Journal of Applied Mathematics
Volume 1 (2001), Issue 2, Pages 69-90
doi:10.1155/S1110757X01000225
Quasi-definiteness of generalized Uvarov transforms of moment functionals
Division of Applied Mathematics, Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea
Received 11 March 2001
Copyright © 2001 D. H. Kim and K. H. Kwon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.