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Mean convergence of orthogonal Fourier series of modified functions


Authors: Martin G. Grigorian, Kazaros S. Kazarian and Fernando Soria
Journal: Trans. Amer. Math. Soc. 352 (2000), 3777-3798
MSC (2000): Primary 42C15, 42C20
DOI: https://doi.org/10.1090/S0002-9947-00-02561-7
Published electronically: April 13, 2000
MathSciNet review: 1695022
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Abstract: We construct orthonormal systems (ONS) which are uniformly bounded, complete, and made up of continuous functions such that some continuous and even some arbitrarily smooth functions cannot be modified so that the Fourier series of the new function converges in the $L^{p} $-metric for any $p > 2. $ We prove also that if $\Phi $ is a uniformly bounded ONS which is complete in all the spaces $L _ {[0,1]} ^{p} , 1 \leq p < \infty $, then there exists a rearrangement $\sigma $ of the natural numbers $\mathbf{N} $such that the system $\Phi _{\sigma }= \{ \phi _{\sigma (n)} \}_{n=1}^{\infty }$ has the strong $L^{p}$-property for all $p>2$; that is, for every $2 \leq p < \infty $ and for every $ f \in L _ {[0,1]} ^{p} $ and $\epsilon > 0 $there exists a function $ f_ \epsilon \in L _ {[0,1]} ^{p} $ which coincides with $f$ except on a set of measure less than $\epsilon $ and whose Fourier series with respect to the system $\Phi _{\sigma }$ converges in $L _ {[0,1]} ^{p} . $


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Additional Information

Martin G. Grigorian
Affiliation: Department of Radiophysics, Yerevan State University, Yerevan 375049, Armenia
Email: gmartin@ysu.am

Kazaros S. Kazarian
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Email: kazaros.kazarian@uam.es

Fernando Soria
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Email: fernando.soria@uam.es

DOI: https://doi.org/10.1090/S0002-9947-00-02561-7
Keywords: Complete orthonormal system, C-strong property, modification of functions, rearrangements of systems, divergence in metric, universal series
Received by editor(s): November 26, 1997
Published electronically: April 13, 2000
Additional Notes: The research described in this publication was made possible in part by Grant PB97/0030 from DGES. The first two authors were also supported by Grant MVR000 from the I.S.F
Article copyright: © Copyright 2000 American Mathematical Society

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