Frames for operators in Banach spaces via semi-inner products

TL;DR

This paper introduces a new framework for operator frames in Banach spaces using semi-inner products, providing characterizations, reconstruction formulas, and perturbation results to advance sampling theory and frame analysis.

Contribution

It generalizes atomic systems to Banach spaces with semi-inner products, offering new characterizations, reconstruction formulas, and perturbation results for operator frames.

Findings

New characterization of atomic systems in Banach spaces

Reconstruction formula for frames in Banach spaces

Perturbation results for frames in Banach spaces

Abstract

In this paper, we propose to define the concept of family of local atoms and then we generalize this concept to the atomic system for operator in Banach spaces by using semi-inner product. We also give a characterization of atomic systems leading to obtain new frames. In addition, a reconstruction formula is obtain. Next, some new results are established. The characterization of atomic systems allows us to state some results for sampling theory in semi-inner product reproducing kernel Banach spaces. Finally, having used frame operator in Banach spaces, new perturbation results are established.