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Vector valued mean-periodic functions on groups

Part of: Harmonic analysis in one variable Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

P. Devaraj  and
Inder K. Rana
P. Devaraj
Affiliation:
Department of Mathematics, Indian Institute of Technology, Powai, Mumbai-76, PIN-400076 India e-mail: devaraj@math.iitb.ac.in
Inder K. Rana
Affiliation:
Department of Mathematics, Indian Institute of Technology, Powai, Mumbai-76, PIN-400076 India e-mail: ikr@math.iitb.ac.in
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Abstract

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Let G be a locally compact Hausdorif abelian group and X be a complex Banach space. Let C(G, X) denote the space of all continuous functions f: G → X, with the topology of uniform convergence on compact sets. Let X′ denote the dual of X with the weak* topology. Let Mc(G, X′) denote the space of all X′-valued compactly supported regular measures of finite variation on G. For a function f ∈ C(G, X) and μ ∈ Mc(G, X′), we define the notion of convolution f * μ. A function f ∈ C(G, X) is called mean-periodic if there exists a non-trivial measure μ ∈ Mc(G, X′) such that f * μ = 0. For μ ∈ Mc(G, X′), let M P(μ) = {f ∈ C(G, X): f * μ = 0} and let M P(G, X) = ∪μ M P(μ). In this paper we analyse the following questions: Is M P(G, X) ≠ 0? Is M P(G, X) ≠ C(G, X)? Is M P(G, X) dense in C(G, X)? Is M P(μ) generated by ‘exponential monomials’ in it? We answer these questions for the groups G = ℝ, the real line, and G = T, the circle group. Problems of spectral analysis and spectral synthesis for C(ℝ, X) and C(T, X) are also analysed.

Keywords

Convolution of vector valued functionsspectrumvector valued mean-periodic functionsspectral synthesis

MSC classification

Secondary: 43A45: Spectral synthesis on groups, semigroups, etc. 42A75: Classical almost periodic functions, mean periodic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 3 , June 2002 , pp. 363 - 388
Copyright
Copyright © Australian Mathematical Society 2002

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