Some Harmonic Analysis

Abstract

The space ℒ consists of smooth functions which together with all their derivatives decay rapidly to zero as x → ∞. It is very useful in Fourier analysis, as it forms an easily manipulated family of functions which is mapped isomorphically onto itself by the Fourier transform. From this starting point, the classical theorems of Plancherel and Hausdorf—Young follow by straightforward completion arguments. This convenient formulation was exploited by S. Bochner. It was the inspired idea of L. Schwartz that an extension by duality, rather than continuity, gives a far-reaching generalization of the Fourier transform which has been crucial in modern analysis ever since. It is the goal of this chapter to give a brief description of these ideas. At the same time we review the basic techniques of the Theory of Distributions. It is assumed that the reader has a modest familiarity with the elementary Theory of Distributions. A brief introduction is presented in Appendix A.