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

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 $\mathcal{L}^{p(\mathbb{R})$,} $1<p<\infty$.