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Izv. RAN. Ser. Mat., 2009, Volume 73, Issue 2, Pages 91–108 (Mi izv1131)  

Fourier series of functions with a non-summable derivative

S. F. Lukomskii

Saratov State University named after N. G. Chernyshevsky, Faculty of Mathematics and Mechanics

Abstract: We consider the convergence of Fourier series in the norm of Orlicz spaces narrower than $L(e^x)$. It is shown that if a continuous function has a non-summable derivative, then its Fourier series is not necessarily convergent in the norm of these Orlicz spaces. We find a condition on a bounded function $f$ under which the Fourier series of $f$ is convergent in the norm of an Orlicz space $L(\varphi)\subset L(e^x)$ and estimate the accuracy of this result.

Keywords: Fourier series, convergence, Lorentz spaces, local modulus of continuity.

DOI: https://doi.org/10.4213/im1131

Full text: PDF file (549 kB)
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English version:
Izvestiya: Mathematics, 2009, 73:2, 301–318

Bibliographic databases:

UDC: 517.51
MSC: 41A25, 41A55, 65L10, 65R20, 42A15, 65T40, 65J105, 46A45, 46BXX
Received: 10.07.2006
Revised: 26.06.2007

Citation: S. F. Lukomskii, “Fourier series of functions with a non-summable derivative”, Izv. RAN. Ser. Mat., 73:2 (2009), 91–108; Izv. Math., 73:2 (2009), 301–318

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  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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