Rational Approximation in the Bergman Spaces

TL;DR

This paper develops a new rational approximation method for Bergman spaces using a pre-orthogonal approach, extending the efficiency of adaptive Fourier decomposition from Hardy spaces to more singular functions.

Contribution

It introduces the Bergman space rational orthogonal (BRO) system and demonstrates its effectiveness as an alternative to AFD in weighted Bergman spaces.

Findings

BRO system effectively approximates functions with boundary singularities.

Theoretical equivalence of the pre-orthogonal method to AFD in Hardy spaces.

Illustrative examples confirm the approximation method's effectiveness.

Abstract

It is known that adaptive Fourier decomposition (AFD) offers efficient rational approxima- tions to functions in the classical Hardy H2 spaces with significant applications. This study aims at rational approximation in Bergman, and more widely, in weighted Bergman spaces, the functions of which have more singularity than those in the Hardy spaces. Due to lack of an effective inner function theory, direct adaptation of the Hardy-space AFD is not performable. We, however, show that a pre-orthogonal method, being equivalent to AFD in the classical cases, is available for all weighted Bergman spaces. The theory in the Bergman spaces has equal force as AFD in the Hardy spaces. The methodology of approximation is via constructing the rational orthogonal systems of the Bergman type spaces, called Bergman space rational orthog- onal (BRO) system, that have the same role as the Takennaka-Malmquist (TM) system in the Hardy spaces. Subsequently, we prove a certain type direct sum decomposition of the Bergman spaces that reveals the orthogonal complement relation between the span of the BRO system and the zero-based invariant spaces. We provide a sequence of examples with different and ex- plicit singularities at the boundary along with a study on the inclusion relations of the weighted Bergman spaces. We finally present illustrative examples for effectiveness of the approximation.