On moduli of continuity and divergence of Fourier series on groups.

Abstract:

Let G be a 0-dimensional, metrizable, compact, abelian group. Then its character group X is a countable, discrete, torsion, abelian group. N. Ja. Vilenkin defined an enumeration for the elements of X and developed part of the Fourier theory on G. Among other things he proved on G a theorem similar to the Dini-Lipschitz test for trigonometric Fourier series. In this note we will show that Vilenkin’s result is in some sense the best possible by proving the existence of a continuous function f on G whose modulus of continuity, ${\theta _k}(f)$, satisfies ${\theta _k}(f) = O({k^{ - 1}})$ as $k \to \infty$ and whose Fourier series diverges at a point of G. The function f will be defined by means of the analogue in X of the classical Fejér polynomials.