Valiron\u2019s Interpolation Formula and a Derivative Sampling Formula in the Mellin Setting Acquired via Polar-Analytic Functions

Carlo Bardaro,Ilaria Mantellini,Paul L Butzer,Gerhard Schmeisser

doi:10.1007/s40315-020-00341-w

Abstract

In this paper, we first recall some recent results on polar-analytic functions. Then we establish Mellin analogues of a classical interpolation of Valiron and of a derivative sampling formula. As consequences a new differentiation formula and an identity theorem in Mellin–Bernstein spaces are obtained. The main tool in the proofs is a residue theorem for polar-analytic functions.

Highlights

The notion of polar-analytic functions was first introduced in [1] as a simple alternative for functions that are analytic on a part of the Riemann surface of the logarithm
It is of interest in the realm of Mellin analysis and the theory of quadrature formulae on the positive real axis as well as in associated computations since it avoids the Dedicated to the memory of Professor Stephan Ruscheweyh
In Mellin analysis polar-analytic functions play a role similar to the one played by analytic functions in Fourier analysis

Summary

Introduction

The notion of polar-analytic functions was first introduced in [1] as a simple alternative for functions that are analytic on a part of the Riemann surface of the logarithm It is of interest in the realm of Mellin analysis and the theory of quadrature formulae on the positive real axis (see [1,2]) as well as in associated computations since it avoids the Dedicated to the memory of Professor Stephan Ruscheweyh. It allowed us to deduce an efficient residue theorem for polar-analytic functions Based on this theorem, we established an analogue of Boas’ differentiation formula for polar Mellin derivatives and, as a consequence, a Bernstein type inequality for polar Mellin derivatives. As a further application of our theory of polar-analytic functions, we employ the new residue theorem of [4] for deducing Mellin analogues of Valiron’s interpolation formula and of a derivative sampling formula.

Preliminaries

Some Basic Results on Polar-Analytic Functions

A Mellin Analogue of Valiron’s Interpolation Formula

Derivative Sampling in the Mellin Setting

The Differentiation Formulae

The Sampling Formula of Theorem 7