No Jackson-type estimates for piecewise $q$-monotone, $q\ge3$, trigonometric approximation

  • D. Leviatan Raymond and Beverly Sackler School Math. Sci., Tel Aviv Univ., Israel
  • O. V. Motorna Taras Shevchenko Nat. Univ. Kyiv, Ukraine
  • I. A. Shevchuk Taras Shevchenko Nat. Univ. Kyiv, Ukraine
Keywords: iecewise $q$-monotone functions, Co-$q$-monotone trigonometric approximation, Degree of approximation


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 $q \ge 3$.

Author Biography

O. V. Motorna, Taras Shevchenko Nat. Univ. Kyiv, Ukraine




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How to Cite
Leviatan , D., O. V. Motorna, and I. A. Shevchuk. “No Jackson-Type Estimates for Piecewise $q$-Monotone, $q\ge3$, Trigonometric Approximation”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 5, June 2022, pp. 662 -75, doi:10.37863/umzh.v74i5.7081.
Research articles