Resultat negatif en theorie d'approximation de compacts fonctionnels par des varietes analytiques et application a un probleme inverse

Abstract

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.