Approximation de compacts fonctionnels par des variétés analytiques

In the theory of approximation there are some problems on approximation of compact sets in functional spaces by analytic families. First, we deal with the case of algebraic varieties, the theorem of Vitushkin, in which we give a new proof based on the method of Warren, with precision of constants. Next, we consider the case of analytic varieties which is as well a negative result: we show that an analytic family with N variables cannot approach the compact Λl,s better than order (N logN)− l s as N increases. We finish by giving some applications in Sturm–Liouville inverse theory. © 2006 Elsevier Inc. Tous droits réservés.