Geodesic
Interpolation by $D^m$-splines and bases in Sobolev spaces
Sbornik. Mathematics, Tome 189 (1998) no. 11, pp. 1657-1684 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $\omega _t(D^kf,N^{-1/n})_{L_p(\Omega )}$. 
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     author = {O. V. Matveev},
     title = {Interpolation by $D^m$-splines and bases in {Sobolev} spaces},
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     pages = {1657--1684},
     year = {1998},
     volume = {189},
     number = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1998_189_11_a3/}
}
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O. V. Matveev. Interpolation by $D^m$-splines and bases in Sobolev spaces. Sbornik. Mathematics, Tome 189 (1998) no. 11, pp. 1657-1684. http://geodesic.mathdoc.fr/item/SM_1998_189_11_a3/

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