Completeness on the worm domain and the M\"untz-Sz\'asz problem for the Bergman space

TL;DR

This paper investigates the completeness of specific orthogonal systems in the Bergman space of the worm domain, linking it to a M"untz-Sz"asz problem, and establishes conditions for completeness.

Contribution

It introduces a M"untz-Sz"asz problem for the Bergman space and connects it to the completeness of orthogonal systems in worm domains, providing new sufficient conditions.

Findings

Certain orthogonal systems are not complete in the worm domain Bergman space.

The union of two specific orthogonal systems is complete.

A sufficient condition for the M"untz-Sz"asz problem in the Bergman space is established.

Abstract

In this paper we are concerned with the problem of completeness in the Bergman space of the worm domain $\mathcal{W}_\mu$ and its truncated version $\mathcal{W}'_\mu$. We determine some orthogonal systems and show that they are not complete, while showing that the union of two particular of such systems is complete. In order to prove our completeness result we introduce the Muentz-Szasz problem for the 1-dimensional Bergman space of the disk $\{\zeta : |\zeta-1|<1\}$ and find a sufficient condition for its solution.