PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 53, Number 1, November 1975 A CHARACTERISATIONOF LIPSCHITZ CLASSES ON 0-DIMENSIONAL GROUPS WALTER R. BLOOM ABSTRACT. This paper is concerned with characterising, in terms of certain properties of their Fourier transforms, the Lipschitz functions of order a. (0 < a < 1) defined on a locally compact metric 0-dimensional Abelian group. 1. Introduction. Let G be a locally compact metric 0-dimensional Abe- lian group with translation-invariant metric d. Its character group will be denoted by Y. We shall characterise the Lipschitz functions of order a (0 < a < l) in terms of certain properties of their Fourier transforms. The results obtained are analogues of classical results due to Jackson and Bernstein (see [5, Chapter 3, Theorems (13.6), (13.20), respectively]). 2. Notation and preliminary results. Choose any strictly-decreasing sequence (fi ) of positive numbers smaller than 1 for which there exists p £ (0, l) such that B + j < pB for all n £ {1, 2, ... }. Consider the open sets V = {x £ G: d(x, 0) < B |. n "n Since \V !°°_. is an open basis at 0, it follows from [3, (7.7)] that all but at most a finite number of the V are contained in a compact subgroup of G. By deleting some of the /3 if necessary, we can assume that all the V enjoy this property. Now put T = A{Y, V ) (the annihilator of V in Y). Then each T r n n 72 72 is a compact open subgroup of Y, and T. C T2 O • • . Furthermore, since (by [3, (24.18)]) every element of Y lies in a compact subgroup of Y, it fol- lows that r = H00 i^ . V"72=I n Let A denote a chosen Haar measure on G. The spectrum (written 1(f)) of / £ L°°(G) will be defined as in [3, (40.21)]. For / £ LP(G) (p £ [l, r»)), we define its spectrum by S(/) = UiS(<£ * /): <f> € C0QiG)\ (where C AG) denotes the space of continuous functions on G with com- pact support). We shall write Received by the editors September 3, 1974. AMS(MOS)subject classifications (1970). Primary 43A15, 43A70. 1 ^q Copyright © 1975, American Mathematical Society