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Zap. Nauchn. Sem. POMI, 2002, Volume 290, Pages 5–26 (Mi znsl1610)  

This article is cited in 3 scientific papers (total in 3 papers)

Sharp Kolmogorov-type inequalities for moduli of continuity and best approximations by trigonometric polynomials and splines

O. L. Vinogradov, V. V. Zhuk

Saint-Petersburg State University

Abstract: In what follows $C$ is the space of $2\pi$-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 $m$th modulus of continuity of a function $f$ with step $h$ and calculated with respect to $P$; $\mathscr K_r=\frac4\pi\sum\limits^{\infty}_{l=0}\frac{(-1)^{l(r+1)}}{(2l+1)^{r+1}}$, $B_r(x)=-\frac{r!}{2^{r-1}\pi^r}\sum\limits^{\infty}_{k-1}\frac{\cos(2k\pi x-r\pi/2)}{k^r}$ $(r\in\mathbb N)$, $B_0(x)=1$, $\gamma_r=\frac{B_r(\frac12)}{r!}$; $(k)=k_1+\cdots+k_m$,
\begin{gather*} K_{r,m}=\{k\in\mathbb Z^m_+:0\le k_{\nu}\le r+\nu-2-k_1-…-k_{\nu-1}\},
A_{r,0}=\frac2{r!}\int^{1/2}_0|B_r(t)-B_r(\frac12)| dt,
A_{r, m}=\sum_{k\in K_{r,m}}(\prod^m_{j=1}|\gamma_{k_j}|)A_{r+m-(k), 0}, \quad \Sigma_{r, m}=\sum^{m-1}_{\nu=0}2^{\nu}A_{r,\nu},
M_{r, m}(f, h)_P=\begin{cases} \Sigma^{-1}_{r,m}\sum\limits^{m-1}_{\nu=0}A_{r,\nu}\omega_{\nu}(f,h)_P,&$r$ is even,
\Sigma^{-1}_{r, m}(\dfrac{A_{r, 0}}2\omega_1(f, h)_P+\sum\limits^{m-1}_{\nu=1}A_{r,\nu}\omega_{\nu}(f,h)_P),&$r$ is odd. \end{cases} \end{gather*}

Theorem 1. \textit{Let $r,m\in\mathbb N$, $n,\lambda>0$, $f\in C^{(r+m)}$. Then}
$$ \begin{gathered} P(f^{(m)})\le\lambda^r\{\Sigma_{r, m}+2^m\sum_{k\in K_{r, m}}(\prod^m_{j=1}|\gamma_{k_j}|)\frac{\mathscr K_{r+m-(k)}}{\lambda^{r+m-(k)}}\}
\times\max\{(\frac{\omega_m(f,\tfrac{\lambda}n)_P}{\mathscr K_{r+m}2^m})^{\frac r{r+m}}M^{\frac m{r+m}}_{r, m},(f^{(r+m)},\frac{\lambda}n), \frac{n^m\omega_m(f,\frac{\lambda}n)_P}{\mathscr K_{r+m}2^m}\}. \end{gathered} $$

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.

Full text: PDF file (262 kB)

English version:
Journal of Mathematical Sciences (New York), 2004, 124:2, 4845–4857

Bibliographic databases:

UDC: 517.5
Received: 22.10.2002

Citation: O. L. Vinogradov, V. V. Zhuk, “Sharp Kolmogorov-type inequalities for moduli of continuity and best approximations by trigonometric polynomials and splines”, Investigations on linear operators and function theory. Part 30, Zap. Nauchn. Sem. POMI, 290, POMI, St. Petersburg, 2002, 5–26; J. Math. Sci. (N. Y.), 124:2 (2004), 4845–4857

Citation in format AMSBIB
\Bibitem{VinZhu02}
\by O.~L.~Vinogradov, V.~V.~Zhuk
\paper Sharp Kolmogorov-type inequalities for moduli of continuity and best approximations by trigonometric polynomials and splines
\inbook Investigations on linear operators and function theory. Part~30
\serial Zap. Nauchn. Sem. POMI
\yr 2002
\vol 290
\pages 5--26
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl1610}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1942534}
\zmath{https://zbmath.org/?q=an:1078.42001}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2004
\vol 124
\issue 2
\pages 4845--4857
\crossref{https://doi.org/10.1023/B:JOTH.0000042445.77567.18}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. O. L. Vinogradov, V. V. Zhuk, “Estimates for functionals with a known finite set of moments in terms of deviations of operators constructed with the use of the Steklov averages and finite differences”, J. Math. Sci. (N. Y.), 184:6 (2012), 679–698  mathnet  crossref
    2. O. L. Vinogradov, V. V. Zhuk, “Estimates for functional with a known finite set of moments in terms of moduli of continuity and behaviour of constants in the Jackson-type inequalities”, St. Petersburg Math. J., 24:5 (2013), 691–721  mathnet  crossref  mathscinet  zmath  isi  elib
    3. O. L. Vinogradov, V. V. Zhuk, “Estimates of functionals by the second moduli of continuity of even derivatives”, J. Math. Sci. (N. Y.), 202:4 (2014), 526–540  mathnet  crossref
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