Approximation of the Hilbert transform on the real semiaxis using Laguerre zeros (Q1602775)

The authors propose two algorithms for the computation of Cauchy principal value integrals on the semiaxis given by \[ H(f w_\alpha,t)=\int^\infty_0 f(x)\;\frac{W_\alpha(x)}{x-t} dx, \] where \(xw_\alpha(x)=x^\alpha e^{-x}\), \(\alpha\geq 0\), \(t>0\). The proposed quadrature rules use zeros of Laguerre polynomials. Next theoretical error estimates are proved. Some approximate values for the integral \(H(f w_\alpha,t)\), \(t\in \mathbb{R}^+\) are obtained by using the algorithms described in the paper.