The $(abc,pqr)$-problem for Approximate Schauder Frames for Banach Spaces

TL;DR

This paper introduces the $(abc,pqr)$-problem for approximate Schauder frames in Banach spaces, extending previous work on Gabor systems and translation operators in $ extit{L}^p$ spaces, to explore new theoretical aspects of frame theory.

Contribution

It formulates a new $(abc,pqr)$-problem for approximate Schauder frames in Banach spaces, expanding the scope of frame theory beyond Hilbert spaces.

Findings

Formulation of the $(abc,pqr)$-problem for Banach spaces.

Extension of $abc$-problem solutions from Hilbert to Banach spaces.

Connection between Gabor systems and approximate Schauder frames.

Abstract

Motivated from the complete solution of important $abc$-problem for Gabor system for the Hilbert space $\mathcal{L}^2(\mathbb{R})$ by Dai and Sun [\textit{Memoirs of Amer. Math. Soc., 2016}] and from the existential result of approximate Schauder frames for $\mathcal{L}^p(\mathbb{R})$ using translation operators on $\mathcal{L}^p(\mathbb{R})$ by Freeman, Odell, Schlumprecht, and Zsak [\textit{Israel. J. Math, 2014}], we formulate $(abc, pqr)$-problem for approximate Schauder frames for Banach spaces $\mathcal{L}^p(\mathbb{R})$, $1<p<\infty$.