Orthogonal polynomials and a Dirichlet problem related to the Hilbert transform

AbstractAn operator closely related to the Hilbert transform on the circle is shown to be unitarily equivalent to a shift realized on a basis of Pollaczek polynomials, a family of orthogonal polynomials with weight supported by all of the real line. There is an associated Dirichlet problem for the disk, where one wants to find harmonic functions with specified boundary values on the upper half circle and with specified constant (possibly complex) direction of the derivative on the real diameter. The Poisson kernel is found and is used to obtain continuous and Lp function existence and boundedness results. The Dirichlet problem is a limiting case of boundary value problems for certain special functions coming from the Heisenberg groups