Geodesic
Approximation of functions of bounded $p$-variation by Euler means
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 3, pp. 300-309 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper we study the Euler means $$e^q_n(f)(x)=\sum^n_{k=0}\binom{n}{k}q^{n-k}(1+q)^{-n}S_k(f)(x), \qquad q\geq 0, \qquad n\in\mathbb Z_+,$$ where $S_k(f)$ is the $k$-th partial trigonometric Fourier sum. For $p$-absolutely continuous functions ($f\in C_p$, $1) we consider their approximation by the Euler means in uniform and $C_p$-metric in terms of moduli of continuity $\omega_k(f)_{C_p}$, $k\in\mathbb N$, and the best approximations by trigonometric polynomials $E_n(f)_{C_p}$. One can note the following inequality for different metrics from Theorem 2: $$\|f-e^q_n(f)\|_\infty\leq C_1(1+q)^{-n} \sum_{j=0}^n\binom{n}{j} q^{n-j}E_j(f)_{C_p}, \quad n\in\mathbb N, $$ which is sharp. Also the following generalization of a result due to C. K. Chui and A. S. Holland is proved. If $\omega$ is a modulus of continuity on $[0,\pi]$ such that $\delta\int^\pi_\delta t^{-2}\omega(t)\,dt=O(\omega(\delta))$, $1 and $f\in C_p$ satisfies two properties 1) $\omega_2(f,t)_{C_p}\leq C\omega(t)$; 2) $\int_{2\pi/(n+1)}^\pi t^{-1}\|\varphi_x(t)-\varphi_x(t+2\pi/(n+1) \|_{C_p}\,dt=O(\omega(1/n))$, where $\varphi_x(t)=f(x+t)+f(x-t)-2f(x)$, then $\|e^1_n(f)-f\|_{C_p}\leq C\omega(1/n)$, $n\in\mathbb N$. Some applications to the approximation in Hölder type metrics are given. 
@article{ISU_2015_15_3_a7,
     author = {A. A. Tyuleneva},
     title = {Approximation of functions of bounded $p$-variation by {Euler} means},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {300--309},
     year = {2015},
     volume = {15},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ISU_2015_15_3_a7/}
}
TY  - JOUR
AU  - A. A. Tyuleneva
TI  - Approximation of functions of bounded $p$-variation by Euler means
JO  - Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
PY  - 2015
SP  - 300
EP  - 309
VL  - 15
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/ISU_2015_15_3_a7/
LA  - ru
ID  - ISU_2015_15_3_a7
ER  - 
%0 Journal Article
%A A. A. Tyuleneva
%T Approximation of functions of bounded $p$-variation by Euler means
%J Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
%D 2015
%P 300-309
%V 15
%N 3
%U http://geodesic.mathdoc.fr/item/ISU_2015_15_3_a7/
%G ru
%F ISU_2015_15_3_a7
A. A. Tyuleneva. Approximation of functions of bounded $p$-variation by Euler means. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 3, pp. 300-309. http://geodesic.mathdoc.fr/item/ISU_2015_15_3_a7/

[1] Terekhin A. P., “The approximation of functions of bounded $p$-variation”, Izv. Vyssh. Uchebn. Zaved. Mat., 1965, no. 2, 171–187 (in Russian) | MR

[2] Bari N. K., Stechkin S. B., “Best approximations and differential properties of two conjugate functions”, Tr. Mosk. Mat. Obs., 5, 1956, 483–522 (in Russian) | MR | Zbl

[3] Hardy G. H., Divergent series, Oxford Univ. Press, Oxford, 1949 | MR | Zbl

[4] Timan A. F., Theory of approximation of functions of a real variable, MacMillan, New York, 1963 | MR

[5] Golubov B. I., “On the best approximation of $p$-absolutely continuous functions”, Some Questions of Function Theory and Functional Analysis, 4, Izd. Tbilisi Univ., Tbilisi, 1988, 85–99 (in Russian)

[6] Zygmund A., Trigonometric series, v. 1, Cambridge Univ. Press, Cambridge, 1959 | MR | MR | Zbl

[7] Volosivets S. S., “Convergence of series of Fourier coefficients of $p$-absolutely continuous functions”, Analysis Math., 26:1 (2000), 63–80 | DOI | MR | Zbl

[8] Tyuleneva A. A., “Approximation of bounded $p$-variation periodic functions by generalized Abel–Poisson means and logarithmic means”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 13:4/1 (2013), 25–32 (in Russian) | Zbl

[9] Chui C. K., Holland A. S. B., “On the order of approximation by Euler and Borel means”, J. Approxim. Theory, 39:1 (1983), 24–38 | DOI | MR | Zbl

[10] Rempulska l., Tomczak K., “On Euler and Borel means of Fourier series in Hölder spases”, Proc. of A. Razmadze Math. Institute, 140 (2006), 141–153 | MR | Zbl