 
Summary: Darboux transformation and perturbation of
linear functionals
M.I. Bueno 1
, F. Marcellán 2
Departamento de Matemáticas, Universidad Carlos III de Madrid.
Avenida Universidad, 30. 28911 Leganés. Spain
Abstract
Let L be a quasidefinite linear functional defined on the linear space of polynomials
with real coefficients. In the literature, three canonical transformations of this func
tional are studied: xL, L + C(x) and 1
xL + C(x) where (x) denotes the linear
functional ((x))(xk) = k,0, and k,0 is the Kronecker symbol. Let us consider the
sequence of monic polynomials orthogonal with respect to L. This sequence satisfies
a threeterm recurrence relation whose coefficients are the entries of the socalled
monic Jacobi matrix. In this paper we show how to find the monic Jacobi matrix
associated with the three canonical perturbations in terms of the monic Jacobi ma
trix associated with L. The main tools are Darboux transformations. In the case
that the LU factorization of the monic Jacobi matrix associated with xL does not
exist and Darboux transformation does not work, we show how to obtain the monic
Jacobi matrix associated with x2L as a limit case. We also study perturbations of
