The deprojection of axisymmetric galaxies

In this paper we prove that, for a class of three-dimensional axisymmetric density distributions expressible as a finite sum of spherical harmonics, the deprojection problem with known inclination can be solved analytically to yield a unique solution. Assuming that the true density distribution of an axisymmetric galaxy can be represented by a smooth, square integrable function of position, we can represent it, to within arbitrary accuracy, by such a finite sum. Our result is in apparent contradiction with the Fourier slice theorem, and we rederive the result in terms of Fourier space. From this we argue that the above assumptions on the density distribution force strong constraints on its Fourier transform, which preclude information being hidden in the cone of ignorance