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Universal symbols on locally compact abelian groups. (English. Russian original) Zbl 1280.43002
Funct. Anal. Appl. 47, No. 1, 1-13 (2013); translation from Funkts. Anal. Prilozh. 47, No. 1, 1-16 (2013).
A symbol on a dual group \(X\) is a function \(f\) such that \(f|Q=\hat{\mu}|Q\), for each compact set \(Q\subset X\), where \(\hat{\mu}\) is the Fourier transform of some measure \(\mu\). For each compact set \(Q\subset X\), let \(B(Q)\) be the Bernstein space of functions whose Fourier transforms are supported on \(Q\). A symbol \(f\) is said to be universal if \(\|f(T)\|=|f(T)|\), for any normal representation \(T\) of a locally compact abelian group \(G\). The aim of the paper is to give a clear description of universal symbols in terms of positive definite functions on groups. It is shown that the universality of the symbol \(f\) is equivalent to the possibility of extending the standard reduction of this symbol to a positive definite function on \(Q\), for every finite set \(Q\subset X\). Moreover, the authors also establish results for connected groups where the universal symbols \(f\) admit the representation \(f=\chi.(f_1\circ\phi)\), where \(\chi\) is a character, \(f_1\) is a one-dimensional universal symbol, and \(\phi\) is a continuous homomorphism into the additive group \(\mathbb R\). Furthermore, this approach is different from the previous approaches in the sense that the authors do not use the existing results on the coincidence of the norm and the spectral radius for individual \(B(Q)\).

MSC:
43A70 Analysis on specific locally compact and other abelian groups
46H30 Functional calculus in topological algebras
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