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

Gilbert Groenewald, Sanne ter Horst and Hugo Woerdeman
Transactions of the American Mathematical Society
19 Mar 2025
DOI: https://doi.org/10.1090/tran/9389

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 H^p and Bergman spaces A p A^p , 1 > p > ∞ 1>p>\infty , on the unit ball in C n \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 f satisfying interpolation conditions f ( z j ) = w j f(z_j)=w_j , j = 1 , … , n j=1,\ldots , n . We also explain the techniques in the setting of ℓ p \ell ^p spaces where the norm is defined via a change of variables and provide numerical examples.

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