Coconvex Approximation of Periodic Functions

Abstract

Let ${{\widetilde{C}}}$ be the space of continuous $2\pi$-periodic functions f, endowed with the uniform norm $\Vert f\Vert :=\max _{x\in {\mathbb {R}}}|f(x)|$, and denote by $\omega _k(f,t)$, the k-th modulus of smoothness of f. Denote by ${{\widetilde{C}}}^r$, the subspace of r times continuously differentiable functions $f\in {{\widetilde{C}}}$, and let ${\mathbb {T}}_n$, be the set of trigonometric polynomials $T_n$ of degree $\le n$ (that is, of order $\le 2n+1$). Given a set $Y_s:=\{y_i\}_{i=1}^{2s}$, of 2s points, $s\ge 1$, such that $-\pi \le y_1<y_2<\cdots<y_{2s}<\pi$, and a function $f\in {{\widetilde{C}}}^r$, $r\ge 3$, that changes convexity exactly at the points $Y_s$, namely, the points $Y_s$ are all the inflection points of f. We wish to approximate f by trigonometric polynomials which are coconvex with it, that is, satisfy

$$\begin{aligned} f''(x)T_n''(x)\ge 0,\quad x\in {\mathbb {R}}. \end{aligned}$$

We prove, in particular, that if $r\ge 3$, then for every $k,s\ge 1$, there exists a sequence $\{T_n\}_{n=N}^\infty$, $N=N(r,k,Y_s)$, of trigonometric polynomials $T_n\in {\mathbb {T}}_n$, coconvex with f, such that

$$\begin{aligned} \Vert f-T_n\Vert \le \frac{c(r,k,s)}{n^r}\omega _k(f^{(r)},1/n). \end{aligned}$$

It is known that one may not take N independent of $Y_s$.