Orthogonal Polynomials and a Dirichlet Problem Related to the Hilbert Transform

摘要

An operator closely related to the Hilbert transform on the circle is shown to be unitarily equivalent to a shift realized on the basis of Pollaczek polynomials, (a family of orthogonal polynomials with weight supported by all the real line). Dirichlet's problem for the disk is associated with harmonic functions with specified boundary values on the upper half circle and with specified constant (possibly complex) directional derivatives on the real diameter. Poisson's Kernel is found and is used to obtain continuous and p-integrable functions. Boundedness results are studied. Dirichlet's problem as a limiting case of boundary value problems for certain special functions from the Heisenberg groups is considered.