The Factorizational Theory of Finite Asymptotic Expansions in the Real Domain: A Survey of the Main Results

Abstract

After studying finite asymptotic expansions in real powers, we have developed a general theory for expansions of type (*) , x → x 0 where the ordered n -tuple forms an asymptotic scale at x 0 , i.e . as x → x 0 , 1 ≤ i ≤ n – 1 , and is practically assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x o . As in previous papers by the author concerning polynomial, real-power and two-term theory, the locution “factorizational theory” refers to the special approach based on various types of factorizations of a differential operator associated to . Moreover, the 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. Some considerations lead to restrict the attention to two sets of operators naturally associated to “canonical factorizations”. 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.