Abstract
Let $G$ be a separable locally compact group with type $I$ left regular representation, $\widehat{G}$ its dual, $A(G)$ its Fourier algebra and $f\in A(G)$ with compact support. If $G=\mathbb{R}$ and the Fourier transform of $f$ is compactly supported, then, by a classical Paley–Wiener theorem, $f=0$. There are extensions of this theorem for abelian and some unimodular groups. In this paper, we prove that if $G$ has no (nonempty) open compact subsets, $\hat{f}$, the regularised Fourier cotransform of $f$, is compactly supported and $\text{Im}\,\hat{f}$ is finite dimensional, then $f=0$. In connection with this result, we characterise locally compact abelian groups whose identity components are noncompact.
Highlights
If G (is nonunimodular and) has type I left regular representation, the Fourier inversion theorem [14, Theorem 4.5] states that a function f ∈ A(G) ∩ L1(G) is recovered from its regularised Fourier cotransform F ( f ) ◦ K by means of f (x) = G Tr[F ( f )(π)Kππ(x)−1] dμ(π), where μ is a Plancherel measure, unique up to equivalence, on G and K = (Kπ)π∈G is a specific measurable field of (unbounded) positive self-adjoint operators
Throughout this paper, G denotes a separable locally compact group with type I left regular representation equipped with a left Haar measure ν
Let ∆ be the modular function of G and A(G) the Fourier algebra of G
Summary
If G (is nonunimodular and) has type I left regular representation, the Fourier inversion theorem [14, Theorem 4.5] states that a function f ∈ A(G) ∩ L1(G) is recovered from its regularised Fourier cotransform F ( f ) ◦ K by means of f (x) = G Tr[F ( f )(π)Kππ(x)−1] dμ(π), where μ is a Plancherel measure, unique up to equivalence, on G and K = (Kπ)π∈G is a specific measurable field of (unbounded) positive self-adjoint operators. G denotes a separable locally compact group with type I left regular representation and fdenotes the regularised Fourier cotransform of f ∈ A(G) [13, page 547].
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