Using a general Parseval relation and the Wiener–Ganelius method, we give sharp Tauberian remainder results for the Hankel transform $F_\nu (x) = \int_0^\infty {\sqrt {xu} } J_\nu (xu)f(u)du$, $\nu \geqq {{ - 1} / 2}$. The remainder of $f(u)$ covers the whole range between $o(1)$ and $O(u^{ - 1} )$ which is a minorant for this transform. Applications to Fourier series and probability theory are possible.