The inversion of the generalized Fourier transform by Abelian summability

Abstract

1. The fundamental ideas of the generalized Fourier transform have been presented by S. Bochner [l; 2], for functions which are either locally integrable or locally square integrable. A parallel line of thought has been followed by N. Wiener [7] and H. R. Pitt [4] in their theory of generalized harmonic analysis. While the parts of the Wiener theory which are connected with the idea of the generalized Fourier transform do not treat functions which are as large at infinity as those treated by Bochner, they do come closer to giving a theory for Lp integrability than does Bochner (see [4]). More recently L. Schwartz [5] has given a treatment of the generalized Fourier transform by quite different methods. One of the weak points of the theory has been the lack of satisfactory inversion formulas, particularly for the generalized transform of order greater than one. The main part of this paper is devoted to showing that by the use of Abelian summability, the generalized Fourier transform may be inverted in the metric of the proper function space. The last part of the paper attempts to unify some of the ideas of generalized harmonic analysis and the generalized Fourier transform and to present a generalization of a well known theorem of Titchmarsh ^6, Theorem 74] The function space Lj, l^p< oo, consists of those functions for which the norm