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 u˜=(x−c)u with SMOP (P˜n)n≥0, we are interested to study the relation between u˜ and u(1)˜, where u(1) is the linear functional for the associated orthogonal polynomials of the first kind (P(1)n)n≥0, and u(1)˜=(x−c)u(1) is its Christoffel transformation. This problem is also studied for canonical Geronimus transformations.