Mellin-Fourier series and the classical Mellin transform
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.
scientific article; zbMATH DE number 1471515