Numerical approximation of weakly singular integrals on the half line
The authors construct a quadrature approximation \(K_m\) to operators of the type \[ K(f,t)=\int_0^\infty {f(x)\over |x-t|^\lambda} \sqrt{w_\alpha(x)} dx, \] where \(w_\alpha\) is the Laguerre weight, \(t\geq 0\) and \(0<\lambda<1\), such that \[ \sup_{t\geq 0} |K(f,t)-K_m(f,t)|\leq {\mathcal C} [E_M(f)_u +e^{-Am}\|fu\|] \log m, \] whre \( E_M(f)_u \) is the error of best approximation in the weighted uniform norm, \(M\sim m\) and the positive constants \({\mathcal C}\) and \(A\) are independent of \(m,f\). The method is based on the Laguerre nodes and on ``truncating'' \(f\) for \(x>4\theta m\), \(0<\theta<1\). The numerical stability is shown, and a method for computing the weights and several numerical examples are given.
scientific article; zbMATH DE number 1758448