A method for the practical evaluation of the Hilbert transform on the real line

We investigate a method for the numerical evaluation of the weighted Hilbert transform over the entire real line, ⨍−∞∞exp(−x2)f(x)(x−t)−1 dx, where the integral is taken in the Cauchy principal value sense. The method is based on a combination of two polynomial approximation processes. In this way, we obtain a procedure that is numerically stable and efficient. The algorithm can be implemented in a fast way, and existing well known software packages may be used. From the point of view of theoretical error bounds, the method is competitive. Generalizations to other approximation methods are also possible.