  We study the problem of an appropriate choice of derivatives associated with
discrete Fourier-Bessel expansions. We introduce a new so-called essential
measure Fourier-Bessel setting, where the relevant derivative is simply the
ordinary derivative. Then we investigate Riesz transforms and Sobolev spaces in
this context. Our main results are $L^p$-boundedness of the Riesz transforms
(even in a multi-dimensional situation) and an isomorphism between the Sobolev
and Fourier-Bessel potential spaces. Moreover, throughout the paper we collect
various comments concerning two other closely related Fourier-Bessel situations
that were considered earlier in the literature. We believe that our
observations shed some new light on analysis of Fourier-Bessel expansions.
