The Hilbert transform (and its inverse) are the integral transform g(y) | = | ℋ[f(x)] = 1/π P V integral_(-∞)^∞ (f(x) d x)/(x - y) f(x) | = | ℋ^(-1)[g(y)] = - 1/π P V integral_(-∞)^∞ (g(y) d y)/(y - x), where the Cauchy principal value is taken in each of the integrals. The Hilbert transform is an improper integral. They will be implemented in a future version of the Wolfram Language as HilbertTransform[f, x, y] and InverseHilbertTransform[g, y, x], respectively.

Abel transform | Fourier transform | improper integral | integral transform | inverse Hilbert transform | Titchmarsh theorem | Wiener-Lee transform