Advances in Operator Theory

On Herz's extension theorem

Antoine Derighetti

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‎We present a self-contained proof of the following famous extension theorem due to Carl Herz‎. ‎A closed subgroup $H$ of a locally compact group $G$ is a set of $p$-synthesis in $G$ if and only if‎, ‎for every $u\in A_p(H)\cap C_{00}(H)$ and for every $\varepsilon > 0$‎, ‎there is $v\in A_p(G)\cap C_{00}(G)$, an extension of $u$, such that \[\|v\|_{A_p(G)} < \|u\|_{A_p(H)}+\varepsilon.\]

Article information

Adv. Oper. Theory, Volume 4, Number 2 (2019), 529-538.

Received: 10 September 2018
Accepted: 16 November 2018
First available in Project Euclid: 1 December 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc.
Secondary: 43A22: Homomorphisms and multipliers of function spaces on groups, semigroups, etc.

locally compact group‎ ‎ ‎set of spectral synthesis ‎ ‎‎‎extension property


Derighetti, Antoine. On Herz's extension theorem. Adv. Oper. Theory 4 (2019), no. 2, 529--538. doi:10.15352/aot.1809-1417.

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