Hilbert Transforms in Yukawan Potential Theory - Duffin, R. J. (1972)
If H denotes the classical Hilbert transform and Hu(x) = v(x), then the functions u(x) and v(x) are the values on the real axis of a pair of conjugate functions, harmonic in the upper half-plane. This note gives a generalization of the above concepts in which the Laplace equation Δu = 0 is replaced by the Yukawa equation Δu = μ(2)u and in which the Cauchy-Riemann equations have a corresponding generalization. This leads to a generalized Hilbert transform H(μ). The kernel function of this new transform is expressable in terms of the Bessel function K(0). The transform is of convolution type.