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Polarizations and differential calculus in affine spaces

Margherita Barile·Fiorella Barone·Wlodzimierz M. Tulczyjew
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Abstract

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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Margherita Barile, Fiorella Barone, Wlodzimierz M. Tulczyjew. 2005-10-18. Polarizations and differential calculus in affine spaces. https://doi.org/10.1080/03081080600643611

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