Hyperseries in the non-Archimedean ring of Colombeau generalized numbers

Diksha Tiwari; Paolo Giordano

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arXiv · 2006.16141

Hyperseries in the non-Archimedean ring of Colombeau generalized numbers

Diksha Tiwari·Paolo Giordano

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Abstract

This article is the natural continuation of the paper: Mukhammadiev A.~et al Supremum, infimum and hyperlimits of Colombeau generalized numbers in this journal. Since the ring $\tilde{R}$ of Robinson-Colombeau is non-Archimedean, a classical series $\sum_{n=0}^{+\infty}a_{n}$ of generalized numbers $a_{n}\in\tilde{R}$ is convergent if and only if $a_{n}\to0$ in the sharp topology. Therefore, this property does not permit us to generalize several classical results, mainly in the study of analytic generalized functions (as well as, e.g., in the study of sigma-additivity in integration of generalized functions). Introducing the notion of hyperseries, we solve this problem recovering classical examples of analytic functions as well as several classical results.

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BibTeX RIS

Diksha Tiwari, Paolo Giordano. 2022-01-14. Hyperseries in the non-Archimedean ring of Colombeau generalized numbers. https://arxiv.org/abs/2006.16141

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