Inverse problems in the theory of orthogonal polynomials can be stated like this: given the orthogonality measure or its moments, say something about the recurrence coefficients or their associated matrix (Jacobi, in the case of orthogonality on \(\mathbb R\), CMV for the circle, Hessenberg in general). In particular, how modifications of the measure of orthogonality impact on the corresponding matrix? There is a beautiful connection of this problem with the classical \(LU\) and \(QR\) factorizations. Namely, if the original measure \(\mu\) corresponds to a Jacobi matrix \(J\), and \(\alpha \in \mathbb R\) lies, for simplicity, to the left of the support of \(\mu\), then the Jacobi matrix \(J_\alpha \) of the new measure \((x-\alpha ) \mu \) is obtained as follows: if \(J-\alpha I= LU\) is the \(LU\) factorization of \(J-\alpha I\), then \(J_\alpha =UL + \alpha I\). Analogously, if \(J=QR\) is the \(QR\) factorization of \(J\), then \(\widetilde J = RQ\) corresponds to the new measure \(x^2 \mu\). Observe that in both cases the original measures are multiplied by a polynomial, i.e.these are particular cases of a polynomial perturbation of the measure.
The main goal of this paper is to extend these results to the case of orthogonality defined by a general symmetric bilinear quasi-definite functional. With this purpose, the authors extend the notion of the polynomial perturbation, and find natural analogues of the results above, establishing the algebraic connection of the original and perturbed Hessenberg matrices, as well as between the corresponding sequences of orthogonal polynomials. In the quasi-definite case they have to cope with some technical subtleties, such as the fact that a quasi-definite functional, multiplied by \(x^2\), is no longer necessarily quasi-definite, and others. As a key tool, different factorizations of the original matrix are used.