The Fourier Transform Identity Theorem
Andrew C. Berry · Annals of Mathematics · 1931
It is known that a function f (x) of integrable square possesses a Fourier transform F(x) also of integrable square and that the Fourier transform of F(x) is precisely f(x). The present paper establishes this identity under more general assumptions and makes use of the Titchmarsh theory only for the derivation of a corollary to the main result. The one-dimensional case will be discussed in detail, and the method of extending the results to n dimensions indicated. 1. Throughout this paper region shall mean point-set, one-dimensional or n-dimensional, measurable in the sense of Lebesgue. A function shall be said to be integrable if it is integrable in the sense of Lebesgue over all finite regions. To simplify the presentation we introduce the following notations. DEFINITION 1. An inteegrable function f (x) wtill be said to be in Le for some e such that 1 ? Q < GO if there exists, as a, finite number,