(1) Contain all real and imaginary parts of bounded holomorphic functions. (2) Be describable as \Poisson integrals" over the Bergman-Shilov boundary against a real kernel (the \Poisson" kernel). (3) Be invariant under all bi-holomorphisms of the domain. (4) Be describable as the nullspace HL of a degenerate-elliptic system L of second order di erential operators. (We refer to HL as the space o...

We study the Bergman kernel and projection on the worm domains Dβ = { ζ ∈ C : Re ( ζ1e −i log |ζ2| 2) > 0, ∣∣ log |ζ2| ∣∣ < β − π 2 } and D β = { z ∈ C : ∣Im z1 − log |z2| ∣∣ < π 2 , | log |z2| | < β − π 2 } for β > π. These two domains are biholomorphically equivalent via the mapping D β ∋ (z1, z2) 7→ (e z1 , z2) ∋ Dβ . We calculate the kernels explicitly, up to an error term that can be contr...

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 t...

We characterize, using the Bergman kernel, Carleson measures of Bergman spaces in strongly pseudoconvex bounded domains in C, generalizing to this setting theorems proved by Duren and Weir for the unit ball. We also show that uniformly discrete (with respect to the Kobayashi distance) sequences give examples of Carleson measures, and we compute the speed of escape to the boundary of uniformly d...

For a smoothly bounded strictly pseudoconvex domain, we describe the boundary singularity of weighted Bergman kernels with respect to weights behaving like a power (possibly fractional) of a defining function, and, more generally, of the reproducing kernels of Sobolev spaces of holomorphic functions of any real order. This generalizes the classical result of Fefferman for the unweighted Bergman...