Functions of Exponential Growth in a Half-Plane, Sets of Uniqueness, and the Muntz-Szasz Problem for the Bergman Space

We introduce and study some new spaces of holomorphic functions on the right half-plane R. In a previous work, S. Krantz, C. Stoppato, and the first named author formulated the Muntz-Szasz problem for the Bergman space, that is, the problem to characterize the sets of complex powers {zeta(lambda j-1)} with Re lambda j > 0 that form a complete set in the Bergman space A(2)(Delta), where Delta = {zeta:vertical bar zeta - 1}. In this paper, we construct a space of holomorphic functions on the right half- plane, that we denote by M-omega(2)(R), whose sets of uniqueness {lambda(j)} correspond exactly to the sets of powers {zeta(lambda j-1)} that are a complete set in A(2)(Delta). We show that M-omega(2)(R) is a reproducing kernel Hilbert space, and we prove a Paley-Wiener-type theorem and several other structural properties. We determine both a necessary and a sufficient condition on a set {lambda(j)} to be a set of uniqueness for M-omega(2)(R), thus providing a condition for the solution of the Muntz-Szasz problem for the Bergman space. Finally, we prove that the orthogonal projection is unbounded on L-p(R, d omega) for all p not equal 2.