Asymptotically uniform functions: a single hypothesis which solves two old problems

IF 0.6 3区数学 Q3 MATHEMATICS

Analysis Mathematica Pub Date : 2024-06-05 DOI:10.1007/s10476-024-00024-x

J.-P. Gabriel, J.-P. Berrut

引用次数: 0

Abstract

The asymptotic study of a time-dependent function ƒ as the solution of a differential equation often leads to the question of whether its derivative \(f'\) vanishes at infinity. We show that a necessary and sufficient condition for this is that \(f'\) is what may be called asymptotically uniform. We generalize the result to higher order derivatives. We also show that the same property for ƒ itself is also necessary and sufficient for its one-sided improper integrals to exist. The article provides a broad study of such asymptotically uniform functions.

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渐近均匀函数：一个假设解决两个老问题

对作为微分方程解的随时间变化的函数 ƒ 的渐近研究常常会引出一个问题：它的导数 \(f'\)是否在无穷远处消失。我们证明，一个必要且充分的条件是 \(f'\)是所谓的渐近均匀的。我们将这一结果推广到高阶导数。我们还证明，ƒ 本身的相同性质也是其单边不完全积分存在的必要且充分条件。文章对这类渐近均匀函数进行了广泛的研究。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Analysis Mathematica MATHEMATICS-

CiteScore

1.00

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： Traditionally the emphasis of Analysis Mathematica is classical analysis, including real functions (MSC 2010: 26xx), measure and integration (28xx), functions of a complex variable (30xx), special functions (33xx), sequences, series, summability (40xx), approximations and expansions (41xx). The scope also includes potential theory (31xx), several complex variables and analytic spaces (32xx), harmonic analysis on Euclidean spaces (42xx), abstract harmonic analysis (43xx). The journal willingly considers papers in difference and functional equations (39xx), functional analysis (46xx), operator theory (47xx), analysis on topological groups and metric spaces, matrix analysis, discrete versions of topics in analysis, convex and geometric analysis and the interplay between geometry and analysis.