  We study the Fourier transform of polynomials in an orthogonal family, taken
with respect to the orthogonality measure. Mastering the asymptotic properties
of these transforms, that we call Fourier--Bessel functions, in the argument,
the order, and in certain combinations of the two is required to solve a number
of problems arising in quantum mechanics. We present known results, new
approaches and open conjectures, hoping to justify our belief that the
importance of these investigations extends beyond the application just
mentioned, and may involve interesting discoveries.
