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

Fernando Albiac; José L. Ansorena; Óscar Ciaurri; Juan L. Varona

doi:10.4171/rmi/1062

JournalsrmiVol. 35, No. 2pp. 561–574

Unconditional and quasi-greedy bases in $L_{p}$ with applications to Jacobi polynomials Fourier series

Fernando Albiac
Universidad Pública de Navarra, Pamplona, Spain
José L. Ansorena
Universidad de la Rioja, Logroño, Spain
Óscar Ciaurri
Universidad de la Rioja, Logroño, Spain
Juan L. Varona
Universidad de la Rioja, Logroño, Spain

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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 $L_{p} (μ)$ , we show that no normalized unconditional basis in $L_{p}$ , $p \neq = 2$ , can be semi-normalized in $L_{q}$ for $q \neq = p$ , thus extending a classical theorem of Kadets and Pełczyński from 1962.

Cite this article

Fernando Albiac, José L. Ansorena, Óscar Ciaurri, Juan L. Varona, Unconditional and quasi-greedy bases in $L_{p}$ with applications to Jacobi polynomials Fourier series. Rev. Mat. Iberoam. 35 (2019), no. 2, pp. 561–574

DOI 10.4171/RMI/1062