Some quadrature formulae with nonstandard weights
The paper deals with the problem of computing integrals of the form \[ \int_0^\infty f(x)w(x)\,dx, \quad \int_0^\infty f(x)K(x,y)u(x)\,dx, \] where \(f\) is continuous. Besides, \(u(x)=x^\alpha e^{-x^\beta/2}/(1+x)^\lambda\), and \(w(x)=x^\alpha e^{-x^\beta}\), with \(\alpha>-1\), \(\beta>1/2\), \(\gamma > -1\), \( \lambda\geq 0\). When \(f\) has exponential growth at infinity, the authors propose the use of truncated quadrature rules to evaluate these integrals. The main results are summarized in two theorems. When evaluating the first integral, an upper estimate for the quadrature error is obtained given in terms of the Sobolev norm of \(f\). Such estimate holds true when it is applied to a Gauss quadrature formula conveniently truncated. The other result deals with the stability and the convergence of a quadrature rule, which is a modification of the previous one, when it is applied to the second integral. The construction of the latter formula is more complicated than the former because of the presence of the kernel \(K\). Hence the end of the article, there is an appendix where the authors discuss the calculation of coefficients for the special case \(\lambda=0\), \(\beta=1\). Some numerical tests are shown to illustrate the performance of the proposed method.
scientific article; zbMATH DE number 5798184