On the analytic form of the discrete Kramer sampling theorem

Antonio G García,María J Muñoz-Bouzo,Miguel A Hernández-Medina

doi:10.1155/s0161171201005385

Abstract

The classical Kramer sampling theorem is, in the subject of self‐adjoint boundary value problems, one of the richest sources to obtain sampling expansions. It has become very fruitful in connection with discrete Sturm‐Liouville problems. In this paper a discrete version of the analytic Kramer sampling theorem is proved. Orthogonal polynomials arising from indeterminate Hamburger moment problems as well as polynomials of the second kind associated with them provide examples of Kramer analytic kernels.

Highlights

The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling theorems [9, 11, 14]
In [6] the authors proved an extension of the Kramer sampling theorem to the case when the kernel is analytic in the sampling parameter λ
The main aim in this paper is to prove the analytic version of the Kramer sampling theorem for the discrete case

Summary

Introduction

The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling theorems [9, 11, 14] The statement of this result is as follows: let K(ω, λ) be a function, defined for all λ in a suitable subset D of R such that, as a function of ω, K(·, λ) ∈ L2(I) for every number λ ∈ D, where I is an interval of the real line. A straightforward discrete version of Kramer’s theorem can be obtained [2, 8] To this end, let K(n, λ) be a kernel such that, as a function of the discrete variable n ∈. The main aim in this paper is to prove the analytic version of the Kramer sampling theorem for the discrete case. We state the discrete version of the analytic Kramer sampling theorem. First we prove that any function defined by (2.1) is an entire function

Consider the function

Schwarz inequality

Schwarz inequality in