Fourier coefficients of functions in power-weightedL2-spaces and conditionality constants of bases in Banach spaces

Abstract

We prove that, given$2< p<\infty$, the Fourier coefficients of functions in$L_2(\mathbb {T}, |t|^{1-2/p}\,{\rm d}t)$belong to$\ell _p$, and that, given$1< p<2$, the Fourier series of sequences in$\ell _p$belong to$L_2(\mathbb {T}, \vert {t}\vert ^{2/p-1}\,{\rm d}t)$. 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$((\log m)^{\alpha })_{m=2}^{\infty }$.