Dautov, R. Z. 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 Keywords:weighted Sobolev space; Besov interpolation space; direct and inverse approximation theorem; Bernstein inequality; inverse inequality PDFBibTeX XMLCite \textit{R. Z. Dautov}, Russ. Math. 66, No. 6, 65--72 (2022; Zbl 1528.41012); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2022, No. 6, 79--86 (2022) Full Text: DOI References: [1] D. Funaro, Polynomial Approximations of Differential Equations (Springer, Berlin, 1992). doi:10.1007/978-3-540-46783-0 · Zbl 0774.41010 [2] Ch. Schwab, p- and hp- Finite Element Methods: Theory and Applications to Solid and Fluid Mechanics (Oxford Univ. Press, Oxford, 1999). · Zbl 0910.73003 [3] Widlund, O., On best error bounds for approximation by piecewise polynomial functions, Numer. Math., 27, 327-338 (1977) · Zbl 0331.41010 · doi:10.1007/BF01396181 [4] Babuška, I.; Kellogg, R. B.; Pitkäranta, J., Direct and inverse error estimates for finite elements with mesh refinements, Numer. Math., 33, 447-471 (1979) · Zbl 0423.65057 · doi:10.1007/BF01399326 [5] Dorr, M. R., The approximation theory for the p-version of the finite element method, SIAM J. Numer. Anal., 21, 1180-1207 (1984) · Zbl 0572.65074 · doi:10.1137/0721073 [6] Babuška, I.; Guo, B., Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spaces. Part I: Approximability of functions in the weighted Besov spaces, SIAM J. Numer. Anal., 39, 1512-1538 (2002) · Zbl 1008.65078 · doi:10.1137/S0036142901356551 [7] I. Babuška and B. Guo, “Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spaces. Part II: Optimal rate of convergence of the p-version finite element solutions,” Math. Models Methods Appl. Sci. (M3AS) 12 (05), 689-719 (2002). doi:10.1142/S0218202502001854 · Zbl 1026.65103 [8] Guo, B.; Babuška, I., Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spaces. Part III: Inverse approximation theorems, J. Approximation Theory, 173, 122-157 (2013) · Zbl 1382.65399 · doi:10.1016/j.jat.2013.05.001 [9] R. A. DeVore and G. G. Lorentz, Constructive Approximation (Springer, Berlin, 1993). · Zbl 0797.41016 [10] G. Szegö, Orthogonal Polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. 23 (Am. Math. Soc., Providence, RI, 1975). · Zbl 0305.42011 [11] Guo, B.-Y.; Shen, J.; Wang, L.-L., Generalized Jacobi polynomials/functions and their applications, Appl. Numer. Math., 59, 1011-1028 (2009) · Zbl 1171.33006 · doi:10.1016/j.apnum.2008.04.003 [12] Guessab, A.; Milovanović, G. V., Weighted L^2-analogues of Bernstein’s inequality and classical orthogonal polynomials, J. Math. Anal. Appl., 182, 244-249 (1994) · Zbl 0799.41017 · doi:10.1006/jmaa.1994.1078 [13] Dautov, R. Z.; Timerbaev, M. R., Sharp estimates for the polynomial approximation in weighted Sobolev spaces, Differ. Equations, 51, 886-894 (2015) · Zbl 1327.41003 · doi:10.1134/S0012266115070071 [14] Triebel, H., Interpolation Theory, Function Spaces, Differential Operators (1978), Amsterdam: North-Holland, Amsterdam · Zbl 0387.46033 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.