Approximation of functions of bounded $p$-variation by Euler means

A. A. Tyuleneva

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 3, pp. 300-309

Résumé

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.