Reproducing kernels of weighted poly-Bergman spaces on the upper half-plane (Q1016747)

The weighted poly-Bergman spaces, \(A^2_n(\Pi,d\mu_\lambda)\), in the upper half plane \(\Pi\) consist of all functions \(f\) in \(L(\Pi,(\lambda+1)(2y)^\lambda)\,dx\,dy)\) which satisfy \((\partial/\partial \bar z)^n f(z)=0\), where \(\lambda>-1\) and \(n\) is a positive integer. In this paper, the author derives explicit formulas for the reproducing kernels for these spaces presented as infinite series. He also studies orthogonal decompositions of \(A^2_n(\Pi,d\mu_\lambda)\) into true \(n\)-analytic functions and derives a number of operator identities relating the different orthogonal projections onto these spases. One of the main tools in this work is the Fourier transform on the real line.