Functions of Exponential Growth in a Half-Plane, Sets of Uniqueness, and the Müntz–Szász Problem for the Bergman Space

We introduce and study some new spaces of holomorphic functions on the right half-plane (Formula presented.) In a previous work, S. Krantz, C. Stoppato, and the first named author formulated the M\ufcntz\u2013Sz\ue1sz problem for the Bergman space, that is, the problem to characterize the sets of complex powers (Formula presented.) with (Formula presented.) that form a complete set in the Bergman space (Formula presented.) where (Formula presented.) In this paper, we construct a space of holomorphic functions on the right half-plane, that we denote by (Formula presented.) whose sets of uniqueness (Formula presented.) correspond exactly to the sets of powers (Formula presented.) that are a complete set in (Formula presented.) We show that (Formula presented.) is a reproducing kernel Hilbert space, and we prove a Paley\u2013Wiener-type theorem and several other structural properties. We determine both a necessary and a sufficient condition on a set (Formula presented.) to be a set of uniqueness for (Formula presented.) thus providing a condition for the solution of the M\ufcntz\u2013Sz\ue1sz problem for the Bergman space. Finally, we prove that the orthogonal projection is unbounded on (Formula presented.) for all (Formula presented.