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

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 [$\mathit{\text{Memoirs}}\mathit{\text{of}}\mathit{\text{Amer.}}\mathit{\text{Math.}}\mathit{\text{Soc.,}}\mathit{\text{2016}}$] 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 [$\mathit{\text{Israel.}}\mathit{\text{J.}}\mathit{\text{Math,}}\mathit{\text{2014}}$], we formulate $(abc,pqr)$-problem for approximate Schauder frames for Banach spaces ${\mathcal{L}}^p(\mathbb{R})$, $1<p<\mathrm{\infty }$.