Quadrature formulas for rational functions
In this interesting paper, the authors study convergence of quadrature formulas that exactly integrate rational functions with given denominators. Let us suppose that we are given monic polynomials \(\pi_n\), \(n\geq 1\), whose zeros lie at least a distance \(\delta\) away from \([-1,1]\) for \(n\geq 1\). Of course \(\delta >0\) should be independent of \(n\). Let \(\omega: [-1,1]\to [0,\infty)\) be a non-negative function, and consider the following quadrature problem: find weights \(\{\lambda_{j,n}\}^n_{j=1}\) and points \(\{x_{jn}\}^n_{j =1}\) such that \[ \int^1_{-1} Rw=\sum^n_{j=1} \lambda_{j,n} R(x_{j,n}) \] for every rational function \(R\) of the form \(R={P \over|\pi_n|^2}\), where \(P\) is a polynomial of degree \(\leq 2n-1\). The authors relate such quadrature formulae to Gauss quadratures (with varying weights) and also present the relationship to multipoint Padé-type approximation. They give conditions for positivity of the weights \(\lambda_{j,n}\) and show how that positivity leads to convergence of the rules as \(n\to \infty\), for bounded Riemann integrable \(f\). More general convergence aspects are explored.
scientific article; zbMATH DE number 1476910