Series of Orthogonal Polynomials as Boundary Values

Abstract

The relations between expansions in orthogonal polynomials of generalized functions on the real axis and certain holomorphic functions in the upper and lower half planes are studied. The holomorphic functions are given by series of functions of the second kind which satisfy the same recurrence formula as the polynomials. A space of generalized functions associated with the polynomials is first introduced. Each element in this space has an analytic representation given by such series whose jump across the real axis is given by the element. Under certain conditions the singularities of the analytic representation may be related to singularities of the associated power series.