Biorthogonal bases and multiple interpolation in weighted Hardy spaces

Abstract

Let {zk=xk+iyk} be a sequence on upper half plane\(\mathbb{R}_ + ^2 \) and {si} be the number of appearence of zk in {z1,z2,...,zk}. Suppose sup si<+∞. Let ω(x) be a weight belonging to A∞ and\(w_j = \smallint _{x_i }^{x_i + y_i } w(x)dx\). We Consider the weighted Hardy space\(H_{ + w}^p H_{ + w}^p (\mathbb{R}_ + ^2 )\) and operator Tp mapping f(z)∈H +w p into a sequence defined by\((T_p f)_j = w_j^{\tfrac{1}{p}} y_j^{s_j - 1} f^{(s_j - 1)} (z_k ),0< p \leqslant + \infty ,j = 1,2, \cdots \), 0<p≤+∞, j=1,2,.... Then Tp(H +w p )=lp if and only if {zk} is uniformly separated. Besides the effective solution for interpolation is obtained.