Stechkin-type estimate for nearly copositive approximation of periodic functions

  • G. A. Dzyubenko Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv

Abstract

Under the conditions that a continuous $2\pi$-periodic function $f$ on the real axis changes its sign at $2s$ points $y_i\colon {-\pi}\le y_{2s}<y_{2s-1}<\ldots <y_1<\pi,$ $s\in\Bbb N,$ the other points $y_i,$ $i\in\Bbb Z,$ are defined by periodicity, and natural $n>N(k,y_i)$, where $N(k,y_i)$ is a constant that depends only on $k\in \Bbb N$ and $\min _{i=1,\ldots ,2s}\{y_i-y_{i+1}\}$, we find a trigonometric polynomial $P_n$ of order $\le n$ such that the signs of $P_n$ and $f$ are the same everywhere with the possible exception for small neighborhoods of the points $y_i\colon  (y_i-\pi/n,y_i+\pi/n),$ $ P_n(y_i)=0,$ $i\in\Bbb Z,$ and $\|f-P_n\|\le c(k,s)\,\omega_k(f,\pi/n),$ where $c(k,s)$ is a constant that depends only on $k$ and $s$; $\omega_k(f,\cdot)$ is the $k$th modulus of smoothness of $f,$ and $\|\cdot\|$ is the max-norm.

References

Dzyadyk, V. K. Введение в теорию равномерного приближения функций полиномами. (Russian) [Introduction to the theory of uniform approximation of functions by polynomials] Nauka, Moscow, 1977. 511 pp.

Lorentz, G. G.; Zeller, K. L. Degree of approximation by monotone polynomials. I. J. Approximation Theory 1 (1968), 501–504. https://doi.org/10.1016/0021-9045(68)90039-7

Dzyubenko, G. A.; Gilewicz, J. Copositive approximation of periodic functions. Acta Math. Hungar. 120 (2008), no. 4, 301–314. https://doi.org/10.1007/s10474-008-6204-0

Pleshakov, M. G.; Popov P. A. Знакосохраняющее приближение периодических функций. (Russian) [Sign-Preserving Approximation of Periodic Functions]. Укр. мат. журн. 55 (2003), no. 8, 1087–1098 [Ukr. Math. J. 55 (2003), no. 8, 1314–1328]. https://doi.org/10.1023/B:UKMA.0000010761.91730.16

Popov, P. A. Один контрприклад у знакозберiгаючому наближеннi перiодичних функцiй. (Ukrainian) [Odyn kontrpryklad u znakozberigajuchomu nablyzhenni periodychnyh funkcij]. Проблеми теорiї наближення функцiй: Зб. праць Iн-ту математики НАН України [Problemy teorii' nablyzhennja funkcij: Zb. prac' In-tu matematyky NAN Ukrai'ny], 2 (2005), no. 2, 176–185.

Dzyubenko, G. A. Поточечная оценка комонотонного приближения. (Russian) [Pointwise estimation of comonotone approximation]. Укр. мат. журн. 46 (1994), no. 11, 1467–1472. [Ukr. Math. J. 46 (1994), no. 11, 1620–1626]. https://doi.org/10.1016/s0021-9045(02)00045-x

Wu, Xiang; Zhou, Song Ping. A counterexample in comonotone approximation in $L^p$ space. Colloq. Math. 64 (1993), no. 2, 265–274. https://doi.org/10.4064/cm-64-2-265-274

Leviatan, D.; Shevchuk, I. A. Nearly comonotone approximation. J. Approx. Theory 95 (1998), no. 1, 53–81. https://doi.org/10.1006/jath.1998.3194

DeVore, R. A.; Leviatan, D.; Shevchuk, I. A. Approximation of monotone functions: a counter example. Curves and surfaces with applications in CAGD (Chamonix–Mont-Blanc, 1996), 95–102, Vanderbilt Univ. Press, Nashville, TN, 1997.

Leviatan, D.; Shevchuk, I. A. Coconvex polynomial approximation. J. Approx. Theory 121 (2003), no. 1, 100–118. https://doi.org/10.1016/s0021-9045(02)00045-x

Dzyubenko, G. A. Nearly comonotone approximation of periodic functions. Anal. Theory Appl. 33 (2017), no. 1, 74–92. https://doi.org/10.4208/ata.2017.v33.n1.7

Dzyubenko, G. A. Майже коопукле наближення неперервних перiодичних функцiй. (Ukrainian) [Almost Coconvex Approximation of Continuous Periodic Functions]. Укр. мат. журн. 71 (2019), no. 3, 353–367. [Ukr. Math. J. 71 (2019), no. 3, 402–418]. https://doi.org/10.1007/s11253-019-01654-3

Dzyubenko, G. A. Поточкова оцiнка майже копозитивного наближення неперервних функцiй алгебраїчними многочленами. (Ukrainian) [Pointwise Estimation of the Almost Copositive Approximation of Continuous Functions by Algebraic Polynomials]. Укр. мат. журн. 69 (2017), no. 5, 641–649. [Ukr. Math. J. 69 (2017), no. 5, 746–756]. https://doi.org/10.1007/s11253-017-1392-9

Whitney, Hassler. On functions with bounded $n$th differences. J. Math. Pures Appl. (9) 36 (1957), 67–95.

Gilewicz, J.; Kryakin, Yu. V.; Shevchuk, I. A. Boundedness by 3 of the Whitney interpolation constant. J. Approx. Theory 119 (2002), no. 2, 271–290. https://doi.org/10.1006/jath.2002.3732

Pleshakov, M. G.; Popov, P. A. Второе неравенство Джексона в знакосохраняющем приближении периодических функций. (Russian) [Second Jackson Inequality in a Sign-Preserving Approximation of Periodic Functions]. Укр. мат. журн. 56 (2004), no. 1, 123–128 [Ukr. Math. J. 56 (2004), no. 1, 153–160]. https://doi.org/10.1023/B:UKMA.0000031710.44467.5e

Dzyubenko, G. A. Комонотонне наближення двiчi диференцiйовних перiодичних функцiй. (Ukrainian) [Comonotone approximation of twice differentiable periodic functions]. Укр. мат. журн. 61 (2009), no. 4, 1435–1451. [Ukr. Math. J. 61 (2009), no. 4, 519]. https://doi.org/10.1007/s11253-009-0235-8

Dzyubenko, G. A. Порядки комонотонного наближення перiодичних функцiй. (Ukrainian) [Porjadky komonotonnogo nablyzhennja periodychnyh funkcij] Теорiя функцiй та сумiжнi питання: Зб. праць Iн-ту математики НАН України (Ukrainian) [Teorija funkcij ta sumizhni pytannja: Zb. prac' In-tu matematyky NAN Ukrai'ny] 10 (2013), no. 1, 110–125.

Shevchuk, I. A. Приближение многочленами и следы непрерывных на отрезке функций. (Russian) [Priblizhenie mnogochlenami i sledy nepreryvnyh na otrezke funkcij]. Nauk. dumka, Kiev (1992).

Stechkin, S. B. О порядке наилучших приближений непрерывных функций. (Russian) [O porjadke nailuchshih priblizhenij nepreryvnyh funkcij]. Izv. AN SSSR, ser. mat. 15 (1951), no. 3, 219–242.

Pleshakov, M. G. Comonotone Jackson's inequality. J. Approx. Theory 99 (1999), no. 2, 409–421. https://doi.org/10.1006/jath.1999.3327

Dzjubenko, G. A.; Pleshakov, M. G. Комонотонное приближение периодических функций. (Russian) [Komonotonnoe priblizhenie periodicheskih funkcij]. Mat. zametki 83 (2008), вып. 2, 199–209. https://doi.org/10.4213/mzm4416

Dzyubenko, G. A.; Gilewicz, J.; Shevchuk, I. A. Piecewise monotone pointwise approximation. Constr. Approx. 14 (1998), no. 3, 311–348. https://doi.org/10.1007/s003659900077

Published
29.04.2020
How to Cite
DzyubenkoG. A. “Stechkin-Type Estimate for Nearly Copositive Approximation of Periodic Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 5, Apr. 2020, pp. 628–634, doi:10.37863/umzh.v72i5.1127.
Section
Research articles