  This paper develops a constructive numerical scheme for Fourier-Bessel
approximations on disks compatible with convolutions supported on disks. We
address accurate finite Fourier-Bessel transforms (FFBT) and inverse finite
Fourier-Bessel transforms (iFFBT) of functions on disks using the discrete
Fourier Transform (DFT) on Cartesian grids. Whereas the DFT and its fast
implementation (FFT) are ubiquitous and are powerful for computing
convolutions, they are not exactly steerable under rotations. In contrast,
Fourier-Bessel expansions are steerable, but lose both this property and the
preservation of band limits under convolution. This work captures the best
features of both as the band limit is allowed to increase. The
convergence/error analysis and asymptotic steerability of FFBT/ iFFBT are
investigated. Conditions are established for the FFBT to converge to the
Fourier-Bessel coefficient and for the iFFBT to uniformly approximate the
Fourier-Bessel partial sums. The matrix form of the finite transforms is
discussed. The implementation of the discrete method to compute numerical
approximation of convolutions of compactly supported functions on disks is
considered as well.
