Analytic theory of finite asymptotic expansions in the real domain. Part I: two-term expansions of differentiable functions

Abstract

Abstract. It is our aim to establish a general analytic theory of asymptotic expansions of type ( * )f (x) = a1φ1(x) + · · · + anφn(x) + o(φn(x)), x → x0 , where the given ordered n-tuple of real-valued functions (φ1 . . . , φn) forms an asymptotic scale at xo ∈ R. By analytic theory, as opposed to the set of algebraic rules for manipulating finite asymptotic expansions, we mean sufficient and/or necessary conditions of general practical usefulness in order that ( * ) hold true. Our theory is concerned with functions which are differentiable (n − 1) or n times and the presented conditions involve integro-differential operators acting on f, φ1, . . . , φn. We essentially use two approaches; one of them is based on canonical factorizations of nth-order disconjugate differential operators and gives conditions expressed as convergence of certain improper integrals, very useful for applications. The other approach, valid for (n − 1)-time differentiable functions starts from simple geometric considerations (as old as Newton's concept of limit tangent) and gives conditions expressed as the existence of finite limits, as x → x 0 ,of certain Wronskian determinants constructed with f, φ1, . . . , φn. There is a link between the two approaches and it turns out that the integral conditions found via the factorizational approach have striking geometric meanings. Our theory extends to general expansions the theory of polynomial asymptotic expansions thoroughly investigated in a previous paper. In the first part of our work we study the case of two comparison functions φ1, φ2. The theoretical background for the two-term theory is much simpler than that for n ≥ 3 and, in addition, it is unavoidable to separate the treatments as the two-term formulas must be explicitly written lest they become unreadable.The present e-paper coincides with the same-titled article published in "Analysis Mathematica, 37(2011), 245-287," except for minor typographical alterations, for the addtion of a last section ( §9) discussing a nontrivial Tauberian problem, and for a list of corrections of misprints reported after the references, misprints that have been corrected in this e-paper.

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

“…1. Condition "f ultimately of one strict sign" is essential both in the general and in our restricted definition.…”

“…1. It is essential to consider the absolute values in order to not impose a-priori restrictions on the signs of the derivatives.…”

“…In a previously-published series of papers ( [1] Balkema, Geluk and de Haan, ( [7]; Lemma 9, p. 410), have shown the equivalence of (1.4), written in the form such a φ being said to have a "power order of growth α at +∞ ".…”

Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part II-B: Solutions of Differential Inequalities and Asymptotic Admissibility of Standard Derivatives

“…This will be also proved true in Part II-C, §15, for a special class of expansions including the real-power case. The same circumstance occurs for a general two-term expansion ( [3]; Remarks, p. 261) but is not a self-evident fact. In each of these three cases direct proofs could be also provided working on the corresponding integral conditions.…”

“…, , 0 on , , 1 ; and we are supposing 3 n ≥ as the two-term theory has been thoroughly studied in [3]. Operators k L and k M are defined in formulas (3.1) to (3.4) in Part II-A; properties of the k L 's are reported in the first few lemmas in §4 and properties of the k M 's are to be found in Proposition 3.1 with the signs specified by (3.19), due to our present assumption (7.3).…”