Sampling basis in reproducing kernel Banach spaces

TL;DR

This paper establishes necessary and sufficient conditions for sampling theorems in semi-inner product reproducing kernel Banach spaces, generalizing known results from Hilbert spaces and emphasizing the role of Riesz bases.

Contribution

It introduces a comprehensive framework for sampling in Banach spaces, extending classical theorems from Hilbert spaces to a broader Banach space setting.

Findings

Sequence must be an $X_d$-Riesz basis for sampling

Sampling basis characterized by necessary and sufficient conditions

Generalization of sampling theorems from Hilbert to Banach spaces

Abstract

We present necessary and sufficient conditions to hold true a Kramer type sampling theorem over semi-inner product reproducing kernel Banach spaces. Under some sampling-type hypotheses over a sequence of functions on these Banach spaces it results necessary that such sequence must be a $X_d$-Riesz basis and a sampling basis for the space. These results are a generalization of some already known sampling theorems over reproducing kernel Hilbert spaces.