On inverse problems for orthogonal polynomials. I (Q1318403)

In this interesting paper, the authors consider the following problem: Let \(\{P_ n\}^ \infty_{n=1}\) and \(\{R_ n\}^ \infty_{n=1}\) be sequences of monic orthogonal polynomials. Suppose that for some fixed polynomial \(\phi\) and non-negative integers \(p\), \(s\) we have \[ \phi(x) R_{n+1}'(x)= \sum^{n+p}_{k= n-s} \lambda_{n,k} P_ k(x),\quad n\geq s. \] Let \(u\) and \(v\) be the linear functionals associated with \(\{P_ n\}^ \infty_{n=1}\) and \(\{R_ n\}^ \infty_{n=1}\), respectively, so that \(u\) and \(v\) are linear functionals on the space of all polynomials satisfying the orthogonality relations \[ u(P_ n P_ m)= k_ n\delta_{mn};\;v(R_ n R_ m)= c_ n \delta_{mn}. \] Here \(k_ n\neq 0\); \(c_ n\neq 0\) \(\forall n\). They show that then there is a polynomial \(h(x)\) such that \[ \phi u= hv. \] The authors give a formula for \(h\). Earlier work of S. Bonan, P. Nevai and the reviewer dealt with the case where \(k_ n> 0\), \(c_ n> 0\), so that we are dealing with orthogonal polynomials with respect to positive measures. The essential new twist in this paper is the non-positivity of the measures/functionals \(u\) and \(v\).