Polarizations and differential calculus in affine spaces

Details Polarizations and differential calculus in affine spaces Polarizations and differential calculus in affine spaces

2005-10-18

arxiv.org/abs/math/0510368v1

Linear and Multilinear Algebra 55 (2007), 121-146

Mathematics

Classical Analysis and ODEs

Scientific paper

10.1080/03081080600643611

Within the framework of mappings between affine spaces, the notion of $n$-th 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 $n$-th polarizations of a function will yield its $n$-th derivatives (without resorting to the usual recursive definition), and the above mentioned properties will be recovered directly in the limit. Polynomial functions will allow us to produce a coordinate free version of Taylor's formula.

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