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.