Multidimensional sampling theorems for multivariate discrete transforms

This paper is devoted to the establishment of two-dimensional sampling theorems for discrete transforms, whose kernels arise from second order partial difference equations. We define a discrete type partial difference operator and investigate its spectral properties. Green’s function is constructed and kernels that generate orthonormal basis of eigenvectors are defined. A discrete Kramer-type lemma is introduced and two sampling theorems of Lagrange interpolation type are proved. Several illustrative examples are depicted. The theory is extendible to higher order settings.


2)
When J is infinite, series (1.2) converges absolutely for t ∈ C and uniformly locally on C.
Let φ(n, t) be the solution of Eq. (1.3) such that M 1 (φ(n, t)) = 0. The eigenvalues of the problem are the zeros of M 2 (φ) and they are simple, where M 2 (φ) is polynomial of degree N . The eigenvalues of the problem are N distinct real numbers which will be denoted by {t k } N k=1 . The corresponding sequence of eigenfunctions is {φ(n, t k )} N k=1 . The sequence {φ(n, t k )} N k=1 , is a set of real-valued functions and it forms an orthogonal basis of l 2 (J); cf. e.g. [16]. Let if zero is not an eigenvalue, ), t 1 = 0, is an eigenvalue. (1.5) One of the sampling results of [4] can be stated as the following Lagrange interpolation theorem. . (1.7) In [1] sampling results were obtained for Eq. (1.3) with the general boundary conditions α 11 y(0) + α 12 y(1) + β 11 y(N) + β 12 y(N + 1) = 0, where α 11 β 22β 12 α 21 = 0.
In this paper we establish two-dimensional sampling theorems associated with a discrete-type Dirichlet problem. For this task we define a second order partial difference operator in the next section. We also impose conditions on the potential which make the problem breakable into two different ordinary Sturm-Liouville discrete systems. This is done in the next section. Section 3 is devoted to the construction of the Green's function of the system and derive its eigenfunctions expansion. Section 4 contains the sampling results of this paper and the last section depicted some worked examples. The theory for 2-D setting can be similarly extended to higher order situation, representing discrete counterpart of the results of both [5,24].

A two-dimensional discrete operator
In this section we define the two-dimensional discrete eigenvalue problem of this paper. α(n, m) 2 , α, β ∈ 2 (I). (2.1) For y ∈ 2 (I), let n and ∇ m be the partial forward and backward difference operators defined, respectively, by Similarly we define m and ∇ m . Let Consider the second order partial difference equation with the separate-type boundary conditions Here h i , l i are real numbers, i = 1, 2. The function Q(n) is also a real-valued function defined on I, and t ∈ C is the eigenvalue parameter.
Assuming that Q(n) = q(n)+p(m), and letting Y (n) = y(n)z(m), make problem (2.2)-(2.3) separable that can be split into two self-adjoint Sturm-Liouville problems with separatetype boundary conditions as follows: where λ + μ = t. Let φ(n, λ) be the solution of D 1 y = λy uniquely determined by the initial conditions and ψ(m, μ) be the solution of D 2 y = λy uniquely determined by the initial conditions Thus [16], both φ(n, λ) and ψ(m, μ) are, respectively, polynomials in λ and μ of degree n -1 and m -1. Noting that the eigenvalues of (2.4) and (2.5) are the zeros of the equations Following the theory developed in [16], the eigenvalues of (2.4) and (2.5) are real distinct and they form the sets respectively. Denote the sets of corresponding eigenvectors by Therefore, a solution of (2.2) is We also conclude that problem (2.2)-(2.3) has the set of eigenvalues with the corresponding real-valued eigenvectors Unlike the case of the one-dimensional problems, the eigenvalues are not necessarily simple. In fact, for all λ k , μ l eigenvalues of the problem (2.4) and (2.5), respectively, λ k + μ l = t; is fixed, then t is an eigenvalue of (2.2)-(2.3) corresponding to all eigenfunctions of the form kl (n, m). Hence, the set of eigenvalues of (2.2)-(2.3) can be listed as {t K } N×M K=1 , where an eigenvalues is repeated according to its (geometric) multiplicity. Note that the eigenvalues (2.11) are real and the eigenvectors (2.12) are real-valued functions. Proof Since all eigenvalues of each of the problems (2.4) and (2.5) are simple, then [16] their eigenvectors {φ k (n)} N k=1 , {ψ l (m)} M l=1 , construct orthogonal bases in 2 (Z N ), 2 (Z M ), respectively. Hence, the eigenvectors of (2.2)-(2.3); { kl (n, m)} N,M k=1,l=1 , are also orthogonal in 2 (I). We have Since 2 (I) has dimension NM, the set { kl } N,M k=1,l=1 is an orthogonal basis of 2 (I).
In the following lemma, we prove that (2.11) and (2.12) are the only eigenvalues and eigenvectors of problem (2.2)-(2.3), which is a discrete counterpart of the classical result of [22, p. 114]. Proof Assume that θ (n, λ) and χ(m, μ) are normalized functions corresponding to φ(n, λ) and ψ(m, μ), respectively. Thus, {θ k (n) = θ (n, λ k )} N k=1 is an orthonormal basis in 2 (Z N ), and {χ l (m) = χ(m, μ l )} M l=1 is an orthonormal basis in 2 (Z M ). Therefore, is an orthonormal basis in 2 (I). Let f (n, m) ∈ 2 (I). Hence, for each m ∈ Z M , we define It is worthwhile to mention that the theory outlined above is a discrete counterpart of Dirichlet boundary-value problem with additive potential; see [5,24] for the treatment of the associated sampling theorems.

Construction of Green's function
In this section we construct Green's function associated with the eigenvalue problem (2.2)-(2.3).
The classical multidimensional Green's functions may be encountered in [22].

Sampling theorems
This section involves the sampling theorems of this paper. We start with introducing a 2-D Kramer-type sampling theorem. Assume that for any (λ, μ), K(n, λ, μ) ∈ 2 (I), and that there exists a set of points such that {K(n, λ k , μ l )} is a complete orthogonal set in 2 (I). The following theorem gives a two-dimensional discrete version of Kramer's sampling theorem. has the sampling expansion The proof is by applying Parseval's relation to (4.1); cf. [17, p. 175]. In the following we will show that the kernel K(n, λ, μ) can arise from solutions of partial difference equations.
Letting λ → λ k , μ → μ l , then (4.7) implies Combining Eqs. (4.5), (4.7) and (4.8), we obtain . (4.9) Since G(λ) and H(μ) are polynomials in λ and μ of degrees N and M with different zeros at {λ k } N k=1 and {μ l } M l=1 , respectively, In the following theorem, where the kernel is the Green's function, we obtain a onedimensional Lagrange interpolation representation and the fundamental polynomial is determined by a polynomial containing both of the sampled values.
Assume that the different eigenvalues of the problem (2.2)-(2.3) are {λ k +μ l } s 1 ,s 2 k=1,l=1 . Since the Green's function (3.1) has simple poles at the eigenvalues, the function (n, t) = P(t)G(n, j 0 , t), is an entire function as a function in t, where j 0 ∈ I is fixed.

Theorem 4.3 For the discrete transform
we have the sampling expansion Proof If the multiplicity of the eigenvalue λ k + μ l is ν kl , with corresponding normalized , then (3.1) will be rewritten as Applying Parseval's relation to (4.11), cf. [17, p. 175], we have (4.14) By the orthogonality property, we get (·, t), i kl (·) = P(t) Also, (4.16) Substituting from (4.15) in (4.14), then using (4.16), one obtains (4.12). where t = λ + μ. The solution of the first problem of (4.19) under the condition y(0) = 0 is The first case of y(n) generates all eigenvalues and eigenvectors of the first problem of (4.19), so we will consider only the case |λ| ≤ 2. Let φ(n, λ) = sin nσ sin σ , then the eigenvalues of the first problem of (4.19) are the zeros of φ(N + 1, λ) = 0, which gives σ k = kπ/(N + 1), then the eigenvalues and the eigenvectors are Here we have , . Also , .