Continuous symmetrized Sobolev inner products of order N (I)

14 pages, no figures.-- MSC2000 codes: 42C05MR#: MR2132890 (2005k:42068)Zbl#: Zbl 1076.42017Given a symmetrized Sobolev inner product of order N, the corresponding sequence of monic orthogonal polynomials {Qn} satisfies that Q_2n(x)=Pn(x2), Q_2n+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.The work of the authors has been partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grants BFM 2003-06335-C03-02 (M.I.B., F.M., J.S.R.) and BFM2001-3878-C02-01 (J.S.R.), NATO collaborative grant PST.CLG.979738 (M.I.B., F.M.), and the Junta de Andalucía research group FQM-0207 (J.S.R).Publicad