Geodesic
Approximation Properties of Some Types of Linear Means in Space $L^{p(x)}_{2\pi}$
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 13 (2013) no. 1, pp. 108-112 
T. N. Shakh-Emirov. Approximation Properties of Some Types of Linear Means in Space $L^{p(x)}_{2\pi}$. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 13 (2013) no. 1, pp. 108-112. http://geodesic.mathdoc.fr/item/ISU_2013_13_1_a26/
@article{ISU_2013_13_1_a26,
     author = {T. N. Shakh-Emirov},
     title = {Approximation {Properties} of {Some} {Types} of {Linear} {Means} in {Space} $L^{p(x)}_{2\pi}$},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {108--112},
     year = {2013},
     volume = {13},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ISU_2013_13_1_a26/}
}
TY  - JOUR
AU  - T. N. Shakh-Emirov
TI  - Approximation Properties of Some Types of Linear Means in Space $L^{p(x)}_{2\pi}$
JO  - Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
PY  - 2013
SP  - 108
EP  - 112
VL  - 13
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/ISU_2013_13_1_a26/
LA  - ru
ID  - ISU_2013_13_1_a26
ER  - 
%0 Journal Article
%A T. N. Shakh-Emirov
%T Approximation Properties of Some Types of Linear Means in Space $L^{p(x)}_{2\pi}$
%J Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
%D 2013
%P 108-112
%V 13
%N 1
%U http://geodesic.mathdoc.fr/item/ISU_2013_13_1_a26/
%G ru
%F ISU_2013_13_1_a26

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

Approximative properties of Norlund $\mathcal{N}_{n}(f,x)$ and Riesz $\mathcal{R}_{n}(f,x)$ means for trigonometric Fourier series in Lebesgue space of variable exponent $L^{p(x)}_{2\pi}$ are considered. Under certain conditions on Norlund and Riesz summation methods it is proved that the estimates $\|f-\mathcal{N}_{n}\|_{p(\cdot)}\le CM\delta^{\alpha}$, $\|f-\mathcal{R}_{n}\|_{p(\cdot)}\le CM\delta^{\alpha}$ hold for $f\in \mathrm{Lip}_{p(\cdot)}(\alpha,M)$ ($0<\alpha\le1$).

[1] Sharapudinov I. I., “Some problems in approximation theory by trigonometric polynomials in $L_{2\pi}^{p(x)}$”, Math. Forum (Itogi nauki. The South of Russia), 5 (2011), 108–118

[2] Guven A., Israfilov D. M., “Trigonometric approximation in Generalized Lebesgue spaces Lp(x)”, J. of Math. Inequalities, 4:2 (2010), 285–299 | DOI | MR | Zbl

[3] Chandra P., “Approximation by Nörlund operators”, Mat. Vestnik, 38 (1986), 263–269 | MR | Zbl

[4] Chandra P., “A note on degree of approximation by Nörlund and Riesz operators”, Mat. Vestnik, 42 (1990), 9–10 | MR | Zbl

[5] Diening L., “Maximal function on generalized Lebesgue spaces Lp($\cdot$)”, Math. Inequal. Appl., 7 (2004), 245–253 | MR | Zbl

[6] Kovacik O., Rakosnik J., “On spaces $L^{p(x)}$ and $W^{k,p(x)}$”, Czechoslovak Math. J., 41:4 (1991), 592–618 | MR | Zbl

[7] DeVore R. A., Lorentz G. G., Constructive Approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 303, Springer-Verlag, Berlin, 1993 | DOI | MR | Zbl