Nonlinear expansions in reproducing kernel Hilbert spaces

TL;DR

This paper introduces a new nonlinear expansion scheme in reproducing kernel Hilbert spaces, generalizing classical series and applying to Hardy spaces, with convergence properties depending on chosen multipliers.

Contribution

It presents a novel nonlinear expansion framework in RKHS and Hardy spaces, extending classical series and analyzing convergence behavior based on multipliers.

Findings

Generalizes Blaschke unwinding series to Hardy spaces

Provides conditions for convergence to functions or projections

Includes explicit examples not covered by previous methods

Abstract

We introduce an expansion scheme in reproducing kernel Hilbert spaces, which as a special case covers the celebrated Blaschke unwinding series expansion for analytic functions. The expansion scheme is further generalized to cover Hardy spaces $H^p$, $1<p<\infty$, viewed as Banach spaces of analytic functions with bounded evaluation functionals. In this setting a dichotomy is more transparent: depending on the multipliers used, the expansion of $f \in H^p$ converges either to $f$ in $H^p$-norm or to its projection onto a model space generated by the corresponding multipliers. Some explicit instances of the general expansion scheme, which are not covered by the previously known methods, are also discussed.