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MATHEMATICS
Optimal subspaces for mean square approximation of classes of differentiable functions on a segment
O. L. Vinogradov, A. Yu. Ulitskaya St. Petersburg State University, 7-9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation
Abstract:
In this paper, we specify a set of optimal subspaces for $L_2$ approximation of three classes of functions in the Sobolev spaces $W_2^{(r)}$, defined on a segment and subject to certain boundary conditions. A subspace $X$ of dimension not exceeding n is called optimal for a function class $A$ if the best approximation of $A$ by $X$ equals the Kolmogorov $n$-width of $A$. These boundary conditions correspond to subspaces of periodically extended functions with symmetry properties. All of the approximating subspaces are generated by equidistant shifts of a single function. The conditions of optimality are given in terms of Fourier coefficients of a generating function. In particular, we indicate optimal spline spaces of all degrees $d \geqslant r-1$ with equidistant knots of several different types.
Keywords:
spaces of shifts, splines, n-widths.
Received: 19.02.2020 Revised: 14.03.2020 Accepted: 19.03.2020
Citation:
O. L. Vinogradov, A. Yu. Ulitskaya, “Optimal subspaces for mean square approximation of classes of differentiable functions on a segment”, Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 7:3 (2020), 404–417; Vestn. St. Petersbg. Univ., Math., 7:3 (2020), 270–281
Linking options:
https://www.mathnet.ru/eng/vspua165 https://www.mathnet.ru/eng/vspua/v7/i3/p404
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Abstract page: | 33 | Full-text PDF : | 4 |
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