Help with Hankel transform filters
I think you misunderstand what this filter is doing. It does not compute approximations of the Bessel functions $J_0(x)$ and $J_1(x)$, but it computes an approximation of the Hankel transform of some input function. The function is expressed as a weighted sum of Bessel functions, and the result of the transform are the weights. The digital filtering approach is an efficient alternative to solving the transform integral by standard methods of numerical integration.