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A direct approach to the mellin transform

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Abstract

The aim of this paper is to present an approach to the Mellin transform that is fully independent of Laplace or Fourier transform theory, in a systematic, unified form, containing the basic properties and major results under natural, minimal hypotheses upon the functions in questions. Cornerstones of the approach are two definitions of the transform, a local and global Mellin transform, the Mellin translation and convolution structure, in particular approximation-theoretical methods connected with the Mellin convolution singular integral enabling one to establish the Mellin inversion theory. Of special interest are the Mellin operators of differentiation and integration, more correctly of anti-differentiation, enabling one to establish the fundamental theorem of the differential and integral calculus in the Mellin frame. These two operators are different than those considered thus far and more general. They are of particular importance in solving differential and integral equations. As applications, the wave equation onℝ + × ℝ+ and the heat equation in a semi-infinite rod are considered in detail. The paper is written in part from an historical, survey-type perspective.

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  1. Lehrstuhl A für Mathematik, RWTH Aachen, Templergraben 55, 52056, Aachen, Germany

    Paul L. Butzer

  2. Mathematiches Seminar, Christian-Albrechts-Univsität Kiel, Ludewig-Meyn-Str. 4, 24098, Kiel, Germany

    Stefan Jansche

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Butzer, P.L., Jansche, S. A direct approach to the mellin transform. The Journal of Fourier Analysis and Applications 3, 325–376 (1997). https://doi.org/10.1007/BF02649101

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