Approximation of periodic functions by Jackson type interpolation sums

V. V. Zhuk

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 23, Tome 357 (2008), pp. 90-114

Résumé

Let $$ \Phi_n(t)=\frac1{2\pi(n+1)}\Biggl(\frac{\sin\frac{(n+1)t}2}{\sin\frac t2}\Biggr)^2 $$ be Fejer's kernel, $C$ be a space of continuous $2\pi$-periodic functions $f$ with the norm $\|f\|=\max_{x\in\mathbb R}|f(x)|$; $$ J_n(f,x)=\frac{2\pi}{n+1}\sum^n_{k=0}f(t_k)\Phi_n(x-t_k),\quad\text{where}\quad t_k=\frac{2\pi k}{n+1}, $$ be Jackson's polynomials of a function $f$, and let $$ \sigma_n(f,x)=\int^\pi_{-\pi}f(x+t)\Phi_n(t)\,dt $$ be Fejer's sums of $f$. The paper establishes upper estimates for the values of the types $$ |f(x)-J_n(f,x)|,\quad|J_n(f,x)-\sigma_n(f,x)|,\quad\|f-J_n(f)\|,\quad\|J_n(f)-\sigma_n(f)\|, $$ which are exact in the order for every function $f\in C$. Special attention is paid to constants occurring in the inequalities obtained. Bibl. – 14 titles.