Comonotone Approximation of Periodic Functions

G. A. Dzyubenko ; M. G. Pleshakov

Matematičeskie zametki, Tome 84 (2008) no. 5, pp. 713-723

Résumé

Suppose that a continuous $2\pi$-periodic function $f$ on the real axis $\mathbb R$  changes its monotonicity at different ordered fixed points $y_i\in[-\pi,\pi)$, $i=1,\dots,2s$, $s\in\mathbb N$. In other words, there is a set $Y:=\{y_i\}_{i\in\mathbb Z}$ of points $y_i=y_{i+2s}+2\pi$ on $\mathbb R$ such that $f$ is nondecreasing on $[y_i,y_{i-1}]$ if $i$ is odd and not increasing if $i$ is even. For each $n\ge N(Y)$, we construct a trigonometric polynomial $P_n$ of order $\le n$ changing its monotonicity at the same points $y_i\in Y$ as $f$ and such that $$ \|f-P_n\|\le c(s)\,\omega_2\biggl(f,\frac\pi n\biggr), $$ where $N(Y)$ is a constant depending only on $Y$, $c(s)$ is a constant depending only on $s$, $\omega_2(f,\,\cdot\,)$ is the modulus of continuity of second order of the function $f$, and ${\|\cdot\|}$ is the $\max$-norm.

Détail

Keywords: periodic function, comonotone approximation, trigonometric polynomial, Jackson-type kernel, Whitney's inequality, modulus of continuity.