No Jackson-type estimates for piecewise $q$-monotone, $q\ge3$, trigonometric approximation
DOI:
https://doi.org/10.37863/umzh.v74i5.7081Keywords:
iecewise $q$-monotone functions, Co-$q$-monotone trigonometric approximation, Degree of approximationAbstract
UDC 517.5
We say that a function $f \in C[a,b]$ is $q$-monotone, $q \ge 3$, if $f\in C^{q-2}(a,b)$ and $f^{(q-2)}$ is convex in $(a,b)$. Let $f$ be continuous and $2\pi$-periodic, and change its $q$-monotonicity finitely many times in $[-\pi,\pi]$. We are interested in estimating the degree of approximation of $f$ by trigonometric polynomials which are co-$q$-monotone with it, namely, trigonometric polynomials that change their $q$-monotonicity exactly at the points where $f$ does. Such Jackson type estimates are valid for piecewise monotone ($q = 1$) and piecewise convex ($q = 2$) approximations. However, we prove, that no such estimates are valid, in general, for co-$q$-monotone approximation, when $q \ge 3$.
References
V. K. Dzyadyk, I. A. Shevchuk, Theory of uniform approximation of functions by polynomials, Walter de Gruyer, Berlin, New York (2008), https://doi.org/10.1515/9783110208245 DOI: https://doi.org/10.1515/9783110208245
G. A. Dzyubenko, Comonotone approximation of twice differentiable periodic functions, Ukr. Math. J., 61, № 4, 519 – 540 (2009), https://doi.org/10.1007/s11253-009-0235-8 DOI: https://doi.org/10.1007/s11253-009-0235-8
D. Leviatan, I. A. Shevchuk, Jackson type estimates for piecewise $q$-monotone approximation, $q geq 3$, are not valid, Pure and Appl. Funct. Anal., 1, 85 – 96 (2016).
G. G. Lorentz, K. L. Zeller, Degree of approximation by monotone polynomials I, J. Approx. Theory, 1, 501 – 504 (1968), https://doi.org/10.1016/0021-9045(68)90039-7 DOI: https://doi.org/10.1016/0021-9045(68)90039-7
M. G. Pleshakov, Comonotone Jacksons inequality, J. Approx. Theory, 99, 409 – 421 (1999), https://doi.org/10.1006/jath.1999.3327 DOI: https://doi.org/10.1006/jath.1999.3327
P. A. Popov, An analog of the Jackson inequality for coconvex approximation of periodic functions, Ukr. Math. J., 53, № 7, 1093 – 1105 (2001), https://doi.org/10.1023/A:1013325131321 DOI: https://doi.org/10.1023/A:1013325131321
A. A. Privalov, Theory of interpolation of functions, Book 1, Saratov Univ. Publ. House, Saratov (1990) (in Russian).
R. S. Varga, A. J. Carpenter, On the Bernstein conjecture in approximation theory, Constr. Approx., 1, 333 – 348 (1985), https://doi.org/10.1007/BF01890040 DOI: https://doi.org/10.1007/BF01890040
V. D. Zalizko, Coconvex approximation of periodic functions, Ukr. Math. J., 59, № 1, 28 – 44 (2007), https://doi.org/10.1007/s11253-007-0003-6 DOI: https://doi.org/10.1007/s11253-007-0003-6
