > 0 are given real number; q x is a given real-valued function, which is continuous in −1, 0 and 0, 1 and has a finite limit q ±0 lim x → ±0 q x ; γ i , δ i , α i , β i , α i , β i i 1, 2 are real numbers; γ i / 0, δ i / 0 i 1, 2 ; ρ and γ are given by In some literature conditions 2.4 are called compatability conditions, see, for example, 24 .
To formulate a theoretic approach to problem 2.1 -2.4 we define the Hilbert space H : where For function f x , which is defined on −1, 0 ∪ 0, 1 and has finite limit f ±0 : lim x → ±0 f x , by f 1 x and f 2 x we denote the functions which are defined on I 1 : −1, 0 and I 2 : 0, 1 , respectively.
In the following we will define the minimal closed operator in H associated with the differential expression , cf. 25, 26 .Let D A ⊆ H be the set of all The eigenvalues and the eigenfunctions of the problem 2.1 -2.4 are defined as the eigenvalues and the first components of the corresponding eigenelements of the operator A, respectively.

6
Abstract and Applied Analysis By two partial integration we obtain where, as usual, by W f, g; x we denote the Wronskian of the functions f and g W f, g; x : f x g x − f x g x .

2.14
Since f x and g x are satisfied the boundary condition 2.2 -2.3 and transmission conditions 2.4 we get

2.15
Finally substituting 2.15 in 2.13 then we have Proof.Formula 2.17 follows immediately from the orthogonality of corresponding eigenelements in the Hilbert space H.

Abstract and Applied Analysis 7
Now, we will construct a special fundamental system of solutions of the equation 2.1 for λ not being an eigenvalue.Let us consider the next initial value problem:

2.20
By virtue of Theorem 1.5 in 27 this problem has a unique solution u ϕ 1 x ϕ 1λ x , which is an entire function of λ ∈ C for each fixed x ∈ −1, 0 .Similarly, employing the same method as in proof of Theorem 1.5 in 27 , we see that the problem −r 2 2 u x q x u x λu x , x ∈ 0, 1 , 2.21 has a unique solution u χ 2 x χ 2λ x which is an entire function of parameter λ for each fixed x ∈ 0, 1 .Now the functions ϕ 2λ x and χ 1λ x are defined in terms of ϕ 1λ x and χ 2λ x as follows: the initial-value problem, −r 2 2 u x q x u x λu x , x ∈ 0, 1 , 2.23 which contains the entire functions of eigenparameter λ in the right-hand side , has unique solution u ϕ 2λ x for each λ ∈ C.
Similarly, the following problem also has a unique solution u χ 1 x χ 1λ x : Since the Wronskians W ϕ iλ , χ iλ ; x are independent on variable x ∈ I i i 1, 2 and ϕ iλ x and χ iλ x are the entire functions of the parameter λ for each x ∈ I i i 1, 2 , then the functions are the entire functions of parameter λ.
Let us construct two basic solutions of 2.1 as

2.28
By virtue of 2.24 and 2.26 these solutions satisfy both transmission conditions 2.4 .Now we may introduce to the consideration the characteristic function ω λ as Theorem 2.6.The eigenvalues of the problem 2.1 -2.4 are coincided zeros of the function ω λ .
Proof.Let ω λ 0 0. Then W ϕ 1λ 0 , χ 1λ 0 ; x 0, and so the functions ϕ 1λ 0 x and χ 1λ 0 x are linearly dependent, that is, Consequently, χ λ 0 x satisfied the boundary condition 2.3 , so the function χ λ 0 x is an eigenfunction of the problem 2.1 -2.4 corresponding to the eigenvalue λ 0 .Now let u 0 x be any eigenfunction corresponding to the eigenvalue λ 0 , but ω λ 0 / 0. Then the functions ϕ iλ 0 x , χ iλ 0 x are linearly independent on I i , i 1, 2. Thus, u 0 x may be represented as in the form where at least one of the constants c i , i 1, 2, 3, 4, is not zero.
Consider the equations as the homogenous system of linear equations of the variables c i , i 1, 2, 3, 4, and taking into account 2.24 and 2.26 , it follows that the determinant of this system is Thus, the system 2.32 has only trivial solution c i 0, i 1, 2, 3, 4, and so we get contradiction which completes the proof.Lemma 2.7.If λ λ 0 is an eigenvalue, then ϕ λ 0 x and χ λ 0 x are linearly dependent.

2.36
By the same procedure from the equality U 4 χ λ 0 0 we can derive that ϕ 2λ 0 0 0.

2.37
From the fact that ϕ 2λ 0 x is a solution of 2.1 on 0, 1 and satisfied the initial conditions 2.36 and 2.37 it follows that ϕ 2λ 0 x 0 identically on 0, 1 , because of the well-known existence and uniqueness theorem for the initial value problems of the ordinary linear differential equations.
for any λ.Since for some k 0 / 0, then

Abstract and Applied Analysis 11
If λ n , n 0, 1, 2, . . .denote the zeros of ω λ , then the three-component vectors are the corresponding eigenvectors of the operator A satisfying the orthogonality relation

2.46
Here {ϕ λ n • } ∞ n 0 will be the sequence of eigenfunctions of 2.1 -2.4 corresponding to the eigenvalues {λ n } ∞ n 0 .We denote by Ψ n • the normalized eigenvectors Because of simplicity of the eigenvalues, we find nonzeros constants k n such that

2.48
To study the completeness of the eigenvectors of A, and hence the completeness of the eigenfunctions of 2.1 -2.4 , we construct the resolvent of A as well as Green's function of problem 2.1 -2.4 .We assume without any loss of generality that λ 0 is not an eigenvalue of A. Otherwise, from discreteness of eigenvalues, we can find a real number c such that c / λ n for all n and replace the eigenparameter λ by λ − c.Now let λ ∈ C not be an eigenvalue of A and consider the inhomogeneous problem and I is the identity operator.Since 51 Now, we can represent the general solution of 2.51 in the following form:

2.53
Applying the method of variation of the constants to 2.53 , thus, the functions A 1 x, λ , B 1 x, λ and A 2 x, λ , B 2 x, λ satisfy the linear system of equations , x ∈ 0, 1 .

2.54
Since λ is not an eigenvalue and W ϕ iλ x , χ iλ x ; x / 0, i 1, 2, each of the linear systems in 2.54 has a unique solution which leads x ∈ 0, 1 .

2.56
Then from 2.52 and the transmission conditions 2.4 we get

2.58
Hence, we have where is the unique Green's function of problem 2.1 -2.4 .Obviously G x, ξ, λ is a meromorphic function of λ, for every x, ξ ∈ −1, 0 ∪ 0, 1 2 , which has simple poles only at the eigenvalues.Although Green's function looks as simple as that of Sturm-Liouville problems, cf., for example, 28 , it is a rather complicated because of the transmission conditions, see the example at the end of this paper.

Lemma 2.10. The operator A is self-adjoint in H.
Proof.Since A is a symmetric densely defined operator, then it is sufficient to show that the deficiency spaces are the null spaces and hence ∈ H and λ is a nonreal number, then taking implies that u ∈ D A .Since G x, ξ, λ satisfies conditions 2.2 -2.4 , then A−λI u x f x .Now we prove that the inverse of A − λI exists.If Au x λu x , then Take λ ±i.The domains of A − iI −1 and A iI −1 are exactly H. Consequently the ranges of A − iI and A iI are also H. Hence the deficiency spaces of A are

2.64
Hence A is self-adjoint.
The next theorem is an eigenfunction expansion theorem, which is similar to that established by Fulton in 29 .

Theorem 2.11. i For
with the series being absolutely and uniformly convergent in the first component for on −1, 0 ∪ 0, 1 and absolutely convergent in the second component.
Proof.The proof is similar to that in 29, pages 298-299 .

Asymptotic Formulas of Eigenvalues and Eigenfunctions
Now we derive first-and second-order asymptotics of the eigenvalues and eigenfunctions similar to the classical techniques of 27, 30 and 29 , see also 25,26 .We begin by proving some lemmas.
Lemma 3.1.Let ϕ λ x be the solutions of 2.1 defined in Section 2, and let λ s 2 .Then the following integral equations hold for k 0 and k 1: q y ϕ 2λ y dy.

3.2
Proof.For proving it is enough substitute s 2 ϕ 1λ y r y ϕ 1λ y and s 2 ϕ 2λ y r y ϕ 2λ y instead of q y ϕ 1λ y and q y ϕ 2λ y in the integral terms of the 3.1 and 3.2 , respectively, and integrate by parts twice.Lemma 3.2.Let λ s 2 , Im s t.Then the functions ϕ iλ x have the following asymptotic representations for |λ| → ∞, which hold uniformly for x ∈ I i (i 1, 2): Proof.Since the proof of the formulae for ϕ 1λ x is identical to Titchmarshs proof of similar results for ϕ λ x see 27, Lemma 1.7 page 9-10 , we may formulate them without proving them here.Therefore we will prove only the formulas for ϕ 2λ x .Let α 2 / 0. Then according to 3.3

3.7
Substituting 3.7 into 3.2 for k 0 , we get

3.10
Denoting M λ : max 0≤x≤1 |F λ x | from the last formula, it follows that for some M 0 > 0. From this, it follows that M λ O 1 as λ → ∞, so 3.12 Substituting 3.12 into the integral on the right of 3.8 yields 3.4 for k 0. The case k 1 of 3.4 follows by applying the same procedure as in the case k 0. The case α 2 0 is proved analogically.
Proof.The proof is immediate by substituting 3.4 and 3.6 into the representation 3.17 Proof.Putting s it t > 0 in the above formulae, it follows that ω −t 2 → ∞ as t → ∞.Hence, ω λ / 0 for λ negative and sufficiently large.
Now we can obtain the asymptotic approximation formula for the eigenvalues of the considered problem 2.1 -2.4 .Since the eigenvalues coincide with the zeros of the entire function ω λ , it follows that they have no finite limit.Moreover, we know from Corollaries 2.2 and 3.4 that all eigenvalues are real and bounded below.Therefore, we may renumber them as λ 0 ≤ λ 1 ≤ λ 2 , . .., listed according to their multiplicity.

3.21
Proof.We will only consider the first case.From 3.13 we have

3.22
We will apply the well-known Rouche theorem, which asserts that if f λ and g λ are analytic inside and on a closed contour C and |g λ | < |f λ | on C, then f λ and f λ g λ have the same number of zeros inside C, provided that each zero is counted according to its multiplicity.It follows that ω λ has the same number of zeros inside the contour as the leading term in 3.22 .If λ 0 ≤ λ 1 ≤ λ 2 , . .., are the zeros of ω λ and λ n s 2 n , we have for sufficiently large n, where |δ n | < π/4, for sufficiently large n.By putting in 3.22 we have δ n O n −1 , so the proof is completed for Case 1.The proof for the other cases is similar.
Then from 3.3 -3.6 for k 0 and the above theorem, the asymptotic behavior of the eigenfunctions

3.25
All these asymptotic formulae hold uniformly for x.

The Sampling Theorem
In this section we derive two sampling theorems associated with problem 2.1 -2.4 .For convenience we may assume that the eigenvectors of A are real valued.

Abstract and Applied Analysis
Then F λ is an entire function of exponential type 2 that can be reconstructed from its values at the points {λ n } ∞ n 0 via the sampling formula The series 4.3 converges absolutely on C and uniformly on compact subset of C.Here ω λ is the entire function defined in 2.29 .
Proof.Relation 4.2 can be rewritten as an inner product of H as follows where Both g • and Φ λ • can be expanded in terms of the orthogonal basis on eigenfunctions, that is, where g n and Φ λ n are the fourier coefficients

4.7
Applying Parseval's identity to 4.4 and using 4.7 , we obtain C not be an eigenvalue and n ∈ N. To prove 4.3 we need to show that , n 0, 1, 2, . . . .

4.9
By the definition of the inner product of H, we have then, from 2.20 and 2.24 , 4.11 becomes From 2.48 , 2.22 , and 2.8 , the Wronskian of ϕ 2λ n and ϕ 2λ at x 1 will be

4.14
Relations 2.48 and R 1 χ λ n γ and the linearity of the boundary conditions yield Letting λ → λ n in 4.16 and since the zeros of ω λ are simple, we have 4.17 Therefore from 4.16 and 4.17 we establish 4.9 .Since λ and n are arbitrary, then 4.3 is proved with a pointwise convergence on C, since the case λ λ n is trivial.Now we investigate the convergence of 4.3 .First we prove that it is absolutely convergent on C. Using Cauchy-Schwarz' inequality for λ ∈ C, Using the same method developed above

4.21
Since −1, 1 × M is compact, then, cf., for example, 31, page 225 , we can find a positive constant C M such that Then uniformly on M. In view of Parseval's equality, Thus σ N λ → 0 uniformly on M. Hence 4.3 converges uniformly on M. Thus F λ is analytic on compact subsets of C and hence it is entire.From the relation and the fact that ϕ 1λ x and ϕ 2λ x are entire function of exponential type 2, we conclude that F λ is also of exponential type 2.
Remark 4.2.To see that expansion 4.3 is a Lagrange-type interpolation, we may replace ω λ by the canonical product The next theorem is devoted to give interpolation sampling expansions associated with problem 2.1 -2.4 for integral transforms whose kernels defined in terms of Green's function.There are many results concerning the use of Green's function in sampling theory, cf., for example, 22, 32-34 .As we see in 2.60 , Green's function G x, ξ, λ of problem 2.1 -2.4 has simple poles at {λ n } ∞ n 0 .Define the function G x, λ to be G x, λ : ω λ G x, ξ 0 , λ , where ξ 0 ∈ −1, 0 ∪ 0, 1 is a fixed point and ω λ is the function defined in 2.29 or it is the canonical product 4.26 .
Then F λ is an entire function of exponential type 2 which admits the sampling representation Series 4.31 converges absolutely on C and uniformly on compact subsets of C.
Proof.The integral transform 4.30 can be written as Applying Parseval's identity to 4.32 with respect to {Φ n • } ∞ n 0 , we obtain From 2.59 and 4.36 we obtain

4.38
Hence 4.38 can be rewritten as

4.39
From the definition of G •, λ , we have

4.40
From formula 2.60 , we get and 2.48 together with 4.40 yields

4.42
Substituting from 4.39 and 4.42 gives As an element of H, G •, λ has the eigenvectors expansion

4.44
Taking the limit when λ → λ n in 4.32 , we get 4.45 The interchange of the limit and summation processes is justified by the uniform convergence of the eigenvector expansion of G x, λ on −1, 1 on compact subsets of C, cf., 2.60 , 3.3 -3.6 , and 3.18 -3.21 .Making use of 4.44 , we may rewrite 4.45 as

4.46
The interchange of the limit and summation is justified by the asymptotic behavior of Φ i x and ω λ .If ϕ λ n ξ 0 / 0, then 4.46 gives Combining 4.43 , 4.47 , and 4.34 we get 4.31 under the assumption that ϕ λ n ξ 0 / 0 for all n.If ϕ λ n ξ 0 0, for some n, the same expansion holds with F λ n 0. The convergence properties as well as the analytic and growth properties can be established as in Theorem 4.1.Now, we give an example exhibiting the obtained results.

Theorem 1 . 2
Paley and Wiener .Let {t k }, k ∈ Z be a sequence of real numbers satisfying

H 1 / 2 . 4 . 18 Since
g • , Φ λ • ∈ H, then both series in the right-hand side of 4.18 converge.Thus series 4.3 converges absolutely on C. For uniform convergence let M ⊂ C be compact.Let λ ∈ M and N > 0. Define σ N λ to be σ N λ : F λ −
representation of the type.
Theorem 1.1 WKS .If f t ∈ PW 2 σ , then it is completely determined from its values at the points t k kπ/σ, k ∈ Z, by means of the formula