Analytic theory of finite asymptotic expansions in the real domain. Part II: the factorizational theory for Chebyshev asymptotic scales

TL;DR

This paper develops a comprehensive theory for finite asymptotic expansions in the real domain using factorizations of differential operators, extending classical concepts like Taylor's formula to Chebyshev asymptotic scales.

Contribution

It introduces a novel factorizational approach to analyze asymptotic expansions within Chebyshev systems, linking differential operator factorizations to formal differentiation properties.

Findings

Establishes a connection between canonical factorizations and formal differentiation of asymptotic expansions.

Identifies classes of generalized convex functions associated with these expansions.

Provides a theoretical framework paralleling classical Taylor expansion results in a more general setting.

Abstract

This paper contains a general theory for asymptotic expansions of type (*) f(x)=a_1 phi_1(x)+...+a_n phi_n(x)+o(phi_n(x)), x tends to x_0, n>=3, where the asymptotic scale phi_1(x)>>phi_2(x)>>...>>phi_n(x), x tends to x_0, is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x_0. "Factorizational theory" refers to proofs being based on various types of factorizations of a differential operator associated to (phi_1,...,phi_n), hence we preliminarly collect various results concerning the concept of Chebyshev asymptotic scale, associated disconjugate operators and canonical factorizations. Another guiding thread of our theory is the property of formal differentiation and we aim at characterizing some n-tuples of asymptotic expansions formed by (*) and (n-1) expansions obtained by formal applications of suitable linear differential operators of orders 1,2,...,n-1. Our second preliminary step will be that of discovering that the class of the operators naturally associated to "canonical" factorizations seems to be the most meaningful to be used in a context of formal differentiation. This gives rise to conjectures whose proofs build an analytic theory of finite asymptotic expansions in the real domain which, though not elementary, parallels the familiar results about Taylor's formula. One of the results states that to each scale of the type under consideration it remains associated an important class of functions (namely that of generalized convex functions) enjoying the property that the expansion (*), if valid, is automatically formally differentiable (n-1) times in two special senses.