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

Groenewald, Gilbert; ter Horst, Sanne; Woerdeman, Hugo

doi:10.1090/tran/9389

Optimal interpolation in Hardy and Bergman spaces: A reproducing kernel Banach space approach
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by Gilbert J. Groenewald, Sanne ter Horst and Hugo J. Woerdeman;

Trans. Amer. Math. Soc. 378 (2025), 4303-4333

DOI: https://doi.org/10.1090/tran/9389

Published electronically: March 19, 2025

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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.

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