The discrete Bessel transform algorithm

We present a general discrete Bessel transform based on the Bessel functions of the first kind for any integer or half-integer order ν. This discrete Bessel transform shares a number of similitudes with the discrete Fourier transform in that we have discretized both the coordinate and momentum continuums, and since the discrete transform of order 1/2 exactly specializes to the discrete sine Fourier transform. We demonstrate that our discrete Bessel transform is comparable to the discrete Fourier transform in terms of both the accuracy and the efficiency. Indeed, our discretization procedure provides an optimal sampling grid for Bessel functions of the first kind, and the accuracy of the transform converges exponentially as the number of grid points is increased. We successfully apply the optimally discretized Bessel methodology to the harmonic oscillator in both cylindrical and spherical coordinates.