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 DiniLipschitz 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 functionf on G whose modulus of continuity, Ok(f), satisfies Ok(f) = O(k-1) as k oo and whose Fourier series diverges at a point of G. The functionf will be defined by means of the analogue in X of the classical Feje'r polynomials. Throughout this paper we will use the terminology and notations of [2] or [4] and we will assume that the reader is familiar with at least one of these papers. Let G and X be as in the abstract. The best-known example of such a group G is fn=l(Z(2))n, which has the system of Walsh functions as its dual group (see [1]). Vilenkin's theorem, mentioned in the abstract, is the following [4, 3.5 ]: If G is a primary group and if f is a continuous function on G whose modulus of continuity, Ok(f), satisfies Ok(f) = o(k-1) as k-+oo, then the Fourier series of f converges uniformly. In [2, Corollary 2 ] Onneweer and Waterman obtained the same result for groups G which satisfy the condition that Supn Pn = p oo. The proof resembles G. Faber's proof of a similar theorem for trigonometric Fourier series [5, p. 302]. It will be preceded by two lemmas. LEMMA 1. Let a0=l and if n satisfies mk<n<mk+1for some k,let an= (-1)(mMk+1-mk)-1. Let S(x) = Enf o anXn(X). Received by the editors February 25, 1970. AMS 1970 subject classifications. Primary 42A56; Secondary 43A75.