Bounded extremal problems in Bergman and Bergman-Vekua spaces

ABSTRACT

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

Keywords:

• Generalized holomorphic functions
• pseudo-holomorphic/pseudo-analytic functions
• Bergman spaces
• Vekua equation
• Bergman-Vekua spaces
• bounded extremal problems

AMS subject classifications:

• 30G20
• 30H20
• 30E10
• 41A29

Acknowledgements

The authors thank the referees for their constructive comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).