Structured coordinate systems in $L^p(\mathbb{R})$

Abstract

We construct Schauder bases of $L^p(\mathbb{R})$ with significant structural constraints on their elements. First, we construct non-negative Schauder bases of $L^p(\mathbb{R})$ for every $1 \leq p < \infty$. Although such bases cannot be unconditional, their basis constants can be made arbitrarily close to one. By duality, we also obtain bases with positive biorthogonal functionals and the same bounds for all $1\leq p<\infty$. Finally, we prove that $L^p(\mathbb{R})$ admits a Schauder basis consisting of translates of a single function for every $2\le p<\infty$. The generating function $f$ can be chosen to be the restriction to $\mathbb{R}$ of an entire function. For $p>2$, the translation parameters can be chosen from any prescribed unbounded subset of $\mathbb{R}$, a conclusion that fails for $p=2$.