Hilbert transforms in yukawan potential theory.
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 Deltau = 0 is replaced by the Yukawa equation Deltau = mu(2)u and in which the Cauchy-Riemann equations have a corresponding generalization. This leads to a generalized Hilbert transform H(mu). The kernel function of this new transform is expressable in terms of the Bessel function K(0). The transform is of convolution type.
Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213.
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Duffin