Interpolating sequences for $H^{\infty}(B_H)$

TL;DR

This paper establishes conditions under which sequences in an infinite-dimensional Hilbert space can be interpolated by bounded holomorphic functions, providing explicit construction and bounds for the interpolation constants.

Contribution

It proves that extended Carleson's condition guarantees linear interpolation in $H^{ty}(B_H)$ for infinite-dimensional Hilbert spaces and constructs explicit interpolating functions.

Findings

Sequences satisfying extended Carleson's condition are interpolatable.

Explicit bounds for interpolation constants are derived.

Constructive methods for interpolating functions are provided.

Abstract

We prove that under the extended Carleson's condition, a sequence $(x_n) \subset B_H$ is linear interpolating for $H^{\infty}(B_H)$ for an infinite dimensional Hilbert space H. In particular, we construct the interpolating functions for each sequence and find a bound for the constant of interpolation.