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

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - S. F. Lukomskii
TI  - Fourier series of functions with a non-summable derivative
JO  - Izvestiya. Mathematics 
PY  - 2009
SP  - 301
EP  - 318
VL  - 73
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2009_73_2_a2/
LA  - en
ID  - IM2_2009_73_2_a2
ER  - 
%0 Journal Article
%A S. F. Lukomskii
%T Fourier series of functions with a non-summable derivative
%J Izvestiya. Mathematics 
%D 2009
%P 301-318
%V 73
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2009_73_2_a2/
%G en
%F IM2_2009_73_2_a2
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/

[1] N. K. Bari, Trigonometricheskie ryady, Fizmatlit, M., 1961 | MR

[2] R. Ryan, “Conjugate functions in Orlicz spaces”, Pacific J. Math., 13:4 (1963), 1371–1377 | MR | Zbl

[3] M. A. Krasnosel'skiǐ, Ja. B. Rutickiǐ, Convex functions and Orlicz spaces, Noordhoff, Groningen, 1961 | MR | MR | Zbl | Zbl

[4] A. Zygmund, Trigonometric series, vols. I, II, Cambridge Univ. Press, New York, 1959 | MR | Zbl | Zbl

[5] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces. II. Function spaces, Ergeb. Math. Grenzgeb., 97, Springer-Verlag, Berlin–Heidelberg–New York, 1979 | MR | Zbl

[6] Sh. Chen, Geometry of Orlicz spaces, Dissertationes Math. (Rozprawy Mat.), 36, Polish Acad. of Sciences, Warsaw, 1996 | MR | Zbl

[7] L. V. Kantorovich, G. P. Akilov, Functional analysis, Pergamon Press, Oxford, 1982 | MR | MR | Zbl | Zbl

[8] B. Sendov, V. A. Popov, Averaged moduli of smoothness, B"lgarski Matematicheski Monografii, 4, Izdatelstvo na B"lgarskata Akademiya na Naukite, Sofiya, 1983 | MR | Zbl