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

Abstract

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.

Convergence and stability of a micro–macro acceleration method: Linear slow–fast stochastic differential equations with additive noise

“…Therefore, Assumption 1 sharpens the superlinear behaviour of tail functions that resulted from assuming Θ 0 = R d , see Remark 1. The inclusion of derivatives in (18) is related to the notion of higher-order regular variation(compare [20] and [19, p. 44]).…”

Complements to the Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions

“…, f AC T ∈ +∞ , f ultimately > 0, its index of asymptotic variation at +∞ is defined as the value of the following 1 , , 1 n f AC T n − ∈ +∞ ≥ is termed "regularly varying at +∞ (in the strong sense) of order n" if each of the functions ( ) 1 ,…”

“…The reader must remember that this is a semiexpository paper like [1] [2] and, as such, some elementary or known facts are explicitly reported or proved to have an exposition self-contained and easily-read.…”

“…§4 contains a comparison between the two main standard approaches to the concept of "type of asymptotic variation": via an asymptotic differential equation or an asymptotic functional equation. The theory developed in [1] [2] is necessarily based on asymptotic differential equations in order to define higher-order types of variation for differentiable functions, whereas the more general Karamata theory is based on asymptotic functional equations. We…”

“…For the reader's convenience we give a list of the special classes of functions characterized in [1] [2] mentioning only the main facts to be used in the present paper.…”