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

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 $A^p$, $1<p<\mathrm{\infty }$, 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 $f(z_j)=w_j$, $j=1,\dots ,n$. We also explain the techniques in the setting of ${\mathrm{\ell }}^p$ spaces where the norm is defined via a change of variables and provide numerical examples.