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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
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.

MSC:
43A45 Spectral synthesis on groups, semigroups, etc.
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