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 R + by an ω-parametric analytic family better than order of (ω ln ω) −(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 [5] some positive results about the best reconstruction formula by giving an almost optimal formula of order of ω −m .
Dans la théorie d'approximation figurent en particulier les problèmes d'approximation de compacts dans les espaces fonctionnels par des familles analytiques. On y traite le cas des variétés algébriques qui est le théorème de Vitushkin, auquel on donne une nouvelle démonstration fondée sur la méthode de Warren, avec précision des constantes. Puis on considère le cas des variétés analytiques dans lequel on établit également un résultat négatif d'approximation qui dit qu'une famille paramétrée analytiquement par N variables ne peut pas approcher le compact Λ l,s mieux qu'à l'ordre (N log N) −l/s , lorsque N augmente. On termine en signalant des applications en problème inverse dans la théorie de Sturm-Liouville.
AbstractIn 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 Λ l,s better than order (N log N) − l s as N increases. We finish by giving some applications in Sturm-Liouville inverse theory.
We deal with a problem of the explicit reconstruction of any holomorphic function f on C 2 from its restricions on a union of complex lines. The validity of such a reconstruction essentially depends on the mutual repartition of these lines, condition that can be analytically described. The motivation of this problem comes also from possible applications in mathematical economics and medical imaging.
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