Continuous symmetrized Sobolev inner products of order N (I)

Given a symmetrized Sobolev inner product of order N, the corresponding sequence of monic orthogonal polynomials {Q(n)} satisfies that Q(2n) (x) = Pn (x(2)), Q(2n+1) (x) = xR(n) (x(2)) for certain sequences of monic polynomials {P-n} and {R-n}. In this paper, we deduce the integral representation of the inner products such that {P-n} and {R-n} 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. (c) 2004 Elsevier Inc. All rights reserved.