Restriction spaces of $A^\infty$

Abstract

In the present paper it is shown that for certain totally disconnected Carleson sets $E$ the restriction space $A_\infty(E)=\{f|_E : f\in A^\infty\}$ has a basis. Its isomorphism type is determined. The result disproves a claim of S. R. Patel in [12]. To prove our result we analyze restriction spaces $C_\infty(E)=\{f|_E : f\in C^\infty(\mathbb{R})\}$ and then, using a result of Alexander, Taylor and Williams, we show that $A_\infty(E)=C_\infty(E)$. Among our examples there are the classical Cantor set and sets of type $E=\{x_n : n\in\mathbb{N}\}\cup\{0\}$, where $(x_n)_{n\in\mathbb{N}}$ is a null sequence in $\mathbb{R}$ with certain properties.