On the convergence of certain Gauss-type quadrature formulas for unbounded intervals

Abstract:

We consider the convergence of Gauss-type quadrature formulas for the integral $\int _0^\infty f(x)\omega (x)\mathrm {d}x$, where $\omega$ is a weight function on the half line $[0,\infty )$. The $n$-point Gauss-type quadrature formulas are constructed such that they are exact in the set of Laurent polynomials $\Lambda _{-p,q-1}=\{\sum _{k=-p}^{q-1} a_k x^k$}, where $p=p(n)$ is a sequence of integers satisfying $0\le p(n)\le 2n$ and $q=q(n)=2n-p(n)$. It is proved that under certain Carleman-type conditions for the weight and when $p(n)$ or $q(n)$ goes to $\infty$, then convergence holds for all functions $f$ for which $f\omega$ is integrable on $[0,\infty )$. Some numerical experiments compare the convergence of these quadrature formulas with the convergence of the classical Gauss quadrature formulas for the half line.