Inverse Function Theorems for Generalized Smooth Functions

TL;DR

This paper develops inverse function theorems for generalized smooth functions, a broad class of functions including distributions, defined on non-Archimedean rings, extending classical analysis to a generalized setting.

Contribution

It introduces inverse function theorems for generalized smooth functions, expanding the theoretical framework beyond classical smooth functions.

Findings

Proved local inverse function theorem for generalized smooth functions

Established global inverse function conditions

Extended classical inverse theorems to non-Archimedean settings

Abstract

Generalized smooth functions are a possible formalization of the original historical approach followed by Cauchy, Poisson, Kirchhoff, Helmholtz, Kelvin, Heaviside, and Dirac to deal with generalized functions. They are set-theoretical functions defined on a natural non-Archimedean ring, and include Colombeau generalized functions (and hence also Schwartz distributions) as a particular case. One of their key property is the closure with respect to composition. We review the theory of generalized smooth functions and prove both the local and some global inverse function theorems.