Mellin-Fourier series and the classical Mellin transform
type: Item
instance of: scholarly article
title: Mellin-Fourier series and the classical Mellin transform
zbMATH Open document ID: 0959.44006
DOI: 10.1016/S0898-1221(00)00139-5
publication date: 2000-11-13T00:00:00Z
review text: This is a continuation of the authors' work [J. Fourier Anal. Appl. 3, No. 4, 325-376 (1997; Zbl 0885.44004)]. Starting from the finite Mellin transformation which defines the Mellin-Fourier coefficients, they carry out a direct development of the Mellin-Fourier series, quite independent of the usual Fourier theory. For functions in the naturally defined normed space of Lebesgue integrable functions \(Y_c\), the convergence theory for the Mellin-Fourier series is developed. Further, the Mellin-Poisson series formula and a Kramer type of exponential sampling theorem of signal analysis are obtained. Along the way ``Mellin derivatives'' and ``Mellin antiderivatives'' are defined and some properties are developed.
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 42A16
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 94C10
zbMATH DE Number: 1471515
zbMATH Keywords: Mellin derivatives
zbMATH Keywords: Mellin antiderivatives
zbMATH Keywords: finite Mellin transformation
zbMATH Keywords: Mellin-Fourier series
zbMATH Keywords: convergence theory
zbMATH Keywords: Mellin-Poisson series formula
zbMATH Keywords: exponential sampling theorem
zbMATH Keywords: signal analysis
author: Paul L. Butzer
author: Stefan Jansche
reviewed by: Robert G. Buschman
MaRDI profile type: Publication
cites work: A direct approach to the Mellin transform
cites work: Q4250570
cites work: Q5799604
cites work: Q5850027
cites work: COMPLEX FOURIER--BESSEL TRANSFORMS
cites work: What is the Laplace Transform?
cites work: Q5624174
cites work: Q5681648
cites work: Q4039797
cites work: Q3735790
cites work: Q3838514
cites work: Adapted wavelet analysis from theory to software
cites work: Q5621638
cites work: Q4343561
cites work: A self-contained approach to mellin transform analysis for square integrable functions; applications
cites work: Exponential-sampling method for Laplace and other dilationally invariant transforms: I. Singular-system analysis
cites work: Exponential-sampling method for Laplace and other dilationally invariant transforms: II. Examples in photon correlation spectroscopy and Fraunhofer diffraction
cites work: Q4839147
cites work: Q3835991
cites work: Q4319646
cites work: The sampling theorem and linear prediction in signal analysis
cites work: Q4524271
cites work: Mellin-type differential equations and associated sampling expansions
cites work: Q4219713
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 1471515