Fourier coefficients of functions in power-weighted $L_2$-spaces and conditionality constants of bases in Banach spaces

TL;DR

This paper establishes a connection between Fourier coefficients in weighted $L_2$ spaces and $ ext{ell}_p$ sequences, then applies these results to construct bases in Banach spaces with controlled conditionality growth.

Contribution

It introduces new relationships between Fourier coefficients and sequence spaces, and constructs Schauder and almost greedy bases with specific conditionality properties in $ ext{ell}_p$ spaces.

Findings

Fourier coefficients of functions in weighted $L_2$ spaces belong to $ ext{ell}_p$ for $2<p< $

Fourier series of $ ext{ell}_p$ sequences belong to weighted $L_2$ spaces for $1<p<2$

Existence of Schauder and almost greedy bases with prescribed conditionality growth in $ ext{ell}_p$

Abstract

We prove that, given $2<p<\infty$, the Fourier coefficients of functions in $L_2(\mathbb{T}, \lvert t \rvert^{1-2/p}\, dt)$ belong to $\ell_p$, and that, given $1<p<2$, the Fourier series of sequences in $\ell_p$ belong $L_2(\mathbb{T}, \lvert t \rvert^{2/p-1}\, dt)$. Then, we apply these results to the study of conditional Schauder bases and conditional almost greedy bases in Banach spaces. Specifically, we prove that, for every $1<p<\infty$ and every $0\le \alpha<1$, there is a Schauder basis of $\ell_p$ whose conditionality constants grow as $(m^\alpha)_{m=1}^\infty$, and there is an almost greedy basis of $\ell_p$ whose conditionality constants grow as $((\log m)^\alpha)_{m=2}^\infty$.