Sharp Kolmogorov-Type Inequalities for Moduli of Continuity and Best Approximations by Trigonometric Polynomials and Splines期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Sharp Kolmogorov-Type Inequalities for Moduli of Continuity and Best Approximations by Trigonometric Polynomials and Splines

Authors:

O. L. Vinogradov V. V. Zhuk

Affiliation:

(1) St.Petersburg State University, Russia

Abstract:

In what follows, $C$ is the space of $$2pi $$

-periodic continuous functions; P is a seminorm defined on C, shift-invariant, and majorized by the uniform norm; $$omega _m (f,h)_P $$

is the mth modulus of continuity of a function f with step h and calculated with respect to P; $mathcal{K}_r = frac{4}{pi }sumlimits_{l = 0}^infty {frac{{( - 1)^{l(r + 1)} }}{{(2l + 1)^{r + 1} }}}$ , $B_r (x) = - frac{{r!}}{{2^{r - 1} pi ^r }}sumlimits_{k = 1}^infty {frac{{cos (2kpi x - rpi /2)}}{{k^r }}}$ ( $$r in mathbb{N}$$

$gamma _r = frac{{B_{r(frac{1}{2})} }}{{r!}};(k) = k_1 + cdots + k_m$

$K_{r,m} = { k in mathbb{Z}_ + ^m :0 leqslant k_nu leqslant r + nu - 2 - k_1 - cdots - k_{nu - 1} } ,$

$A_{r,0} = frac{2}{{r!}}int_0^{{1 mathord{left/ {vphantom {1 2}} right. kern-nulldelimiterspace} 2}} {|B_r (t) - B_r (frac{1}{2})| } dt,$

$A_{r,m} = sumlimits_{k in K_{r,m} } {left( {prodlimits_{j = 1}^m {|gamma _{k_j } |} } right)} A_{r + m - (k),0, } sumnolimits_{r,m} { = sumlimits_{nu = 0}^{m - 1} {2^nu A_{r,nu } ,} }$

$M_{r,m} (f,h)_P = left{ {_{sumnolimits_{r,m}^{ - 1} {left( {frac{{A_{r,0} }}{2}omega _1 (f,h)_P + sumlimits_{nu = 1}^{m - 1} {A_{r,nu } omega _nu (f,h)_P } } right), r is odd.} }^{sumnolimits_{r,m}^{ - 1} {sumlimits_{nu = 0}^{m - 1} {A_{r,nu } omega _nu (f,h)_P , r is even} } } } right.$

Theorem 1.Let $r,m in mathbb{N},n,lambda > 0,f in C^{(r + m)}$ . Then

$P(f^{(m)} ) leqslant lambda ^r left{ {sumnolimits_{r,m} { + 2^m } sumlimits_{k in K_{r,m} } {left( {prodlimits_{j = 1}^m {|gamma _{k_j } |} } right)} frac{{mathcal{K}_{r + m - (k)} }}{{lambda ^{r + m - (k)} }}} right}$

$times maxleft{ {left( {frac{{omega _m (f,frac{lambda }{n})_P }}{{mathcal{K}_{r + m} 2^m }}} right)^{frac{r}{{r + m}}} M_{r,m}^{frac{m}{{r + m}}} left( {f^{(r + m)} ,frac{lambda }{n}} right)_P ,frac{{n^m omega _m left( {f,frac{lambda }{n}} right)_P }}{{mathcal{K}_{r + m} 2^m }}} right}$

For some values of $$lambda $$

and seminorms related to best approximations by trigonometric polynomials and splines in the uniform and integral metrics, the inequalities are sharp. Bibliography: 6 titles.

Keywords:

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