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

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.