Abel transform

A geometrical interpretation of the Abel transform in two dimensions. An observer (I) looks along a line parallel to the x axis a distance y above the origin. What the observer sees is the projection (i.e. the integral) of the circularly symmetric function f(r) along the line of sight. The function f(r) is represented in gray in this figure. The observer is assumed to be located infinitely far from the origin so that the limits of integration are ±∞.

In two dimensions, the Abel transform F(y) can be interpreted as the projection of a circularly symmetric function f(r) along a set of parallel lines of sight at a distance y from the origin. Referring to the figure on the right, the observer (I) will see

${\displaystyle F(y)=\int _{-\infty }^{\infty }f\left({\sqrt {x^{2}+y^{2}}}\right)\,dx,}$

where f(r) is the circularly symmetric function represented by the gray color in the figure. It is assumed that the observer is actually at x = ∞, so that the limits of integration are ±∞, and all lines of sight are parallel to the x axis. Realizing that the radius r is related to x and y as r² = x² + y², it follows that

${\displaystyle dx={\frac {r\,dr}{\sqrt {r^{2}-y^{2}}}}}$

for x > 0. Since f(r) is an even function in x, we may write

${\displaystyle F(y)=2\int _{0}^{\infty }f\left({\sqrt {x^{2}+y^{2}}}\right)\,dx=2\int _{|y|}^{\infty }f(r)\,{\frac {r\,dr}{\sqrt {r^{2}-y^{2}}}},}$

which yields the Abel transform of f(r).

The Abel transform may be extended to higher dimensions. Of particular interest is the extension to three dimensions. If we have an axially symmetric function f(ρ, z), where ρ² = x² + y² is the cylindrical radius, then we may want to know the projection of that function onto a plane parallel to the z axis. Without loss of generality, we can take that plane to be the yz plane, so that

${\displaystyle F(y,z)=\int _{-\infty }^{\infty }f(\rho ,z)\,dx=2\int _{y}^{\infty }{\frac {f(\rho ,z)\rho \,d\rho }{\sqrt {\rho ^{2}-y^{2}}}},}$

which is just the Abel transform of f(ρ, z) in ρ and y.

A particular type of axial symmetry is spherical symmetry. In this case, we have a function f(r), where r² = x² + y² + z². The projection onto, say, the yz plane will then be circularly symmetric and expressible as F(s), where s² = y² + z². Carrying out the integration, we have

${\displaystyle F(s)=\int _{-\infty }^{\infty }f(r)\,dx=2\int _{s}^{\infty }{\frac {f(r)r\,dr}{\sqrt {r^{2}-s^{2}}}},}$

which is again, the Abel transform of f(r) in r and s.

Assuming ${\displaystyle f}$ is continuously differentiable and ${\displaystyle f}$, ${\displaystyle f'}$ drop to zero faster than ${\displaystyle 1/r}$, we can set ${\displaystyle u=f(r)}$ and ${\displaystyle v={\sqrt {r^{2}-y^{2}}}}$. Integration by parts then yields

${\displaystyle F(y)=-2\int _{y}^{\infty }f'(r){\sqrt {r^{2}-y^{2}}}\,dr.}$

Differentiating formally,

${\displaystyle F'(y)=2y\int _{y}^{\infty }{\frac {f'(r)}{\sqrt {r^{2}-y^{2}}}}\,dr.}$

Now substitute this into the inverse Abel transform formula:

${\displaystyle -{\frac {1}{\pi }}\int _{r}^{\infty }{\frac {F'(y)}{\sqrt {y^{2}-r^{2}}}}\,dy=\int _{r}^{\infty }\int _{y}^{\infty }{\frac {-2y}{\pi {\sqrt {(y^{2}-r^{2})(s^{2}-y^{2})}}}}f'(s)\,dsdy.}$

By Fubini's theorem, the last integral equals

${\displaystyle \int _{r}^{\infty }\int _{r}^{s}{\frac {-2y}{\pi {\sqrt {(y^{2}-r^{2})(s^{2}-y^{2})}}}}\,dyf'(s)\,ds=\int _{r}^{\infty }(-1)f'(s)\,ds=f(r).}$

Consider the case where ${\displaystyle F(y)}$ is discontinuous at ${\displaystyle y=y_{\Delta }}$, where it abruptly changes its value by a finite amount ${\displaystyle \Delta F}$. That is, ${\displaystyle y_{\Delta }}$ and ${\displaystyle \Delta F}$ are defined by ${\displaystyle \Delta F\equiv \lim _{\epsilon \rightarrow 0}[F(y_{\Delta }-\epsilon )-F(y_{\Delta }+\epsilon )]}$. Such a situation is encountered in tethered polymers (Polymer brush) exhibiting a vertical phase separation, where ${\displaystyle F(y)}$ stands for the polymer density profile and ${\displaystyle f(r)}$ is related to the spatial distribution of terminal, non-tethered monomers of the polymers.

The Abel transform of a function f(r) is under these circumstances again given by:

${\displaystyle F(y)=2\int _{y}^{\infty }{\frac {f(r)r\,dr}{\sqrt {r^{2}-y^{2}}}}.}$

Assuming f(r) drops to zero more quickly than 1/r, the inverse Abel transform is however given by

${\displaystyle f(r)=\left[{\frac {1}{2}}\delta (r-y_{\Delta }){\sqrt {1-(y_{\Delta }/r)^{2}}}-{\frac {1}{\pi }}{\frac {H(y_{\Delta }-r)}{\sqrt {y_{\Delta }^{2}-r^{2}}}}\right]\Delta F-{\frac {1}{\pi }}\int _{r}^{\infty }{\frac {dF}{dy}}{\frac {dy}{\sqrt {y^{2}-r^{2}}}}.}$

where ${\displaystyle \delta }$ is the Dirac delta function and ${\displaystyle H(x)}$ the Heaviside step function. The extended version of the Abel transform for discontinuous F is proven upon applying the Abel transform to shifted, continuous ${\displaystyle F(y)}$, and it reduces to the classical Abel transform when ${\displaystyle \Delta F=0}$. If ${\displaystyle F(y)}$ has more than a single discontinuity, one has to introduce shifts for any of them to come up with a generalized version of the inverse Abel transform which contains n additional terms, each of them corresponding to one of the n discontinuities.

• GPS radio occultation

• Bracewell, R. (1965). The Fourier Transform and its Applications. New York: McGraw-Hill. ISBN 0-07-007016-4.