Fourier Series and Integrals

Abstract

The unit circle T consists of all complex numbers of modulus 1. It is a compact abelian group under multiplication. If ƒ is a function on T, we can define a periodic function F on the real line R by setting F(x) = f(e ^ix). It does not matter whether we study functions on T or periodic functions on R; generally we shall write functions on T. Everyone knows that in this subject the factor 2π appears constantly. However most of these factors can be avoided if we replace Lebesgue measure dx on the interval (0, 2π) by dσ(x) = dx/2π. Since σ is a measure on that interval, we can also omit the limits of integration when integrating with respect to a; they are always 0 and 2π.