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An exact Jackson-Stechkin inequality for \(L^2\)-approximation on the interval with the Jacobi weight and on projective spaces. (English. Russian original) Zbl 0938.41015
Izv. Math. 62, No. 6, 1095-1119 (1998); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 62, No. 6, 27-52 (1998).
The author investigates best approximation on \([0,\pi]\) by cosine-polynomials in the Hilbert space \(L^2_{\alpha,\beta}\). defined by the Jacobi-weight belonging to the indices \(\alpha>\beta\geq -1/2\). An exact Jackson-Stechkin inequality is proved where the minimal deviation is estimated by a generalized modulus of continuity of (fractional) order \(r\geq 1\). By the results similar inequalities for multivariate functions are obtained, for instance on spheres or projective spaces.

41A50 Best approximation, Chebyshev systems
41A10 Approximation by polynomials
42A10 Trigonometric approximation
41A25 Rate of convergence, degree of approximation
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiń≠-type inequalities)
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