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

Abstract

We construct a linear method of the approximation $ \{Q_{n,\psi} \}_{n \in {\mathbb N}}$ in the unit disk of classes of holomorphic functions $A^{\psi}_p$ that are the Hadamard convolutions of unit balls of the Bergman space $A_p$ with reproducing kernels $\psi(z) = \sum^\infty_{k=0}\psi_k z^k.$ We give conditions on $\psi$ under which the method $ \{Q_{n,\psi} \}_{n \in {\mathbb N}}$ approximate the class $A^{\psi}_p$ in metrics of the Hardy space $H_s$ and Bergman space $A_s,\; 1 \leq s \leq p,$ with error that coincides in order with a value of the best approximation by algebraic polynomials.