On the Uniformly Discrete and Minimal System for a Banach Subspace in $$L_p({\mathbb{R}}^{d})$$

Abstract

In this note, it is shown that if $$(f_i,g_i)_{i=1}^\infty \in L_p({\mathbb {R}}^d) \times L_q({\mathbb {R}}^d)$$ is a Schauder frame for a closed subspace X of $$L_p({\mathbb {R}}^d)$$, then X embeds almost isometrically into $$l_p$$. Also, the same conclusion holds, if for $$f\in L_p({\mathbb {R}}^d)$$, the translations f by $$\{x_i:x_i \in {\mathbb {R}}^d\}$$ is a bounded minimal system for X. A basis (frame) for the Banach space $$L_p[0,1]^2$$, $$1\le p< \infty$$ is constructed.