Functions of exponential growth in a half-plane, sets of uniqueness and the M"untz--Sz'asz problem for the Bergman space

TL;DR

This paper introduces a new space of holomorphic functions on the right half-plane, linking sets of uniqueness to completeness of powers in Bergman space, and provides conditions for solving the M"untz--Sz'asz problem.

Contribution

It constructs a new function space on the half-plane with properties that relate to the M"untz--Sz'asz problem for Bergman space, including a reproducing kernel structure and a Paley--Wiener type theorem.

Findings

Established a space of holomorphic functions with sets of uniqueness matching complete power sets in Bergman space.

Proved the space is a reproducing kernel Hilbert space.

Provided a sufficient condition for sets of powers to be uniqueness sets in this space.

Abstract

We introduce and study some new spaces of holomorphic functions on the right half-plane. In a previous work, S. Krantz, C. Stoppato and the first named author formulated the M"untz--Sz'asz problem for the Bergman space, that is, the problem to characterize the sets of complex powers that form a complete set the unweighted Bergman space of a disc. In this paper, we construct a space of holomorphic functions on the right half-plane, whose sets of uniqueness correspond exactly to the sets of powers that are a complete set in Bergman space. We show that this space is a reproducing kernel Hilbert space and we prove a Paley--Wiener type theorem among several other structural properties. Moreover, we determine a sufficient condition on a set of powers to be a set of uniqueness for this space, thus providing a sufficient condition for the solution of the M"untz--Sz'asz problem for the Bergman space.