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

In this paper we are concerned with the problem of completeness in the Bergman space of the worm domain Wµ and its truncated version Wµ0 . 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 ζ : |ζ − 1| < 1 and find a sufficient condition for its solution.