Quasi-definiteness of generalized Uvarov transforms of moment functionals

Abstract

When σ is a quasi-definite moment functional with the monic orthogonal polynomial system {P n (x)}n=0∞, we consider a point masses perturbation τ of σ given by τ:=σ+λΣl=1 mΣk=0 ml((−1)kulk/k!)δ (k)(x − c l), where λ,ulk, and cl are constants with ci≠cj for i≠j. That is, τ is a generalized Uvarov transform of σ satisfying A(x) τ=A(x) σ, where A(x)=∏l=1m(x−cl)ml+1. We find necessary and sufficient conditions for τ to be quasi-definite. We also discuss various properties of monic orthogonal polynomial system {Rn (x)}n=0∞ relative to τ including two examples.