INTERPOLATION OF WEIGHTED $\ell^q$ SEQUENCES BY $H^p$ FUNCTIONS

Abstract

Let $(z_n)$ be a sequence of points in the open unit disc $D$ and $\rho_n = \prod_{m \ne n}|(z_n - z_m)(1 - \bar{z}_mz_n)^{-1}| \gt 0$. Let $a = (a_j)^\infty_{j = 1}$ be a sequence of positive numbers and $\ell^s(a) = \{(w_j);~(a_jw_j) \in \ell^s\}$ where $1 \le s \le \infty$. When $1 \le p \le \infty$ and $1/p + 1/q = 1$, we show that $\{(f(z_n));~f \in H^p\} \supset \ell^s(a)$ if and only if there exists a finite positive constant $\gamma$ such that $\left\{{\sum^\infty_{n = 1}} (a_n \rho_n)^{-t}(1 - |z_n|^2)^t|f(z_n)|^t \right\}^{1/t} \le \gamma \|f\|_q ~(f \in H^q)$, where $1/s + 1/t = 1$. As results, we show that $\{(f(z_j));~f \in H^p\} \supset \ell^1(a)$ if and only if ${\sup_n}(a_n \rho_n)^{-1}(1 - |z_n|^2)^{1/p} \lt \infty$, and $\{(f(z_n));~f \in H^1\} \supset \ell^\infty(a)$ if and only if ${\sum_n}(a_n \rho_n)^{-1}(1 - |z_n|^2)\delta_{z_n}$ is finite measure on $D$. These are also proved in the case of weighted Hardy spaces.