Investigations in harmonic analysis

Abstract

This paper is concerned with the theory of ideals in the algebra L1 of integrable functions on a locally compact abelian group. After some preliminaries an analytical proof is given of the known theorem that an analytic function of a Fourier transform represents again a Fourier transform (p. 406). Then, in part I, the continuous homomorphisms of closed ideals I of L1 upon C, the field of complex numbers, are studied. Any such homomorphism is given by a Fourier transform and, if I o is its kernel, the quotient-algebra I/Io, normed in the usual way, is not only algebraically isomorphic, but also isometric with C (Theorem 1.2). Another result states that homomorphic groups have homomorphic L'-algebras and that a corresponding property of isometry holds (Theorem 1.3). In part II, which may be read independently of part I, a theorem of S. Mandelbrojt and S. Agmon, which generalizes Wiener's theorem on the translates of a function in Ll, is extended to groups (Theorem 2.2). Several generalizations of Wiener's classical theorem have been published in the past few years; references to the literature are given on p. 422. The rest of part II is devoted to some applications (pp. 422-425). In conclusion it should be said that the work is carried out in abstract generality, with the methods, and in the spirit, of analysis, which is then applied to algebra. To Professors S. Mandelbrojt, B. L. van der Waerden, and A. Weil I owe my mathematical education. The inspiration which I have received in their lectures, in letters, and above all in personal contact, is at the base of this work ; may I here express my gratitude.