On the convergence of the order-preserving weak greedy algorithm for subspaces generated by the Szegö kernel in the Hardy space

In this article we consider representing properties of subspaces generated by the Szegö kernel. We examine under which conditions on the sequence of points of the unit disk the order-preserving weak greedy algorithm for appropriate subspaces generated by the Szegö kernel converges. Previously, we constructed a representing system based on discretized Szegö kernels. The aim of this paper is to find an effective algorithm to get such representation, and we draw on the work of Silnichenko that introduced the notion of the order-preserving weak greedy algorithm. By selecting a special sequence of discretization points we refine one of Totik's results on the approximation of functions in the Hardy space using Szegö kernels. As the main result we prove the convergence criteria of the order-preserving weak greedy algorithm for subspaces generated by the Szegö kernel in the Hardy space.