Abstract
This article is the natural continuation of the paper: Mukhammadiev et al. Supremum, infimum and hyperlimits of Colombeau generalized numbers in this journal. Since the ring of Robinson-Colombeau is non-Archimedean and Cauchy complete, a classical series sum _{n=0}^{+infty }a_{n} of generalized numbers is convergent if and only if a_{n}rightarrow 0 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.
Highlights
In this article, the study of supremum, infimum and hyperlimits of Colombeau generalized numbers (CGN) we carried out in [13] is applied to the introduction of a corresponding notion of hyperseries
The point of view of the present work is that in a non-Archimedean ring such as ρR, the notion of hyperseries n∈ρN an, i.e. where we sum over the set of hyperfinite natural numbers ρN, yields results which are more closely related to the classical ones, e.g. in studying analytic functions, sigma additivity and limit theorems for integral calculus, or in possible generalization of the Cauchy-Kowalevski theorem to generalized smooth functions (GSF; see e.g. [7])
The possibility to prove the integral test for hyperseries is constrained by the existence of a notion of generalized function that can be defined on an unbounded interval, e.g. of the form [0, +∞) ⊆ ρR
Summary
The study of supremum, infimum and hyperlimits of Colombeau generalized numbers (CGN) we carried out in [13] is applied to the introduction of a corresponding notion of hyperseries. A series of CGN an converges ⇐⇒ an → 0 (in the sharp topology) Once again, this is a well-known property of every ultrametric space, cf., e.g., [11]. The point of view of the present work is that in a non-Archimedean ring such as ρR, the notion of hyperseries n∈ρN an, i.e. where we sum over the set of hyperfinite natural numbers ρN, yields results which are more closely related to the classical ones, e.g. in studying analytic functions, sigma additivity and limit theorems for integral calculus, or in possible generalization of the Cauchy-Kowalevski theorem to generalized smooth functions The ideas presented in the present article, which needs only [13] as prior knowledge, can surely be useful to explore similar ideas in other non-Archimedean Cauchy complete settings, such as [2,3,11,19]
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