  In the theory of approximation there are some problems on approximation of
compacts in functional spaces by nonlinear families : first we deal with the
polynomial case, and then we consider the analytic case. We demonstrate a
negative result in which we claim that an analytic familie of functions with
$N$ parameters can not approach the compact $\Lambda_l(I^s)$ closer than of
order $(N\log N)^{\frac{l}{s}}$, when $N$ increases. As applied to an inverse
problem in Sturm-Liouville theory, this assertion provides an answer to a
question about the best possible reconstruction of the negative potential $Q$
with $m+1$ integrable derivatives, from its eigenvalues and characteristic
values of the equation $-y''+\omega^2Qy=\lambda y$, when $\omega$ increases :
we show that it is impossible to get an analytic approximating formula with
precision better than of order $(\omega\log\omega)^{-(m+1)}$. Moreover there is
from Henkin-Novikova formulas which are almost optimal.
