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Hyper-power series and generalized real analytic functions.

Diksha Tiwari¹, Akbarali Mukhammadiev¹, Paolo Giordano¹

¹Faculty of Mathematics, University of Vienna, Wien, Austria.

Monatshefte Fur Mathematik

|September 30, 2024

Summary

This study introduces generalized real analytic functions, extending Colombeau theory by analyzing hyper-power series and their convergence properties. These new functions offer greater flexibility than classical or Colombeau approaches.

Keywords:

Colombeau generalized numbers Generalized functions Non-Archimedean rings

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Area of Science:

Non-Archimedean analysis
Generalized functions
Hyper-power series

Background:

Builds upon prior work on hyperseries in Colombeau generalized numbers.
Classical analytic functions have limitations in certain mathematical domains.

Purpose of the Study:

To define and investigate one-variable generalized real analytic functions.
To analyze hyper-power series, including their radius of convergence and algebraic properties.

Main Methods:

Analysis of radius of convergence for hyper-power series.
Proving classical results for hyper-power series (algebraic operations, composition, reciprocal).
Defining and studying generalized real analytic functions, including derivation, integration, and identity theorem.

Main Results:

Established classical results for hyper-power series.
Introduced generalized real analytic functions with enhanced flexibility.
Demonstrated that Colombeau real analytic functions are a subset of generalized real analytic functions.

Conclusions:

Generalized real analytic functions are less rigid than classical and Colombeau theories.
These functions can encompass non-analytic smooth functions and distributions like the Dirac delta.
The study expands the scope of analytic function theory within generalized number systems.