  We present in this work alternative analytic formulations for the fan-beam
tomographic backprojection operation and its associated adjoint transform in
standard (equiangular) and linear (equidistant) detector geometries. The
proposed formulations are obtained from a recent backprojection theorem in
parallel tomography. Such formulations are written as a Bessel-Neumann series
in the frequency domain that can be implemented as an $O(N^{2.3729})$ matrix
multiplication. Proofs are provided together with numerical simulations
compared with conventional fan-beam $O(N^{3})$ backprojection representations
showing more robustness when dealing with highly noisy data.
