A function $f$ on an interval $[0,b]$ can be expanded as a sum of Bessel functions, using the inner product
$$\int_0^b f(x) g(x) x\mathrm dx$$
under which these functions are orthogonal, for example $$\int_0^b J_{\nu}(a_1 x)J_{\nu}(a_2 x) x\mathrm dx =0$$ if $a_1\neq a_2$ and $J_{\nu}(a_1 b)=J_{\nu}(a_2 b)=0$
This is completely analogous to the Fourier series, of course, but calculating this inner product requires us to evaluate the function $f$ at all points in $[0,b]$. What if we only have samples of $f$?
In the case of the Fourier transform, an inner product can be constructed that uses only values at regularly spaced discrete points, while maintaining exact orthogonality of the basis functions under this inner product. Can the same thing be done for the Fourier-Bessel series?