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Mat. Sb., 1998, Volume 189, Number 11, Pages 75–102 (Mi msb371)  

This article is cited in 1 scientific paper (total in 1 paper)

Interpolation by $D^m$-splines and bases in Sobolev spaces

O. V. Matveev

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: Approximation of functions of several variables by $D^m$-interpolating splines on irregular grids is considered. Sharp in order estimates (of various kinds) of the error of the approximation of functions $f\in W^k_p(\Omega )$ in the seminorms ${\|D^l\cdot \|_{L_q}}$ are obtained in terms of the moduli of smoothness in $L_p$ of the $k$-th derivatives of $f$. As a consequence, for a bounded domain $\Omega$ in $\mathbb R^n$ with minimally smooth boundary and for each $t\in \mathbb N$ a basis in the Sobolev space $W^k_p(\Omega )$ is constructed such that the error of the approximation of $f\in W^k_p(\Omega )$ by the $N$-th partial sum of the expansion of $f$ with respect to this basis has an estimate in terms of its $t$-th modulus of smoothness $\omega _t(D^kf,N^{-1/n})_{L_p(\Omega )}$.

DOI: https://doi.org/10.4213/sm371

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English version:
Sbornik: Mathematics, 1998, 189:11, 1657–1684

Bibliographic databases:

UDC: 517.518
MSC: 41A15, 46E35
Received: 30.01.1997 and 15.04.1998

Citation: O. V. Matveev, “Interpolation by $D^m$-splines and bases in Sobolev spaces”, Mat. Sb., 189:11 (1998), 75–102; Sb. Math., 189:11 (1998), 1657–1684

Citation in format AMSBIB
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\by O.~V.~Matveev
\paper Interpolation by $D^m$-splines and bases in Sobolev spaces
\jour Mat. Sb.
\yr 1998
\vol 189
\issue 11
\pages 75--102
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\crossref{https://doi.org/10.4213/sm371}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1685920}
\zmath{https://zbmath.org/?q=an:0934.41008}
\transl
\jour Sb. Math.
\yr 1998
\vol 189
\issue 11
\pages 1657--1684
\crossref{https://doi.org/10.1070/sm1998v189n11ABEH000371}
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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. V. Matveev, “Bases in Sobolev Spaces on Bounded Domains with Lipschitzian Boundary”, Math. Notes, 72:3 (2002), 373–382  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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