Geodesic
Approximation of periodic functions by Jackson type interpolation sums
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 23, Tome 357 (2008), pp. 90-114 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {V. V. Zhuk},
     title = {Approximation of periodic functions by {Jackson} type interpolation sums},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {90--114},
     year = {2008},
     volume = {357},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_357_a6/}
}
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V. V. Zhuk. Approximation of periodic functions by Jackson type interpolation sums. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 23, Tome 357 (2008), pp. 90-114. http://geodesic.mathdoc.fr/item/ZNSL_2008_357_a6/

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