Formal orthogonal polynomials revisited. Applications (Q1911446)

Let \(c^{(0)}\) be a linear functional on the space of all polynomials defined via its moments \(c^{(0)}(x^i)= c^{(0)}_i\) \((i\geq 0)\). Consider the associated Hankel matrices \(M^{(0)}_k= (c^{(0)}_{i+ j})^{k- 1}_{i, j= 0}\) as well as an associated sequence of formal orthogonal polynomials \(P^{(0)}_k\) with \(\deg P^{(0)}_k= k\) and \(c^{(0)}(x^j \cdot P^{(0)}_k)= 0\) for \(0\leq j\leq k- 1\). Motivated by the Laczos method, the \(G\) and \(\varepsilon\)-algorithm as well as by generalizations by C. Brezinski, M. Redivo-Zaglia and H. Sadok, the author investigates interrelations between the systems \(c^{(0)}\), \(P^{(0)}_k\), \(M^{(0)}_k\) and the modified systems \(c^{(s)}\), \(P^{(s)}_k\), \(M^{(s)}_k\) for \(s\geq 1\), where the functionals \(c^{(s)}\) are constructed recursively by \(c^{(s- 1)}(P)= c^{(s)} (P\cdot (x- \alpha_s))\); here \((\alpha_s)_{s\geq 1}\) is an arbitrary sequence of real numbers. These interrelations are expressed in terms of regularity or singularity of the Hankel matrices and certain formal orthogonality relations.