Spectral synthesis in certain Beurling algebras on the unit circle. (Synthèse spectrale dans certaines algèbres de Beurling sur le cercle unité.) (French) Zbl 0751.43006

Zarrabi, Mohamed

Bull. Soc. Math. Fr. 118, No. 2, 241-249 (1990).

A subset \(S\) of \(\Gamma\) is of type \(ZA^ +\) if there is a nontrivial function \(f\) in the Fourier algebra \(A(\Gamma)\) with \(\widehat f(n)=0\) for \(n<0\) vanishing on \(S\). Consider the Beurling algebras \(A_ w(\Gamma)\) of continuous functions on the unit circle \(\Gamma\) satisfying \(\sum_{n\in\mathbb{Z}}| \widehat f(n)| w_ n<\infty\) with a weight \((w_ n)_{n\in\mathbb{Z}}\) with \(w_ n=1\) for \(n\geq 0\) and \(\limsup_{n\to-\infty}(\log w(n)/\sqrt{-n})=0\). The author shows that among closed subsets of \(\Gamma\) of measure 0 and type \(ZA^ +\) only countable sets are sets of spectral synthesis for all Beurling algebras \(A_ w\) satisfying the conditions above.

Reviewer: M.Blümlinger (Wien)

MSC:

43A45

Spectral synthesis on groups, semigroups, etc.

Keywords:

Fourier algebra; Beurling algebras; unit circle; sets of spectral synthesis

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