Functions and their Fourier transforms with supports of finite measure for certain locally compact groups
Let G be a locally compact group, and f an integrable function on G. The Fourier transform of f, an operator-valued function on the unitary dual \(\hat G\) of G is written \(\hat f.\) We let \(A_ f=\{x\in G:\) \(f(x)\neq 0\}\) and \(B_ f=\{\lambda \in \hat G:\) \(\hat f(\lambda)\neq 0\}\). The authors investigate the significance of the two inequalities \(m(A_ f)<m(G)\) and \(\mu (B_ f)<\mu (\hat G)\), where m is Haar measure on G and \(\mu\) is Plancherel measure on \(\hat G,\) or related inequalities. It is shown that, for many groups G, if these two conditions hold, then \(f=0\) a.e., generalizing a theorem of \textit{M. Benedicks} [J. Math. Anal. Appl. 106, 180-183 (1985; Zbl 0576.42016)] for \({\mathbb{R}}^ n\). The groups G considered include compact Lie groups, the group of Euclidean motions of \({\mathbb{R}}^ 2\), the \(``ax+b\)-group'', the Heisenberg group, and SL(2,\({\mathbb{R}})\).
scientific article; zbMATH DE number 4075768
scientific article