The Mellin–Parseval formula and its interconnections with the exponential sampling theorem of optical physics. Issue 1 (2nd January 2016)

• Record Type: Journal Article
• Authors: Bardaro, Carlo; Butzer, Paul L.; Mantellini, Ilaria
• Is Part Of: Integral transforms and special functions. Volume 27:Issue 1(2016)
• Page Start: 17
• Page End: 29
• Publication Date: 2016-01-02
• Subjects: Mellin transform -- Mellin bandlimited functions -- exponential sampling theorem -- reproducing kernel formula -- Mellin–Parseval formula; 30D10 -- 94A20 -- 42C15 -- 46E22; Integral transforms -- Periodicals; Transcendental functions -- Periodicals; Transformations (Mathematics) -- Periodicals; Calculus, Integral -- Periodicals; 515
• DOI: 10.1080/10652469.2015.1087401
• Languages: English
• ISSNs: 1065-2469

Abstract

ABSTRACT: In this paper, we establish a Mellin version of the classical Parseval formula of Fourier analysis in the case of Mellin bandlimited functions, and its equivalence with the exponential sampling formula (ESF) of signal analysis, in which the samples are not equally spaced apart as in the classical Shannon theorem, but exponentially spaced. Two quite different examples are given illustrating the truncation error in the ESF. We employ Mellin transform methods for square-integrable functions.