Direct and Inverse Theorems of the Approximation of Functions by Algebraic Polynomials and Splines in the Norms of the Sobolev Space

Abstract 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.

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