33 pages, no figures.-- MSC1991 code: Primary 42C05.MR#: MR1680116 (2000b:42021)Zbl#: Zbl 0936.42012Starting from a sequence {Pn}n≥0 of monic polynomials orthogonal with respect to a linear functional u, we find a linear functional v such that {Qn}≥0, with either Q2n(x)=Pn(T(x)) or Q2n+1(x)=(x−a)Pn(T(x)) where T is a monic quadratic polynomial and a\in\C, is a sequence of monic orthogonal polynomials with respect to v. In particular, we discuss the case when u and v are both positive definite linear functionals. Thus, we obtain a solution for an inverse problem which is a converse, for quadratic mappings, of one analyzed in [11].This paper was finished with financial support of a Grant from Junta Nacional de Investigação Científica e Tecnológica (JNITC) - BD976 - and Centro de Matemática da Universidade de Coimbra (CMUC) of Portugal. The work of the first author was supported by the Dirección General de Enseñanza Superior (DGES) of Spain - PB96-0120-C03-01.Publicad