Revista Matemática Iberoamericana


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Published online first: 2019-02-05
DOI: 10.4171/rmi/1062

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

Fernando Albiac[1], José L. Ansorena[2], Óscar Ciaurri[3] and Juan L. Varona[4]

(1) Universidad Pública de Navarra, Pamplona, Spain
(2) Universidad de la Rioja, Logroño, Spain
(3) Universidad de la Rioja, Logroño, Spain
(4) Universidad de la Rioja, Logroño, Spain

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 $L_p$, $p \neq 2$, can be semi-normalized in $L_q$ for $q \neq p$, thus extending a classical theorem of Kadets and Pełczyński from 1962.

Keywords: Thresholding greedy algorithm, unconditional basis, quasi-greedy basis, $L_p$-spaces, Jacobi polynomials

Albiac Fernando, Ansorena José, Ciaurri Óscar, Varona Juan: Unconditional and quasi-greedy bases in $L_p$ with applications to Jacobi polynomials Fourier series. Rev. Mat. Iberoam. Electronically published on February 5, 2019. doi: 10.4171/rmi/1062 (to appear in print)