The interpolation of $l^r$ sequences by $H^p$ functions

S. V. Shvedenko

Matematičeskie zametki, Tome 21 (1977) no. 4, pp. 503-508.

Résumé

The sequence space $H^p(Z)=\{\{f(z_k)\}:f\in H^p\}$ is defined for a fixed sequence $Z=\{z_k\}$ of different points of the open unit disk and the Hardy class $H^p$ of analytic functions in the disk. For an arbitrary p $p\in[1,\infty)$ is constructed a point sequence $Z=\{z_k\}$ such that $l^1\subset H^p(Z)$, but $l^r\not\subset H^p(Z)$ for $r>1$. It follows from a well-known result of L. Carleson that the inclusions $l^r\subset H^\infty(Z)$ for all $r\in[1,\infty]$ are equivalent.