  We give here some negative results in Sturm-Liouville inverse theory, meaning
that we cannot approach any of the potentials with $m+1$ integrable derivatives
on $\mathbb{R}^+$ by an $\omega$-parametric analytic family better than order
of $(\omega\ln\omega)^{-(m+1)}$.
  Next, we prove an estimation of the eigenvalues and characteristic values of
a Sturm-Liouville operator and some properties of the solution of a certain
integral equation. This allows us to deduce from [Henkin-Novikova] some
positive results about the best reconstruction formula by giving an almost
optimal formula of order of $\omega^{-m}$.
