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Closed ideals in certain Beurling algebras, and synthesis of hyperdistributions. (English) Zbl 0864.46025
Summary: We consider the ideal structure of two topological Beurling algebras which arise naturally in the study of closed ideals of \(A^+\). Even in the case of closed ideals \(I\) such that \(h(I)=E_{1/p}\), the perfect symmetric set of constant ratio \(1/p\), some questions remain open, despite the fact that closed ideals \(J\) of \(A^+\) such that \(h(J)=E_{1/p}\) can be completely described in terms of inner functions. The ideal theory of the topological Beurling algebras considered in this paper is related to questions of synthesis for hyperdistributions such that \(\limsup_{n\to-\infty}|\widehat\varphi(n)|<\infty\) and such that \(\limsup_{n\to\infty}(\log^+|\widehat\varphi(n)|)/\sqrt n<\infty\).

46H10 Ideals and subalgebras
43A20 \(L^1\)-algebras on groups, semigroups, etc.
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