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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.

MSC:
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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