Closed ideals in certain Beurling algebras, and synthesis of hyperdistributions

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 $lim sup_{n→-∞}$ |\hatφ(n)| < ∞$ and such that $lim sup_{n→∞} (log^{+}|\hatφ(n)|)/√n < ∞$