Optimal interpolation in Hardy and Bergman spaces: a reproducing kernel Banach space approach

TL;DR

This paper develops a framework using reproducing kernel Banach spaces to solve optimal interpolation problems in Hardy and Bergman spaces, providing a method to find minimal norm solutions and illustrating with numerical examples.

Contribution

It introduces a reproducing kernel Banach space approach to interpolation in Hardy and Bergman spaces, extending existing methods to a broader functional analytic setting.

Findings

Derived a procedure for minimal norm interpolation in Hardy and Bergman spaces.

Extended the framework to $ ext{ell}^p$ spaces with numerical demonstrations.

Provided a unified approach connecting kernel methods and Banach space theory.

Abstract

After a review of the reproducing kernel Banach space framework and semi-inner products, we apply the techniques to the setting of Hardy spaces $H^p$ and Bergman spaces $A^p$, $1<p<\infty$, on the unit ball in $\mathbb{C}^n$, as well as the Hardy space on the polydisk and half-space. In particular, we show how the framework leads to a procedure to find a minimal norm element $f$ satisfying interpolation conditions $f(z_j)=w_j$, $j=1,\ldots , n$. We also explain the techniques in the setting of $\ell^p$ spaces where the norm is defined via a change of variables and provide numerical examples.