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This article is cited in 2 scientific papers (total in 2 papers)
Mathematics
Approximation of functions of bounded $p$-variation by Euler means
A. A. Tyuleneva Saratov State University, 83, Astrakhanskaya st., 410012, Saratov, Russia
Abstract:
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
($f\in C_p$, $1<p<\infty$) 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 $\delta\int^\pi_\delta
t^{-2}\omega(t)\,dt=O(\omega(\delta))$, $1<p<\infty$ 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.
Key words:
functions of bounded $p$-variation, $p$-absolutely continuous functions, Euler means, best approximation, modulus of continuity.
Citation:
A. A. Tyuleneva, “Approximation of functions of bounded $p$-variation by Euler means”, Izv. Saratov Univ. Math. Mech. Inform., 15:3 (2015), 300–309
Linking options:
https://www.mathnet.ru/eng/isu596 https://www.mathnet.ru/eng/isu/v15/i3/p300
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