Differences of functions and groups generators for compact Abelian groups

Abstract

LetG be a Hausdorff compact Abelian group andC be the component of the identity element ofG. We consider a special class, ℋ(G), of functions inL ² (G) whose Fourier series satisfy certain convergence conditions (stronger than absolute convergence). We show thatG/C is topologically generated by not more thann elements if and only if, for each functionf in ℋ(G), there area ₁,...,a _n inG and functionf ₁,...f _n in ℋ(G) such that

$$f = \sum\limits_{j = 1}^n {(f_j - \delta _{aj} * f_j ),}$$

where * is convolution defined in the usual sense, and δ _a denotes the Dirac measure ataεG.