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Direct and inverse theorems of the approximation of functions by algebraic polynomials and splines in the norms of the Sobolev space. (English. Russian original) Zbl 1528.41012

Russ. Math. 66, No. 6, 65-72 (2022); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2022, No. 6, 79-86 (2022).
Summary: In the one-dimensional case, interpolation weighted Besov spaces have been defined, for the functions from which the direct and inverse estimates of the approximation error by algebraic polynomials and splines in the Sobolev norms are valid. In several cases, the estimates have made it possible to obtain the exact values of the considered constants. These results, as well as the inverse inequalities proved in the paper, can be used to justify the \(p\)- and \(hp\)-finite element methods for solving boundary value problems in the case of one-dimensional differential equations of order \(2m\).

MSC:

41A10 Approximation by polynomials
41A15 Spline approximation
41A27 Inverse theorems in approximation theory
41A81 Weighted approximation
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