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Zap. Nauchn. Sem. POMI, 2017, Volume 456, Pages 55–76 (Mi znsl6421)  

Sharp estimates of linear approximations by nonperiodic splines in terms of linear combinations of moduli of continuity

O. L. Vinogradov, A. V. Gladkaya

St. Petersburg State University, St. Petersburg, Russia

Abstract: Suppose that $\sigma>0$, $r,\mu\in\mathbb N$, $\mu\geqslant r+1$, $r$ is odd, $p\in[1,+\infty]$, $f\in W^{(r)}_p(\mathbb R)$. We construct linear operators $\mathcal X_{\sigma,r,\mu}$ whose values are splines of degree $\mu$ and of minimal defect with knots $\frac{k\pi}\sigma$ ($k\in\mathbb Z$) such that
\begin{gather*} \|f-\mathcal X_{\sigma,r,\mu}(f)\|_p
\leqslant(\frac\pi\sigma)^r\{\frac{A_{r,0}}2\omega_1(f^{(r)},\frac\pi\sigma)_p+\sum_{\nu=1}^{\mu-r-1}A_{r,\nu}\omega_\nu(f^{(r)},\frac\pi\sigma)_p\}
+(\frac\pi\sigma)^r( \frac{\mathcal K_r}{\pi^r}-\sum_{\nu=0}^{\mu-r-1}2^\nu A_{r,\nu})2^{r-\mu}\omega_{\mu-r}(f^{(r)},\frac\pi\sigma)_p, \end{gather*}
where for ${p=1,+\infty}$ the constants cannot be reduced on the class $W^{(r)}_p(\mathbb R)$. Here $\mathcal K_r=\frac4\pi\sum_{l=0}^\infty\frac{(-1)^{l(r+1)}}{(2l+1)^{r+1}}$ are the Favard constants, the constants $A_{r,\nu}$ are constructed explicitly, $\omega_\nu$ is a modulus of continuity of order $\nu$. As a corollary, we get the sharp Jackson type inequality
$$ \|f-\mathcal X_{\sigma,r,\mu}(f)\|_p\leqslant\frac{\mathcal K_r}{2\sigma^r}\omega_1(f^{(r)},\frac\pi\sigma)_p. $$


Key words and phrases: best approximation, nonperiodic splines, Jackson type inequalities.

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English version:
Journal of Mathematical Sciences (New York), 2018, 234:3, 303–317

UDC: 517.5
Received: 02.05.2017

Citation: O. L. Vinogradov, A. V. Gladkaya, “Sharp estimates of linear approximations by nonperiodic splines in terms of linear combinations of moduli of continuity”, Investigations on linear operators and function theory. Part 45, Zap. Nauchn. Sem. POMI, 456, POMI, St. Petersburg, 2017, 55–76; J. Math. Sci. (N. Y.), 234:3 (2018), 303–317

Citation in format AMSBIB
\Bibitem{VinGla17}
\by O.~L.~Vinogradov, A.~V.~Gladkaya
\paper Sharp estimates of linear approximations by nonperiodic splines in terms of linear combinations of moduli of continuity
\inbook Investigations on linear operators and function theory. Part~45
\serial Zap. Nauchn. Sem. POMI
\yr 2017
\vol 456
\pages 55--76
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6421}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2018
\vol 234
\issue 3
\pages 303--317
\crossref{https://doi.org/10.1007/s10958-018-4006-7}


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