An algorithm for fast Hilbert transform of real functions (Q489778)

The authors propose an algorithm for the discretization of the Hilbert transform on the real line of the function \( f(x)\) \[ H_{R}f(x)=\frac{1}{\pi} p. v.\int^{\infty}_{- \infty}\frac{f(y)dy}{x-y}, \] using the linear interpolation. The complexity of this algorithm is reduced to \( O(N \log N)\) comparatively with other known methods of complexity \( O( N^{2})\), where \( N \) is a number of grid points.