Szasz Analytic Functions and Noncompact Toric Varieties

We relate the classical approximations SN (f)(x) of O.Szasz to the Bergman kernel of the Bargmann-Fock space H(C, e |z| 2 dm(z)). This relation is the analogue for compact toric varieties of the relation between Bernstein polynomials and Bergman kernels on compact toric Kähler varieties of S. Zelditch. The relation is then used to generalize the Szasz analytic functions to any infinite volume toric Kähler variety. Further, we show that the Szsaz analytic function is the universal scaling limit of the Bernstein polynomial as a point approaches the boundary. Applications to summing lattice points over infinite volume polytopes are given.