Hyper-power series and generalized real analytic functions

TL;DR

This paper extends the theory of hyper-power series and generalized real analytic functions, analyzing convergence, algebraic operations, and properties, including classical functions and distributions within a non-Archimedean Colombeau framework.

Contribution

It introduces a new approach to generalized real analytic functions, broadening classical and Colombeau theories by allowing more flexible convergence and including distributions.

Findings

Classical results like algebraic operations and composition hold for hyper-power series.

Generalized real analytic functions include classical smooth functions with flat points and distributions.

The new framework is less rigid than classical and Colombeau theories, capturing a wider class of functions.

Abstract

This article is a natural continuation of the paper Tiwari, D., Giordano, P., Hyperseries in the non-Archimedean ring of Colombeau generalized numbers in this journal. We study one variable hyper-power series by analyzing the notion of radius of convergence and proving classical results such as algebraic operations, composition and reciprocal of hyper-power series. We then define and study one variable generalized real analytic functions, considering their derivation, integration, a suitable formulation of the identity theorem and the characterization by uniform upper bounds of derivatives on functionally compact sets. On the contrary with respect to the classical use of series in the theory of Colombeau real analytic functions, we can recover several classical examples in a non-infinitesimal set of convergence. The notion of generalized real analytic function reveals to be less rigid both with respect to the classical one and to Colombeau theory, e.g. including classical non-analytic smooth functions with flat points and several distributions, such as the Dirac delta. On the other hand, each Colombeau real analytic function is also a generalized real analytic function.