Quasi-orthogonal polynomials, quadrature, and interpolation
Starting from a sequence of orthogonal polynomials \(\{p_ n(d\alpha,x)\}\) derived from a positive Borel measure \(\alpha\) with finite moments on \(\mathcal R\), the author turns to quasi-orthogonal polynomials \[ q_ n(d\alpha,x)= p_ n(d\alpha,x)+ \rho_ 1 p_{n-1} (d\alpha,x)+\cdots+ \rho_ r p_{n-r} (d\alpha,x), \] where the \(\rho\)'s are real numbers and \(n> r\geq 1\). The existence of a \((2n-1-r,n)\) quadrature formula is equivalent to the existence of the sequence of \(q\)'s subject to the non-vanishing of the Coates-numbers for the Lagrange interpolation building blocks. The author then gives representations of quasi-orthogonal polynomials as characteristic polynomial of symmetric matrices (no longer tri-diagonal of course), thus showing that the location of the zeros can be studied by solving the eigenvalue problem for certain symmetric matrices. New results for the zeros of these quasi-orthogonal polynomials are derived. Moreover, the mean convergence of Lagrange interpolation is studied. In the paper, the case \(r=3\) is studied explicitly and the author states that ``most of the results could have been carried over to \(r> 3\), but the problem is mainly computational and the formulae for the \(\tau_ k\dots\) very soon become too complicated to tackle''.
scientific article; zbMATH DE number 584090