About Calculation of the Hankel Transform Using Preliminary Wavelet Transform

The purpose of this paper is to present an algorithm for evaluating Hankel transform of the null and the first kind. The result is the exact analytical representation as the series of the Bessel and Struve functions multiplied by the wavelet coefficients of the input function. Numerical evaluation of the test function with known analytical Hankel transform illustrates the proposed algorithm.

The Hankel transform is a very useful instrument in a wide range of physical problems which have an axial symmetry [1] . The influence of the Laplasian on a function in a cylindrical coordinates is equal to the product of the parameter of the transformation squared and the transform of the function: The Hankel transform of the null (n = 0) and the first (n = 1) kind are represented as Besides that integrals like (2) are connected with the problems of geophysics and cosmology, for example [3,4].
But practical calculation of direct and inverse Hankel transform is connected with two problems. The first problem is based on the fact that not every transform in the real physical situation has analytical expression for result of inverse Hankel transform. The second one is determination of functions as a set of their values for numerical calculations. Large bibliography on those issues can be found in [5] The classical trapezoidal rule, Cotes rule and other rules connected with replacement of integrand by sequence of polynoms have high accuracy if integrand is a smooth function. But f (r)J n (pr)r (or F p (p)J n (pr)p) is a quick oscillating function if r (or p) is large. There are two general methods of the effective calculation in this area. The first is the Fast Hankel transform [6]. The specification of that method is transforming the function to the logarithmical space and fast Fourier transform in that space. This method needs a smoothing of the function in log-space. The second method is based on the separation of the integrand into product of slowly varying component and a rapidly oscillating Bessel function [7]. But it needs the smoothness of the slow component for its approximation by low-order polynoms.
The goal of this article is to apply wavelet transform with Haar bases to (2). The both direct and inverse transforms (2) are symmetric. Let us consider only one of them, for example, direct transform. Let's denote f (r)r as g(r). Then Hankel transform is The expansion g(r) ∈ L 2 (R) into wavelet series with the Haar bases is [8] : .
The most sufficient result is that equations (5) and (6)   The detail coefficients are The formulas (5) and (6) allow us to get full analytical solution if integrals above have close form solution. In the opposite case the solution must be numerical but this method provides an effective algorithm for that. It's obvious that d jk decrease very quickly if g(r) is a smooth function. One can practically use d jk > ε, where ε is small. But if g(r) has steps, sharp vertices or discontinues then the detail coefficients concentrate around these points and one can appropriate they are equal to the zero in other areas. Let us consider for example a function with known analytical Hankel transform ∞ 0 e −a 2 r 2 rJ 1 (pr)rdr = p 4a 4 e −p 2 /4a 2 .
The approximation and detail coefficients may be calculated analytically in closed form Thus (6) with the coefficients (8) is the exact representation of the Hankel transform. Let us consider the approximate solution. Suppose the function is known (7) only in the segment [0, h]. Then there is the series instead of (4): If J → ∞ then (9) is exact for this truncated function. But practically we only use several first levels. For example we can see the original function (the replacement r to x = r/h is used) (7) and the transform at the Fig.1. One can see that exact transform (solid line) and the transform at level J = 3 (dotted line) coincide at  Fig.2 (left). It's oblivious that the error is small in comparison with the values of the F 1 (p). The absolute error at the level J = 3 in a wide range of p is plotted in the right side of the Fig.2. One can see that this error has quasi-periodic oscillations because the function is truncated. But they decrease with the growth of p (and J) when oscillations in classical Fast Hankel Transform [4] increase.