III - Fourier–Stieltjes Coefficients, Fourier–Stieltjes Transforms and Characteristic Functions

Abstract

This chapter discusses Fourier–Stieltjes coefficients, Fourier–Stieltjes transforms, and characteristic functions. It also discusses the average of Fourier–Stieltjes coefficients, unicity theorem for Fourier–Stieltjes coefficients, Fourier–Stieltjes transform and characteristic function, the Fourier–Stieltjes transform of a convolution, and the Bernoulli convolution. The Fourier–Stieltjes coefficients determine the function uniquely up to additive constants. The chapter discusses some inequality relations for characteristic functions and also presents some inequality relations of elementary character. It also presents the notion of the vector sum of two point sets. The vector sum of two point sets A and B is the set of all points that are represented as a + b, a ∈ A, b ∈ B. If one of A and B is empty, the vector sum is interpreted to be empty.