Geodesic
Fourier series of functions with a non-summable derivative
Izvestiya. Mathematics , Tome 73 (2009) no. 2, pp. 301-318

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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, Lorentz spaces, local modulus of continuity.
Mots-clés : convergence 
@article{IM2_2009_73_2_a2,
     author = {S. F. Lukomskii},
     title = {Fourier series of functions with a non-summable derivative},
     journal = {Izvestiya. Mathematics },
     pages = {301--318},
     publisher = {mathdoc},
     volume = {73},
     number = {2},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2009_73_2_a2/}
}
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S. F. Lukomskii. Fourier series of functions with a non-summable derivative. Izvestiya. Mathematics , Tome 73 (2009) no. 2, pp. 301-318. http://geodesic.mathdoc.fr/item/IM2_2009_73_2_a2/