It is well-known that the notion of limit in the sharp topology of sequences of Colombeau generalized numbers $$\widetilde{\mathbb {R}}$$ R ~ does not generalize classical results. E.g. the sequence $$\frac{1}{n}\not \rightarrow 0$$ 1 n ↛ 0 and a sequence $$(x_{n})_{n\in \mathbb {N}}$$ ( x n ) n ∈ N converges if and only if $$x_{n+1}-x_{n}\rightarrow 0$$ x n + 1 - x n → 0 . This has several deep consequences, e.g. in the study of series, analytic generalized functions, or sigma-additivity and classical limit theorems in integration of generalized functions. The lacking of these results is also connected to the fact that $$\widetilde{\mathbb {R}}$$ R ~ is necessarily not a complete ordered set, e.g. the set of all the infinitesimals has neither supremum nor infimum. We present a solution of these problems with the introduction of the notions of hypernatural number, hypersequence, close supremum and infimum. In this way, we can generalize all the classical theorems for the hyperlimit of a hypersequence. The paper explores ideas that can be applied to other non-Archimedean settings.

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 local uniform upper bounds of derivatives. 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.

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Using the existence of infinite numbers k in the non-Archimedean ring of Robinson-Colombeau, we define the hyperfinite Fourier transform (HFT) by considering integration extended to [−k, k] n instead of (−∞, ∞) n . In order to realize this idea, the space of generalized functions we consider is that of generalized smooth functions (GSF), an extension of classical distribution theory sharing many nonlinear properties with ordinary smooth functions, like the closure with respect to composition, a good integration theory, and several classical theorems of calculus. Even if the final transform depends on k, we obtain a new notion that applies to all GSF, in particular to all Schwartz's distributions and to all Colombeau generalized functions, without growth restrictions. We prove that this FT generalizes several classical properties of the ordinary FT, and in this way we also overcome the difficulties of FT in Colombeau's settings. Differences in some formulas, such as in the transform of derivatives, reveal to be meaningful since allow to obtain also non-tempered global unique solutions of differential equations.