The Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions. Part I: Higher-Order Regular, Smooth and Rapid Variation  

作　　者：Antonio Granata Antonio Granata(Department of Mathematics and Computer Science, University of Calabria, Rende (Cosenza), Italy)

机构地区：[1]Department of Mathematics and Computer Science, University of Calabria, Rende (Cosenza), Italy

出　　处：《Advances in Pure Mathematics》2016年第12期776-816,共42页理论数学进展（英文）

摘　　要：Motivated by a general theory of finite asymptotic expansions in the real domain for functions f of one real variable, a theory developed in a previous series of papers, we present a detailed survey on the classes of higher-order asymptotically-varying functions where “asymptotically” stands for one of the adverbs “regularly, smoothly, rapidly, exponentially”. For order 1 the theory of regularly-varying functions (with a minimum of regularity such as measurability) is well established and well developed whereas for higher orders involving differentiable functions we encounter different approaches in the literature not linked together, and the cases of rapid or exponential variation, even of order 1, are not systrematically treated. In this semi-expository paper we systematize much scattered matter concerning the pertinent theory of such classes of functions hopefully being of help to those who need these results for various applications. The present Part I contains the higher-order theory for regular, smooth and rapid variation.Motivated by a general theory of finite asymptotic expansions in the real domain for functions f of one real variable, a theory developed in a previous series of papers, we present a detailed survey on the classes of higher-order asymptotically-varying functions where “asymptotically” stands for one of the adverbs “regularly, smoothly, rapidly, exponentially”. For order 1 the theory of regularly-varying functions (with a minimum of regularity such as measurability) is well established and well developed whereas for higher orders involving differentiable functions we encounter different approaches in the literature not linked together, and the cases of rapid or exponential variation, even of order 1, are not systrematically treated. In this semi-expository paper we systematize much scattered matter concerning the pertinent theory of such classes of functions hopefully being of help to those who need these results for various applications. The present Part I contains the higher-order theory for regular, smooth and rapid variation.

关 键 词：Higher-Order Regularly-Varying Functions Higher-Order Rapidly-Varying Functions Smoothly-Varying Functions Exponentially-Varying Functions Asymptotic Functional Equations 

分 类 号：O17[理学—数学]