Within the framework of mappings between affine spaces, the notion of nth polarization of a function will lead to an intrinsic characterization of polynomial functions. We prove that the characteristic features of derivations, such as linearity, iterability, Leibniz and chain rules are shared – at the finite level – by the polarization operators. We give these results by means of explicit general formulae, which are valid at any order n, and are based on combinatorial identities. The infinitesimal limits of the nth polarizations of a function will yield its nth derivatives (without resorting to the usual recursive definition), and the afore-mentioned properties will be recovered directly in the limit. Polynomial functions will allow us to produce a coordinate free version of Taylor's formula.
Polarizations and differential calculus in affine spaces
BARILE, Margherita;BARONE, Fiorella;
2007-01-01
Abstract
Within the framework of mappings between affine spaces, the notion of nth polarization of a function will lead to an intrinsic characterization of polynomial functions. We prove that the characteristic features of derivations, such as linearity, iterability, Leibniz and chain rules are shared – at the finite level – by the polarization operators. We give these results by means of explicit general formulae, which are valid at any order n, and are based on combinatorial identities. The infinitesimal limits of the nth polarizations of a function will yield its nth derivatives (without resorting to the usual recursive definition), and the afore-mentioned properties will be recovered directly in the limit. Polynomial functions will allow us to produce a coordinate free version of Taylor's formula.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
