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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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##### References:
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