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Finite dimensional H-invariant spaces

Published online by Cambridge University Press:  17 April 2009

K.E. Hare  and
J.A. Ward
K.E. Hare
Affiliation:
Department of Pure MathematicsUniversity of WaterlooWaterloo, Ontario N2L 3G1Canada e-mail: kehare@math.uwaterloo.ca
J.A. Ward
Affiliation:
School of Mathematical and Physical SciencesMurdoch UniversityMurdoch WA 6155Australia e-mail: ward@prodigal.murdoch.edu.au
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Abstract

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A subset V of M(G) is left H-invariant if it is invariant under left translation by the elements of H, a subset of a locally compact group G. We establish necessary and sufficient conditions on H which ensure that finite dimensional subspaces of M(G) when G is compact, or of L∞(G) when G is locally compact Abelian, which are invariant in this weaker sense, contain only trigonometric polynomials. This generalises known results for finite dimensional G-invariant subspaces. We show that if H is a subgroup of finite index in a compact group G, and the span of the H-translates of μ is a weak*-closed subspace of L∞(G) or M(G) (or is closed in Lp(G)for 1 ≤ p < ∞), then μ is a trigonometric polynomial.

We also obtain some results concerning functions that possess the analogous weaker almost periodic condition relative to H.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 3 , December 1997 , pp. 353 - 361
Copyright
Copyright © Australian Mathematical Society 1997

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