Vol. 54, No. 2, 1974

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Invariant means and ergodic sets in Fourier analysis

Gordon S. Woodward

Vol. 54 (1974), No. 2, 281–299

Let ϕ be a umiformly continuous function on the locally compact, non-compacl abelian group G. Denote the dual of G by Γ. Then ϕ is ergodic on E rif ϕγ has a unique (translation-invariant) mean value for each γ E. Theorem: If E is closed and scattered, then there is a sequence {pn} of ffigonometric polynomials and a sequence {gn}⊂ L1(G) with suppĝn E = Φ such that gϕ + pn ϕ uniformly. Now suppose E = Γ∖{γ} and let {pβ} be any net of almost periodic functions on G with bounded L1 (G)- norm. Theorem: All cluster points 20 {Mypβ(⋅− y)ϕ(y)}β in the topoloy of uniform convergence on compact subsets of G are ergodic. Here M is any invariant mean on G. A compact set E Γ is ergodic if each ϕ Φ(E), the L(G)-weak* closure of span E, is (equal a.e. to) a function ergodic on Γ. Theorem: If E ergodic and S is compact scattered, then E S is ergodic. For E’s which are Helson sets this implies that the elements of Φ(E S) decompose into μ + u where μ Mc(E) and u is almost periodic.

Mathematical Subject Classification 2000
Primary: 43A60
Secondary: 43A07
Received: 3 January 1973
Revised: 12 August 1974
Published: 1 October 1974
Gordon S. Woodward