Spectral transformations and associated linear functionals of the first kind

Abstract

Given a quasi-definite linear functional u in the linear space of polynomials with complex coefficients, let us consider the corresponding sequence of monic orthogonal polynomials (SMOP in short) (Pn)n≥0. For a canonical Christoffel transformation ue = (x − c)u with SMOP (Pen)n≥0, we are interested to study the relation between ue and ug(1) , where u (1) is the linear functional for the associated orthogonal polynomials of the first kind (P (1) n )n≥0, and ug(1) = (x − c)u (1) is its Christoffel transformation. This problem is also studied for canonical Geronimus transformations.