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Approximation of classes of functions defined by a generalized \(k\)-modulus of smoothness. (English) Zbl 0932.41006
The authors’ aim is to characterize the best approximation by algebraic polynomials in the space \(L_{p,\alpha,\beta}, 1 \leq p < \infty\), equipped with the norm \(\|f \|_{p,\alpha,\beta}:=(\int_{-1}^1 (1-x)^{\alpha}(1+x)^{\beta}|f(x)|^pdx)^{1/p}\) in terms of a generalized \(k\)-th order modulus of smoothness, based on an asymmetric translation operator. The first order modulus of this type was defined in a recent paper of M. K. Potapov [On the coincidence of the classes of functions defined by the operator of generalized translation or by the order of the best approximation by algebraic polynomials, Mat. Zametki, to appear (in Russian)], where a corresponding characterization theorem has been proved. In the present paper analogous results are obtained for the generalized modulus of smoothness of order \(k\).
Reviewer: P.Petrov (Sofia)

41A10 Approximation by polynomials
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
41A25 Rate of convergence, degree of approximation