Mathematical Research Letters

Volume 26 (2019)

Number 1

Completeness on the worm domain and the Müntz–Szász problem for the Bergman space

Pages: 231 – 251



Steven G. Krantz (Department of Mathematics, Washington University, St. Louis, Missouri, U.S.A.)

Marco M. Peloso (Dipartimento di Matematica, Università degli Studi di Milano, Italy)

Caterina Stoppato (Istituto Nazionale di Alta Matematica, Università di Firenze, Firenze, Italy)


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 $\mathcal{W}^\prime_\mu$. We determine some orthogonal systems and show that they are not complete, while showing that the union of two particular such systems is complete.

In order to prove our completeness result we introduce the Müntz–Szász problem for the $1$-dimensional Bergman space of the disk $\lbrace \zeta : \lvert \zeta-1 \rvert \lt 1 \rbrace$ and find a sufficient condition for its solution.

The second author supported in part by the 2010-11 PRIN grant “Real and Complex Manifolds: Geometry, Topology and Harmonic Analysis” of the Italian Ministry of Education (MIUR).

The third author supported by the same PRIN grant; by the FIRB grant “Differential Geometry and Geometric Function Theory” of the MIUR; and by the research group GNSAGA of the Istituto Nazionale di Alta Matematica.

Received 5 October 2015

Accepted 31 October 2018

Published 7 June 2019