Bounded Extremal Problems in Bergman and Bergman-Vekua spaces

TL;DR

This paper studies bounded extremal approximation problems in Bergman and generalized Bergman-Vekua spaces, focusing on solutions to the Vekua equation in the unit disc, with initial constructive results for p=2.

Contribution

It introduces a new class of extremal problems in Bergman and Vekua spaces and provides preliminary constructive solutions for the case p=2.

Findings

Formulation of extremal approximation problems in Bergman and Vekua spaces.

Initial constructive solutions provided for the case p=2.

Analysis of approximation constraints in generalized analytic function spaces.

Abstract

We analyze Bergman spaces A p f (D) of generalized analytic functions of solutions to the Vekua equation $\partial$w = ($\partial$f /f)w in the unit disc of the complex plane, for Lipschitz-smooth non-vanishing real valued functions f and 1 < p < $\infty$. We consider a family of bounded extremal problems (best constrained approximation) in the Bergman space A p (D) and in its generalized version A p f (D), that consists in approximating a function in subsets of D by the restriction of a function belonging to A p (D) or A p f (D) subject to a norm constraint. Preliminary constructive results are provided for p = 2.