 
Summary: Continuous symmetrized Sobolev inner
products of order N (I)
M. Isabel Bueno a,1
, Francisco Marcellán a,2
,
Jorge SánchezRuiz a,b,3
aDepartamento de Matemáticas, Universidad Carlos III de Madrid,
Avda. de la Universidad 30, 28911 Leganés, Madrid, Spain
bInstituto Carlos I de Física Teórica y Computacional, Universidad de Granada,
18071 Granada, Spain
Abstract
Given a symmetric Sobolev inner product of order N, the corresponding sequence
of monic orthogonal polynomials {Qn} satisfies that Q2n(x) = Pn(x2), Q2n+1(x) =
xRn(x2) for certain sequences of monic polynomials {Pn} and {Rn}. In this paper, we
deduce the integral representation of the inner products such that {Pn} and {Rn} are
the corresponding sequences of orthogonal polynomials. Moreover, we state a relation
between both inner products which extends the classical result for symmetric linear
functionals.
Key words: Sobolev inner product, orthogonal polynomials, symmetrization
process.
