@MISC{The_interpolation, author = {Alain Lascoux The}, title = {Interpolation}, year = {} }

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Abstract

mial g of degree n such that (Lgr1) a 2 A ) g(a) = f(a) Denote by R(A ;B ) the product (a \Gamma b), and by A n B the set difference. One remarks that for every a 2 A the polynomial R(x;A n a) vanishes in all the points of A other than a. It is clear that by linear combination of these n + 1 polynomials of degree n, one can express any element of V , and thus these polynomials constitute the basis adapted to the functionals L a : f ! L a (f) := f(a), modulo normalisation. In other words, one has Lagrange formula (J. Ecole Polyt., II,p.277 ) : (Lgr2) g(x) = R(x;A n a) R(a;A n a) : The application f ! g is a projector on the space Pol(n) of polynomials of degree n. This projector is no other than the "Remainder modulo R(x;A )" since it is the identity on Pol(n) and since it vanishes on every multiple of R(x;A ) (and thus its values on every polynomial is well determined). Lagrange interpolation allows decomposition of rational fractions : dividing both members of (Lgr2) b