Continuous symmetrized Sobolev inner products of order N (I)

Abstract

Given a symmetrized Sobolev inner product of order N, the corresponding sequence of monic orthogonal polynomials { Q n } satisfies that Q 2 n ( x ) = P n ( x 2 ) , Q 2 n + 1 ( x ) = x R 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.