The order of comonotone approximation of differentiable periodic functions

Author:

Dzyubenko German1,Yushchenko Lyudmyla2

Affiliation:

1. Institute of Mathematics of the NAS of Ukraine, Kyiv, Ukraine

2. University of Toulon, La Garde, France

Abstract

Let $\Dely$ be a set of all $2\pi$-periodic functions $f$ that are continuous on the real axis $R$\ and\ change their monotonicity at various fixed points $y_{i}\in\lbrack-\pi,\pi),\ i=1,...,2s,\ s\in N$ (i.e., there is a set $Y:=\{y_{i}\}_{i\in\mathbb{Z}}$ of points $y_{i}=y_{i+2s}+2\pi$ on $R$ such that $f$ are nondecreasing on $[y_{i},y_{i-1}]$ if $i$ is even, and nonincreasing if $i$ is odd). In the article, a function $f_{Y}=f\in C^{(1)}\cap\Dely$ has been constructed such that \[ \lim_{n\rightarrow\infty}\sup\frac{n\,E_{n}^{(1)}(f)}{\omega_{4}(f^{\prime },\pi/n)}=\infty, \] where $E_{n}^{(1)}(f)$ is the error of the best uniform approximation of the function $f\in\Dely$ by trigonometric polynomials of order $n\in N$, which also belong to the set $\Dely$, and $\omega_{4}(f^{\prime},\cdot)$ is the $4$-th modulus of smoothness of the function $f^{\prime}.$ So, for a certain constant $c$, the inequality $E_{n}^{(1)}(f)\leq\frac{c}{n}\omega _{3}(f^{\prime},\pi/n)$ is the best with respect to the order of the modulus of smoothness.

Publisher

Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine

Reference14 articles.

1. Dzyadyk, V.K. (1977). Introdution to the Theory of Uniform Approximation of Functions by Polynomials. Moscow, Nauka (in Russian).

2. Dzyubenko, G.A., & Pleshakov, M.G. (2008). Comonotone approximation of periodic functions. Mat. Zamet., 83 (2), 199-209. https://doi.org/10.1134/s0001434608010203

3. Dzyubenko, G.A. (2009). Comonotone approximation of twice differentiable periodic functions. Ukr. Mat. Zh., 61(4), 435-451. https://doi.org/10.1007/s11253-009-0235-8

4. Dzyubenko, G.A. (2013). Orders of comonotone approximation of periodic functions. Collected Papers of the Institute of Mathematics of the NAS of Ukraine “Function Theory and Related Issues”, 10(1), 110-125.

5. Pleshakov, M.G. (1997). Comonotone Approximation of Periodic Functions from the Sobolev Classes. Ph.D. thesis. Saratov, Saratov State University (in Russian).

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