Rational Approximation in a Class of Weighted Hardy Spaces

Abstract

This study aims at rational approximation of a class of weighted Hardy spaces, including the classical Bergman space, the weighted Bergman spaces, the Hardy space, the Dirichlet space and the Hardy–Sobolev spaces. We will mainly concentrate in the Bergman cases in the unit disc context. The methodology of the approximation is a pre-orthogonal method, called Pre-Adaptive Fourier Decomposition. The new idea is that a function is not expanded into a basis but an orthonormal system adapted to the given function. In such way by using a unified method we obtain efficient approximations in our sequence of spaces while avoiding discussions on basis and uniqueness sets, etc. The type of function decompositions is related to direct sum decompositions of the underlying spaces into the closure of the span of a sequence of repeating reproducing kernels and the corresponding zero-based invariant subspaces that arises deep studies.