Convergence Accelerator Approach to the Numerical Evaluation of Hilbert Transforms Based on Expansions in Hermite Functions

The evaluation of the Hilbert transform of a function is a very important problem that appears widely in science and engineering, specially in signal analysis. In this work, the numerical evaluation of a class of Hilbert transforms using an expansion in terms of Hermite functions, eigenfunctions of the Fourier transform, is performed. Judicious selection of convergence accelerators allows for the efficient evaluation of the resulting series. The approach is particularly accurate for functions having a Gaussian-like asymptotic behavior. For more slowly decreasing functions, the accuracy of the evaluation decreases.