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# Wells' construction of interpolants in C^{1,1}(R^n).

le 7 décembre 2011

14H - Groupe de travail "Applications des Mathématiques"

ENS Rennes Bâtiment Sauvy, Salle 5 (rdc)

Séminaire de Matthew Hirn (Yale University) au groupe de travail "Applications des mathématiques"

In this talk we present John C. Wells' construction of interpolants (from 1973) for the following problem. We are given a set E = {p_1, . . . , p_m} included in R^n, a finite field of first degree polynomials {T_{p_1} , . . . , T_{p_m} } defined on R^n that are indexed by E, and a constant M R. Does there exist a function f in C^{1,1}(R^n) such that 1. J_{p_i}f = T_{p_i}    for all i = 1,...,m, where J_{p_f} is the first order jet of f at p, 2. \Vert \nabla f(x)- \nabla f(y)\Vert leq M\Vert x-y \Vert forall x,y in R^n. Wells' gives a sufficient condition for such an interpolant to exist, and then gives a construction of one such interpolant. Furthermore, the condition given by Wells' is in fact equivalent to the formula proved by Le Gruyer for the minimal constant M, and so when we combine the two results we obtain a construction for the interpolant with minimal Lipschitz constant M for finite fields of affine polynomials. After presenting Wells' construction, we will discuss it from a computational perspective. In particular, is it possible to compute such an interpolant effi- ciently on a computer? Here we would like to minimize the amount of memory needed for storage and the number of computations.

Thématique(s)
Recherche - Valorisation
Contact
Erwan Faou et Yannick Privat

Mise à jour le 2 décembre 2011