ABEL - Functions to compute the discrete ABEL transform:

abelalloc	allocate and return a pointer to an Abel transformer
abelfree 	free an Abel transformer
abel		compute the Abel transform

Function prototypes:
void *abelalloc (int n);
void abelfree (void *at);
void abel (void *at, float f[], float g[]);

Input:
ns		number of samples in the data to be transformed
f[]		array of floats, the function being transformed

Output:
at		pointer to Abel transformer returned by abelalloc(int n)
g[]		array of floats, the transformed data returned by 
		abel(*at,f[],g[])

Notes:
The Abel transform is defined by:

	         Infinity
	g(y) = 2 Integral dx f(x)/sqrt(1-(y/x)^2)
		   |y|

Linear interpolation is used to define the continuous function f(x)
corresponding to the samples in f[].  The first sample f[0] corresponds
to f(x=0) and the sampling interval is assumed to be 1.  Therefore, the
input samples correspond to 0 <= x <= n-1.  Samples of f(x) for x > n-1
are assumed to be zero.  These conventions imply that 

	g[0] = f[0] + 2*f[1] + 2*f[2] + ... + 2*f[n-1]

References:
Hansen, E. W., 1985, Fast Hankel transform algorithm:  IEEE Trans. on
Acoustics, Speech and Signal Processing, v. ASSP-33, n. 3, p. 666-671.
(Beware of several errors in the equations in this paper!)

Authors:  Dave Hale and Lydia Deng, Colorado School of Mines, 06/01/90