General Walsh Series and Fourier-Stieltjes Series. Questions on Uniqueness of Representations of Functions by Walsh Series

Abstract

In this chapter we shall consider general Walsh series, i. e., series whose coefficients are not necessarily Walsh-Fourier coefficients of some function. For the study of these series, the function which is the sum of series obtained by term by term integration of the given series plays an important role. This function allows us to formulate necessary and sufficient conditions for a given series to be a Walsh-Fourier series or a Walsh-Fourier-Stieltjes, i. e., a Walsh series with coefficients of the form

$${{a}_{i}} = \int_{0}^{1} {{{w}_{i}}(x)d\psi (x).}$$

(This integral is understood as a Stieltjes integral (see A.4.3).) In the process we shall show that any Walsh series can be interpreted as a Walsh-Fourier-Stieltjes series if we generalize the Stieltjes integral suitably. This generalization of the Stieltjes integral will be based on binary nets which were introduced in §1.1.