Unconditional and quasi-greedy bases in $L_p$ with applications to Jacobi polynomials Fourier series

TL;DR

This paper investigates the convergence properties of Fourier series with Jacobi polynomial bases in Lp spaces and establishes limitations on the existence of unconditional bases across different Lp spaces.

Contribution

It demonstrates non-convergence of Jacobi polynomial Fourier series in Lp for p≠2 and extends classical results on unconditional bases in Lp spaces.

Findings

Fourier series with Jacobi polynomials do not converge in Lp unless p=2.

No normalized unconditional basis in Lp can be semi-normalized in Lq for p≠q.

Extension of Kadets and Pe{}czy{\'n}ski's theorem to broader Lp contexts.

Abstract

We show that the decreasing rearrangement of the Fourier series with respect to the Jacobi polynomials for functions in $L_p$ does not converge unless $p=2$. As a by-product of our work on quasi-greedy bases in $L_{p}(\mu)$, we show that no normalized unconditional basis in $L_p$, $p\not=2$, can be semi-normalized in $L_q$ for $q\not=p$, thus extending a classical theorem of Kadets and Pe{\l}czy{\'n}ski from 1968.