Kramer-type sampling theorems associated with higher-order differential equations

For many decades, Kramer’s sampling theorem has been attracting enormous interest in view of its important applications in various branches. In this paper we present a new approach to a Kramer-type theory based on spectral differential equations of higher order on an interval of the real line. Its novelty relies partly on the fact that the corresponding eigenfunctions are orthogonal with respect to a scalar product involving a classical measure together with a point mass at a finite endpoint of the domain. In particular, a new sampling theorem is established, which is associated with a self-adjoint Bessel-type boundary value problem of fourth-order on the interval [0, 1]. Moreover, we consider the Laguerre and Jacobi differential equations and their higher-order generalizations and establish the Green-type formulas of the differential operators as an essential key towards a corresponding sampling theory.


Introduction
In 1959, Kramer [29] established his famous sampling theorem by generalizing a well-known result due to Whittaker, Shannon, and Kotel'nikov. Since then Kramer's theorem and its far-reaching generalizations have proved to be extremely useful in various fields of mathematics and related sciences, notably in sampling theory, interpolation theory or signal analysis. In particular, significant results were achieved by Paul Leo Butzer and-often in a very fruitful cooperation-by his students, colleagues Communicated by Gianluca Vinti.
w (I ). (1.1) Moreover, let a so-called Kramer kernel K (x, λ) : I × R → R be given with the properties (i) K (·, λ) ∈ L 2 w (I ), λ ∈ R, (ii) There exists a monotone increasing, unbounded sequence of real numbers {λ k } k∈N 0 , such that the functions {K (x, λ k )} k∈N 0 form a complete, orthogonal set in L 2 w (I ).
If a function F(λ), λ ∈ R, can be given, for some function g ∈ L 2 w (I ), as

2)
then F can be reconstructed from its samples in terms of an absolutely convergent series, Due to the completeness of {χ k }, this yields for all λ ∈ R, Now apply the Cauchy-Schwarz inequality to get the pointwise convergence As to the absolute convergence, it follows by means of and of the Bessel inequality for orthogonal systems in Hilbert spaces that K (·, λ) ∧ (k) 2 1/2 = g w K (·, λ) w < ∞.
Moreover, the series (1.3) is uniformly convergent on any subset of the real line where K (·, λ) w is bounded by a constant C independent of λ, for then As noticed by Kramer [29] already and later by Campbell [7], the richest source to reveal a suited Kramer kernel is to study a self-adjoint boundary value problem on an interval I ⊂ R, which is generated by a linear differential operator L of some even order. Then the functions K (x, λ k ), x ∈ I , k ∈ N 0 , arise as the eigenfunctions of L, where the corresponding eigenvalues are given by the required countable set {λ k } k∈N 0 .
In addition, it turns out that Kramer's sampling expansion (1.3) may be seen as a generalized Lagrange interpolation formula, if the kernel K (x, λ) arises from a Sturm-Liouville boundary value problem, see e.g. [17,19,21,35,37,38]. In fact, if G(λ) denotes a function with {λ k } k∈N 0 as its simple zeros, then the sampling functions may be represented as Starting off from a regular boundary value problem of order n = 2m, m ∈ N, Butzer and Schöttler [4] and Zayed et al. [36,37] utilized Kramer's approach in a more general framework. Nevertheless, almost all relevant examples given so far in the literature, are associated with a Sturm-Liouville problem built upon the singular, second-order Bessel, Laguerre, and Jacobi equations, see e.g. [1,18,35,38]. More specifically, let the (normalized) Bessel functions of the first kind, {J α λ } λ>0 , the Laguerre polynomials {L α n } n∈N 0 , and the Jacobi polynomials {P α,β n } n∈N 0 be respectively given in terms of a (generalized) hypergeometric function with one or two parameters α > −1, β > −1 by (as usual, (a) 0 = 1, (a) n = a(a + 1) · · · (a + n − 1), a ∈ C). Then their spectral differential equations are of the common form [9, Secs. 7, 10] where the entries are respectively given by (1.9) Furthermore, there exists a discrete eigenfunction system associated with the Bessel equation, as well. In fact, when restricting the range of the equation to the finite interval (0, 1] and imposing the boundary condition J α λ (1) = 0 at the right endpoint, the parameter λ of the spectrum takes the discrete values {γ α k } k∈N 0 , which arise as the consecutive positive zeros of the Bessel function J α (λ). The corresponding eigenfunctions {J α γ α k (x)} k∈N 0 , usually called the Fourier-Bessel functions, form an orthogonal system in the Hilbert space L 2 w (0, 1) with weight function w(x) = x 2α+1 , see e.g. [9,Sec.7.15], [33,Sec.18].
In the present paper, all three equations will serve as a starting point towards a sampling theorem. But even more, they give rise to far-reaching extensions: While it is well-known that the Laguerre and Jacobi polynomials belong to the very few orthogonal polynomial systems satisfying a spectral differential equation of second order, there is a long history in searching for polynomials which solve a higher-order equation, cf. [11]. Successfully, this led to the 'Laguerre-type' polynomials {L α,N n (x)} n∈N 0 and the 'Jacobi-type' polynomials {P α,β,N n (x)} n∈N 0 , which are orthogonal with respect to the inner products (1), (1.10) respectively. Here, N > 0 is an additional point mass, and the (normalized) Laguerre and Jacobi weight functions are given by (1.11) Moreover, both polynomial systems are the eigenfunctions of a differential operator of order 2α + 4, provided that α ∈ N 0 , see Prop. 3.2 and Prop. 4.1.
Similarly, by adding a point mass N > 0 to the (normalized) Bessel weight function at the origin, Everitt and the author [12] introduced a continuous system of Bessel-type functions solving an equation of the same higher order. Furthermore, by restricting the Bessel-type equation to the finite interval (0, 1] and imposing a suited boundary condition at x = 1, we obtained a discrete eigenfunction system which generalizes the Fourier-Bessel functions, cf. [15,30,31]. One major aim of this paper is to determine the spectrum of their eigenvalues and to investigate, to which extent this approach gives rise to a new sampling theorem. In the proof of a sampling theorem associated with a classical Sturm-Liouville differential operator L x in (1.6), the knowledge of the corresponding Green formula turned out to be essential: Given two real-valued functions f , g defined on an appro- (1.13) Imposing now, if necessary, certain boundary conditions on the functions f , g, such that [ f , g] x vanishes at both endpoints of the interval, the operator L x becomes (formally) self-adjoint. As a simple consequence, two eigenfunctions of Eq. (1.6) with eigenvalues 1 = 2 , say, are orthogonal in So, when looking for a sampling theory associated with a higher-order differential equation, one has to establish a Green-type formula that generalizes the classical formula (1.13), appropriately. It will be another purpose of the paper to provide such formulas for of all three higher-order equations of Bessel-, Laguerre-, and Jacobi-type.
The paper is organized as follows. In Section 2, we first state the higher-order Bessel-type differential equation on the half line and introduce the continuous system of Bessel-type functions together with the eigenvalues. In particular, we present the Green-type formula for the corresponding differential operator B α,N x , α ∈ N 0 , N > 0, with respect to the inner product (1.14) where the weight function ω α (x) is given in (1.12), but now restricted to 0 < x ≤ 1, see Thm. 2.1. Then we impose a boundary condition at x = 1 to be fulfilled by the discrete system of the so-called Fourier-Bessel-type functions. The evaluation of the inner product (1.14) involving both a Bessel-type function as well as a (discrete) Fourier-Bessel-type function, will play an essential role in determining a corresponding sampling theory. As a first new example, a Kramer-type theorem is explicitly established in case of the fourth-order Bessel-type equation on the interval (0, 1]. Sections 3 is devoted to the Whittaker equation as the continuous counterpart of the Laguerre equation. Again we proceed from the Green formula (1.13), now associated with the Laguerre differential operator L α x . But in general, the eigenfunctions of the Whittaker equation do not belong to the corresponding Hilbert space L 2 w α (0, ∞). So similarly as in the Fourier-Bessel case, we restrict the differential equation to the finite interval (0, 1] and impose a boundary condition at the right endpoint to be fulfilled by its solutions. Actually we succeeded to determine a discrete spectrum of eigenvalues such that the eigenfunctions are orthogonal in the space L 2 w α ((0, 1]). This gives rise to a new sampling theorem, see Thm. 3.1. Furthermore we present the higher-order differential equation for the Laguerre-type polynomials and introduce the corresponding Whittaker-type equation and its eigenfunctions. Then we establish the Green-type formula for the higher-order differential operator which may serve as an essential tool towards an extended theory.
Finally, in Section 4, we first recall a known sampling theorem associated with the classical Jacobi equation (1.6),(1.9), see [18,35] In a second step, we then define their continuous counterparts, cf. (4. [10][11][12], and close with a new Green-type formula for the Jacobi-type differential operator L α,β,N x . 2 Sampling theory associated with the higher-order Bessel-type differential equation In view of (1.6-1.7), the Bessel (eigen)functions y λ (x) = J α λ (x), satisfy, for any α > −1, the differential equation With the new parameter N > 0, the Bessel-type functions are then given in the three equivalent ways, see [12], (2.2) Proposition 2.1 [12,31] For N > 0 and provided that α ∈ N 0 , the Bessel-type functions {J α,N λ (x)} λ>0 satisfy the differential equation where B α x is the classical Bessel operator given in (2.1) or, equivalently, by and T α x denotes the differential operator of order 2α + 4,

(2.4)
Similarly, the eigenvalue parameter splits up into α,N Here and in the following, Notice that the continuous system of Bessel-type functions gives rise to a generalized Hankel transform [10], but do not belong to L 2 ω α,N (0, ∞). So, being interested in a possible sampling theorem, we reduce the range of Eq. (2.3) to the finite interval I = (0, 1] as in the classical Fourier-Bessel case.
For functions f : [0, 1] → C belonging to the domain let the Bessel-type differential operatorB α,N x be defined by Notice that for f (x) = J α,N λ (x), the definition at the origin is justified since An essential feature of the Bessel-type operator is its Green-type formula.
Furthermore, by observing that δ 2 j Here, the remaining integrals have cancelled each other by symmetry in f , g. Concerning the value of the sum H ( f , g) at the origin, it turns out that only the term for j = α + 1 does not vanish. Hence, by some straightforward calculations, we get and thus This expression, however, annihilates the term Putting things together and observing that complete the proof of Thm. 2.1.
As an immediate consequence of this Green-type formula, the Bessel-type differential operatorB α,N x can be made (formally) self-adjoint with respect to the inner product (·, ·) ω α,N by imposing a boundary condition at the right endpoint x = 1, namely as two distinct eigenfunctions of the operatorB α,N x and using the abbreviation (2.10) Since N > 0 is arbitrary, both terms in (2.10) must vanish simultaneously. This however holds, if there exists a constant c = c(α, N ) ∈ R not depending on the eigenvalue parameter, such that (2.11) In the limit case N = 0, it is easy to see that (2.11) is satisfied if λ, μ are zeros of the Bessel function J α (x) or, more generally, of the combination (c+α+2)J α (x)+x J α (x). This actually gives rise to the classical Fourier-Bessel-and Fourier-Dini-functions, respectively, cf. [9, 7.10.4], [33,Chap. 18].
If N > 0, however, it is a crucial task to determine the constant c(α, N ) and to restrict the eigenvalue parameter such that (2.11) is fulfilled. In order to simplify the derivatives occurring in the sum of (2.11) up to the order 2α + 3, a promising strategy is to start off from a λ-dependent, second-order differential equation for the Bessel-type functions {J α,N λ (x)} λ>0 . Then, after deriving this equation sufficiently often and evaluating each of the resulting equations at the endpoint In the following, we illustrate our strategy in the first non-trivial case α = 0. For simplicity we drop this parameter, as long as there is no confusion. Here, the differential equation (2.3) for the Bessel type functions y λ (x) := J 0,N λ (x) becomes the fourth-order equation and eigenvalue parameter N λ = λ 2 + (N /8) λ 4 . Notice that Eq. (2.12), when multiplied by −8x/N , N > 0, takes the Lagrange symmetric form In a number of papers by Everitt and the author [13][14][15], we used this equation to develop a spectral theory for the Bessel-type differential operator on the half line and, in particular, on the finite interval 0 < x ≤ 1. In this latter case, we found a discrete spectrum of eigenvalues, which are based on the real zeros of an even function ϕ N (x) involving the classical Bessel functions J 0 (x), J 1 (x), see (2.19). By using the above approach, we obtain the function ϕ N (x) as follows: The Bessel- (2.9), a straightforward calculation yields (1), for simplicity, we obtain, for x = 1, (2.14) Hence, the boundary condition in (2.11) is equivalent to This, however, holds for any values of λ, μ, if c 2 = 4/N with the two solutions c = c ± := ±2/ √ N . So it remains to determine the eigenvalues such that In view of the representation (2.2) and the classical Bessel equation (2.1), there holds and thus (2.16) Henceforth we focus on the case c + = 2/ √ N , while the case c − = −2/ √ N can be treated similarly.
where the domain is restricted by two boundary conditions involving c = 2/ √ N , i.e., Furthermore, let {γ k,N } ∞ k=0 denote the unbounded sequence of positive, strictly increasing zeros of In particular, they satisfy the two boundary conditions in (2.18).
But in view of (2.14) with c 2 = 4/N , this is true since For N = 0, 1, 4, the first five zeros of ϕ N (λ) (up to 6 decimals) are listed in Now we can state a Kramer-type theorem with respect to the Fourier-Bessel-type functions.
(a) If a function F(λ), λ > 0, is given, for some function g ∈ L 2 ω N ((0, 1]), as then F can be reconstructed from its samples via the absolutely convergent series Proof (a) This is an immediate application of Kramer's Theorem 1.1, see (1.2-1.3).
(b) One way to determine the functions S N k (λ) is to utilize the fact that the inner product of two Bessel-type functions is related to that of the related classical Bessel functions by, see [12,Thm. 4.2], In view of the Bessel equation (2.1) and the Green formula for the operator B 0 x , there holds Dividing this expression by the constant h N k in (2.21) then yields (2.23). A second proof is of its own interest, for it proceeds directly from the Bessel-type equation (2.12) and the Green-type formula in Thm. 2.1 for α = 0. In view of (2.10), we have Due to the relationships (2.14) and the fact that Y (1) γ Inserting these four quantities into (2.24) and collecting coefficients of Y γ k,N , we get But since, in view of (2.15-2.16), Deviding by the normalization constant h N k in (2.21), we arrive at the same result as in (2.23), which settles the proof of Thm. 2.2. .
The result then follows by applying part (a).
. (2.25) Hence, this expansion furnishes an interpolation formula in accordance with Cor.

Sampling theory associated with the Laguerre equation
In order to establish a Kramer-type sampling theorem based on the Laguerre equation (1.6),(1.8), it is natural to look for an appropriate Kramer kernel K (x, ), 0 < x < ∞, with continuous eigenvalue parameter , i.e., This is a confluent hypergeometric equation [9, Chap.6], whose two linear independent solutions are denoted by (− , α+1; x) ≡ 1 F 1 (− ; α+1; x) and (− , α+1; x), respectively. When transforming Eq.(3.1) into Whittaker's standard [9, 6.1(4)] and setting = λ − (α + 1)/2, the two resulting solutions are the Whittaker functions In Example 6 of his treatise on sampling theorems, Zayed [35] used a slightly different version of the Whittaker equation to introduce his so-called continuous Laguerre transform. But while the function obviously reduces to the (normalized) Laguerre polynomials R α n (x), if λ = n + (α + 1)/2, n ∈ N 0 , it was pointed out in [35] that, in general, Consequently, Zayed used the second function α λ (x) as a Kramer kernel to built upon a sampling theorem, provided that the function space is suitably restricted to justify the absolute convergence of the integral Nevertheless, in case of the regular solution α λ (x), there has also been attempts to avoid the lack of convergence. In particular, Jerri [24] derived a kind of sampling expansion by means of a Laguerre-transform with kernel (−λ, α + 1; νx/(ν − 1)), α > −1, λ ≥ 0, −1 < ν < 1.
In the following, we propose a different approach to a Laguerre sampling theorem, now for functions on the bounded interval (0, 1]. To this end, we make use of the Green formula (1.13) associated with the differential operator L α x defined in (3.1). Observing that for any (α + 1)/2 ≤ λ, μ < ∞, the two functions α λ , α μ belong to L 2 w α ((0, 1]), we obtain 4) since lim a→0+ α λ , α μ a = 0 for any α > −1. The task is now to find a condition under which the resulting value at x = 1 vanishes, as well. This is indeed possible by restricting the eigenvalue parameters according to the following feature.

increasing zeros of the Bessel function J α (λ), the zeros of ψ α (λ) possess the following lower and upper bounds for k
and form an orthogonal system with respect to the discrete measure (β) x c x /x!, x = 0, 1, ...∞. Now let β = α + 1 and choose c = c n := n/(n + 1) ∈ (0, 1), where n ∈ N coincides with the degree of the Meixner polynomial M n . Substituting, moreover, x = λ − (α + 1)/2, we obtain In fact, the inequality is satisfied for any > 0, provided that n is chosen large enough. To see this, just fix N = N ( , α, λ) > 2 such that Then there exists an N 1 > N such that for any n > N 1 , Moreover it is well known that each Meixner polynomial M n (x; β, c) has n distinct zeros on the positive half-line, which have been studied in detail by various authors, see, e.g., [8,22,23,25,26]. So, if for any n ∈ N, the zeros of M n (x; α + 1, c n ) are denoted by M α,c n n,1 < M α,c n n,2 < · · · < M α,c n n,n < ∞, the required behaviour of the zeros {δ α k } k∈N 0 follows by taking the limit In particular, given the zeros of R α n (x) by l α n,1 < · · · < l α n,k < · · · < l α n,n , it was shown in [23, Thm. 6.2] that So if c = c n := n n+1 , the estimates (3.10) imply that n(n + 1) l α n,k < M α,c n n,k + (3.11) The zeros l α n,k of the Laguerre polynomials, in turn, can be estimated, partly in terms of the zeros of the Bessel function J α (λ), by 18.16.10-11]. When inserting these lower and upper bounds into the two-sided inequality (3.11) and taking the limit n → ∞, we finally arrive at the estimate (3.6) by virtue of the limit relation (3.9).
(c) By definition of ψ α , it follows that More refined asymptotic formulas for the zeros of the Meixner polynomials can be found in [25,, while a sharp upper bound for the highest zero of a Meixner polynomial was given already in [22,Thm. 6]. As to the Laguerre polynomials, various inequalities for their smallest and largest zeros are presented in [8, 3.3-4].

Proposition 3.1 Let {δ α
k } k∈N 0 be the strictly increasing zeros of the function ψ α (λ) in (3.5). Then the functionŝ satisfy the orthogonality relation (3.12) Proof Proceeding from the Green-type formula (3.4) and assuming that λ = δ α k and μ = δ α n are two distinct zeros of ψ α , the right-hand side of (3.4) vanishes sincê under the integral. Then we arrive at (δ α n − δ α k ) ˆ α k ,ˆ α n w α = 0, which yields the orthogonality of the two eigenfunctions. As to the normalization constantĥ α k , it follows again by (3.4) that This yields the representation (3.12).
Motivated by the result in Sec. 2 associated with the Fourier-Bessel-type functions, one may think of a Kramer-type theory associated with the Laguerre-type polynomials {L α,N n (x)} n∈N 0 , as well. When normalized in the same way as their classical counterparts R α n (x) in (1.5), the Laguerre-type polynomials possess the representations, see [27,28,31], [34,Sec. 3.3], where a α,N n : (3.15) Moreover, they form an orthogonal polynomial system in the space L 2 w α,N (0, ∞) with respect to the inner product ( f , g) w α,N defined in (1.10), with orthogonality relation, for k, n ∈ N 0 ,

16)
where L α x is the classical operator in (3.1) and T α x denotes the differential operator Furthermore, the eigenvalue parameter has the two components As it was shown by Wellman [34,Sec 5], an equivalent form of the differential expression L α,N x gives rise to a unique self-adjoint operator A α,N . Moreover, the so-called right-definite space L 2 w α,N (0, ∞) has the Laguerre-type polynomials as a complete set of eigenfunctions.
Our next crucial step is to determine a continuous system of functions { α,N λ (x)} λ>0 , which are eigenfunctions of the Laguerre-type operator L α,N x and reduce, as N → 0+, to the functions α λ (x) defined in (3.2). Since L α,N x is independent of the eigenvalue parameter, it suffices to look for continuous counterparts of the Laguerre-type polynomials and their eigenvalues. Observing that for α ∈ N 0 , b n := (α + 2) n−1 /(n − 1)! = (n) α+1 /(α + 1)!, we first rewrite the representations (3.15) by (3.18) Replacing now n ∈ N 0 by λ − α+1 2 , the required Whittaker-type functions and their spectral differential equation follow by analytic continuation.

Proposition 3.3
For α ∈ N 0 , N > 0, and e α λ := λ − α+1 2 ≥ 0, let the Whittaker-type functions be defined on 0 ≤ x < ∞ by Then there holds, with L α,N x as in (3.16), In particular, As in the case N = 0, it is clear that for N > 0 and any λ = n + (α + 1)/2, the Whittaker-type functions α,N λ (x) do not belong to L 2 w α,N (0, ∞). So once more, we restrict the differential equation (3.19) to the finite interval (0, 1] and define the operator for functions in the domain In particular, we note that α,N λ (x) ∈ D(L α,N x ), and The operatorL α,N x satisfies the following Green-type formula. (3.21) Proof We consider the two components of the operator L α,N T α x , separately. According to (3.4), the first part yields while, after applying an (α + 2)-fold integration by parts, the second part becomes since the resulting integrals have cancelled each other by symmetry in f , g. Evaluating the latter expression at the origin, we find that only the term for j = α + 1 contributes to the sum. In fact, it is easy to see that for any h ∈ D(L α,N x ), So we arrive at This, however, annihilates the term N (L α,N on the right-hand side of (3.21), which settles the proof of Thm. 3.2.

(3.22)
It is still an open problem whether the approach just outlined is applicable to find a new Kramer-type theorem for N > 0. In case α = 0, for instance, the Whittaker-type functions 0,N λ (x), 0 ≤ x ≤ 1, λ ≥ 1/2, may serve as a Kramer kernel K (x, λ). In order to generalize the sampling function S 0 k (λ) in (3.14) to a function S 0,N k (λ) via the Green-type formula (3.22), we have to find a sequence of eigenvalues δ 0,N k , k ∈ N 0 , say, such that the functions {K (x, δ 0,N k )} k∈N 0 form a complete orthogonal system in the respective Hilbert space.

Lemma 4.2 The expansion (4.4) is equivalent to
Proof Due to well-known properties of the Jacobi polynomials [9, 10.8], there holds while Eq.(4.1) and the Green formula for the operator L α,β Evaluating this expression at x = ±1, it follows by (4.2) that the only non-vanishing part is Hence we arrive at In view of the property of the -function [9, 1.2], the representation (4.4) follows.