Geodesic
Weighted integrability of sums of series with respect to multiplicative systems
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 2, pp. 129-136.

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

A necessary and sufficient condition for $L^p$-integrability with power weight of a function $f$ represented by the series with respect to multiplicative systems with generalized monotone coefficients is obtained. The integrability of the majorant of partial sums of a representing series is also described by the same conditions. In addition we study the integrability of difference quotient $(f(x)-f(0))/x$. 
@article{ISU_2014_14_2_a1,
     author = {S. S. Volosivets and R. N. Fadeev},
     title = {Weighted integrability of sums of series with respect to multiplicative systems},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {129--136},
     publisher = {mathdoc},
     volume = {14},
     number = {2},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ISU_2014_14_2_a1/}
}
TY  - JOUR
AU  - S. S. Volosivets
AU  - R. N. Fadeev
TI  - Weighted integrability of sums of series with respect to multiplicative systems
JO  - Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
PY  - 2014
SP  - 129
EP  - 136
VL  - 14
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ISU_2014_14_2_a1/
LA  - ru
ID  - ISU_2014_14_2_a1
ER  - 
%0 Journal Article
%A S. S. Volosivets
%A R. N. Fadeev
%T Weighted integrability of sums of series with respect to multiplicative systems
%J Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
%D 2014
%P 129-136
%V 14
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ISU_2014_14_2_a1/
%G ru
%F ISU_2014_14_2_a1
S. S. Volosivets; R. N. Fadeev. Weighted integrability of sums of series with respect to multiplicative systems. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 2, pp. 129-136. http://geodesic.mathdoc.fr/item/ISU_2014_14_2_a1/

[1] Golubov B. I., Efimov A. V., Skvortsov V. A., Walsh series and transforms. Theory and applications, Kluwer, Dordrecht, 1991 | MR | MR | Zbl | Zbl

[2] Tikhonov S., “Trigonometric series with general monotone coefficients”, J. Math. Anal. Appl., 326:1 (2007), 721–735 | DOI | MR | Zbl

[3] Leindler L., “A new class of numerical sequences and its application to sine and cosine series”, Analysis Math., 28:4 (2002), 279–286 | DOI | MR | Zbl

[4] Konyushkov A. A., “The best approximation by trigonometrical polynomials and Fourier coefficients”, Mat. sbornik, 44(86):1 (1958), 53–84 | MR | Zbl

[5] Boas R. P., Integrability theorems for trigonometric transforms, Springer, Berlin, 1967, 68 pp. | MR | Zbl

[6] Dyachenko M., Tikhonov S., “Integrability and continuity of functions represented by trigonometric series: coefficients criteria”, Studia Math., 193:3 (2009), 285–306 | DOI | MR | Zbl

[7] Moricz F., “On Walsh series with coefficients tending monotonically to zero”, Acta Math. Hung., 38:1–4 (1983), 183–189 | MR

[8] Timan M. F., Rubinstein A. I., “On embedding of function classes defined on zero-dimensional groups”, Soviet Math., 1980, no. 6, 66–76 | MR | Zbl

[9] Volosivets S. S., “On certain conditions in the theory of series with respect to multiplicative systems”, Analysis Math., 33:3 (2007), 227–246 | DOI | MR | Zbl

[10] Volosivets S. S., Fadeev R. N., “Estimates of best approximations in integral metrics and Fourier coefficients with respect to multiplicative systems”, Analysis Math., 37:3 (2011), 215–238 | DOI | MR | Zbl

[11] Askey R., Wainger S., “Integrability theorems for Fourier series”, Duke Math. J., 33:2 (1966), 223–228 | DOI | MR | Zbl

[12] Vukolova T. M., Dyachenko M. I., “On the properties of trigonometric series sums with monotone coefficients”, Vestnik Mosk. Universiteta. Ser. Matem. mech., 1994, no. 3, 22–31 | MR

[13] Agaev G. N., Vilenkin N. Ya., Dzhafarli G. M., Rubinstein A. I., Multiplicative systems of functions and harmonic analysis on zero-dimensional groups, Elm, Baku, 1980 | MR

[14] Hardy G. H., Littlewood J. E., Polya G., Inequalities, Cambridge University Press, Cambridge, 1934 | Zbl

[15] Leindler L., “Inequalities of Hardy–Littlewood type”, Analysis Math., 2:2 (1976), 117–123 | DOI | MR | Zbl

[16] Leindler L., “Generalization of inequalities of Hardy and Littlewood”, Acta Sci. Math. (Szeged), 31:1–2 (1970), 279–285 | MR | Zbl

[17] Iofina T. V., Volosivets S. S., “On the degree of approximation by means of Fourier–Vilenkin series in Hölder and $L^p$ norm”, East J. Approximations, 15:2 (2009), 143–158 | MR