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Approximating periodic functions by interpolation sums of Jackson type

Authors:

V. V. Zhuk

Affiliation:

(1) St. Petersburg State University, St. Petersburg, Russia

Abstract:

Let

$Phi _{n} {left( t right)} = frac{1} {{2pi {left( {n + 1} right)}}}{left( {frac{{sin frac{{{left( {n + 1} right)}t}} {2}}} {{sin frac{t} {2}}}} right)}^{2}$

be the Fejér kernel, C be the space of contiuous 2π-periodic functions f with the norm ${left| f right|} = {mathop {max }limits_{x in mathbb{R}} }{left| {f{left( x right)}} right|}$ , let

$J_{n} {left( {f,x} right)} = frac{{2pi }} {{n + 1}}{sumlimits_{k = 0}^n {f{left( {t_{k} } right)}Phi _{n} {left( {x - t_{k} } right)}} },quad wherequad t_{k} = frac{{2pi k}} {{n + 1}},$

be the Jackson polynomials of the function f, and let

$sigma _{n} {left( {f,x} right)} = {intlimits_{ - pi }^pi {f{left( {x + t} right)}Phi _{n} {left( t right)}dt} }$

be the Fejér sums of f. The paper presents upper bounds for certain quantities like

${left| {f{left( x right)} - J_{n} {left( {f,x} right)}} right|},quad {left| {J_{n} {left( {f,x} right)} - sigma _{n} {left( {f,x} right)}} right|},quad {left| {f - J_{n} {left( t right)}} right|},quad {left| {J_{n} {left( t right)} - sigma n{left( f right)}} right|},$

which are exact in order for every function f ∈ C. Special attention is paid to the constants occurring in the inequalities obtained. Bibliography: 14 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 90–114.

Keywords:

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