J. Austral. Math. Soc.  72 (2002), 363-388
Vector valued mean-periodic functions on groups

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


Abstract
Let $G$ be a locally compact Hausdorff abelian group and $X$ be a complex Banach space. Let $C(G, X)$ denote the space of all continuous functions $f: G\to X$, with the topology of uniform convergence on compact sets. Let $X'$ denote the dual of $X$ with the weak* topology. Let $M_c(G, X')$ denote the space of all $X'$-valued compactly supported regular measures of finite variation on $G$. For a function $f\in C(G, X)$ and $\mu\in M_c(G, X')$, we define the notion of convolution $f\star\mu$. A function $f\in C(G, X)$ is called mean-periodic if there exists a non-trivial measure $\mu\in M_c(G, X')$ such that $f\star\mu=0$. For $\mu\in M_c(G, X')$, let $MP(\mu)=\{f\in C(G, X) : f\star\mu=0\}$ and let $MP(G, X)=\bigcup_{\mu}MP(\mu)$. In this paper we analyse the following questions:
Is $MP(G, X)\neq \emptyset$?
Is $MP(G, X)\neq C(G, X)$?
Is $MP(G, X)$ dense in $C(G, X) $?
Is $MP(\mu)$ generated by `exponential monomials' in it?
We answer these questions for the groups $G=\mathbb{R}$, the real line, and $G=\mathbb{T}$, the circle group. Problems of spectral analysis and spectral synthesis for $C(\mathbb{R}, X)$ and $C(\mathbb{T}, X)$ are also analysed.
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