The Discrete Hankel Transform

The Hankel transform is an integral transform and is also known as the Fourier-Bessel transform. Until recently, there was no established discrete version of the transform that observed the same sort of relationship to its continuous counterpart as the discrete Fourier transform does to the continuous Fourier transform. Previous definitions of a discrete Hankel transform (DHT) only focused on methods to approximate the integrals of the continuous Hankel integral transform. Recently published work has remedied this and this chapter presents this theory. Specifically, this chapter presents a theory of a DHT that is shown to arise from a discretization scheme based on the theory of Fourier-Bessel expansions. The standard set of shift, modulation, multiplication, and convolution rules are shown. In addition to being a discrete transform in its own right, this DHT can approximate the continuous forward and inverse Hankel transform.