Approximation of a weighted Hilbert transform by using perturbed Laguerre zeros. (English) Zbl 1372.65334

Occorsio, Donatella

Dolomites Res. Notes Approx. 9, Spec. Iss., 45-58 (2016).

Summary: In the present paper is proposed a numerical method to approximate Hilbert transforms of the type \[ H(fw,t)=\int^{+\infty}_0\frac{f(x)}{x-t}w(x)dx,\qquad t>0, \] where \(w(x)=e^{-x}x^\alpha\), \(\alpha>-1\) is a Laguerre weight, by means of a new Lagrange interpolation process essentially based on the midpoints between two consecutive zeros of Laguerre polynomials. Theoretical error estimates are proved in some weighted uniform spaces and some numerical tests which confirm the theoretical estimates are shown.

MSC:

65R10	Numerical methods for integral transforms
44A15	Special integral transforms (Legendre, Hilbert, etc.)