Linear methods for approximation of some classes of holomorphic functions from the Bergman space

We construct a linear method {ie910-01} for the approximation (in the unit disk) of classes of holomorphic functions {ie910-02} that are the Hadamard convolutions of the unit balls of the Bergman space A _p with reproducing kernels {ie910-03}. We give conditions for ψ under which the method {ie910-04} approximates the class {ie910-05} in the metrics of the Hardy space H _s and the Bergman space A _s , 1 ≤ s ≤ p, with an error that coincides in order with the value of the best approximation by algebraic polynomials. Translated from in Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 783–795, June, 2008.