Tiwari Diksha, Mukhammadiev Akbarali, Giordano Paolo
Faculty of Mathematics, University of Vienna, Wien, Austria.
Mon Hefte Math. 2024;203(2):475-508. doi: 10.1007/s00605-023-01849-8. Epub 2023 Mar 29.
This article is a natural continuation of the paper Tiwari, D., Giordano, P., 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.
本文是发表于本期刊的论文《蒂瓦里，D.，乔达诺，P.》的自然延续。我们通过分析收敛半径的概念并证明诸如超幂级数的代数运算、复合和倒数等经典结果，来研究单变量超幂级数。然后我们定义并研究单变量广义实解析函数，考虑它们的求导、积分、恒等定理的适当表述以及通过导数的局部一致上界进行的刻画。与在科洛姆博实解析函数理论中对级数的经典用法相反，我们可以在一个非无穷小的收敛集里恢复几个经典例子。广义实解析函数的概念相较于经典概念和科洛姆博理论而言，显得不那么严格，例如包括具有平坦点的经典非解析光滑函数以及诸如狄拉克δ函数等几种分布。另一方面，每个科洛姆博实解析函数也是一个广义实解析函数。
