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 Zap. Nauchn. Sem. POMI, 2008, Volume 357, Pages 90–114 (Mi znsl2121)

Approximation of periodic functions by Jackson type interpolation sums

V. V. Zhuk

Saint-Petersburg State University

Abstract: Let
$$\Phi_n(t)=\frac1{2\pi(n+1)}(\frac{\sin\frac{(n+1)t}2}{\sin\frac t2})^2$$
be Fejer's kernel, $C$ be a space of continuous $2\pi$-periodic functions $f$ with the norm $\|f\|=\max_{x\in\mathbb R}|f(x)|$;
$$J_n(f,x)=\frac{2\pi}{n+1}\sum^n_{k=0}f(t_k)\Phi_n(x-t_k),\quadwhere\quad t_k=\frac{2\pi k}{n+1},$$
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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English version:
Journal of Mathematical Sciences (New York), 2009, 157:4, 592–606

Bibliographic databases:

UDC: 517.5

Citation: V. V. Zhuk, “Approximation of periodic functions by Jackson type interpolation sums”, Analytical theory of numbers and theory of functions. Part 23, Zap. Nauchn. Sem. POMI, 357, POMI, St. Petersburg, 2008, 90–114; J. Math. Sci. (N. Y.), 157:4 (2009), 592–606

Citation in format AMSBIB
\Bibitem{Zhu08} \by V.~V.~Zhuk \paper Approximation of periodic functions by Jackson type interpolation sums \inbook Analytical theory of numbers and theory of functions. Part~23 \serial Zap. Nauchn. Sem. POMI \yr 2008 \vol 357 \pages 90--114 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl2121} \zmath{https://zbmath.org/?q=an:05659055} \transl \jour J. Math. Sci. (N. Y.) \yr 2009 \vol 157 \issue 4 \pages 592--606 \crossref{https://doi.org/10.1007/s10958-009-9346-x}