Polynomial asymptotic expansions in the real domain: the geometric, the factorizational, and the stabilization approaches

Abstract

A b s t r a c t . The problem of the existence of an asymptotic expansion of type f (x) = anx n + an−1x n−1 + · · · + aix i + o(x i ), x → +∞, is thoroughly studied, comparing and completing the known results obtained through the three different approaches mentioned in the title. A unifying thread is provided by the canonical factorizations of the differential operator D n . Particularly meaningful are several characterizations of the polynomial asymptotic expansions of an nth order convex function.

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

“…This has been proved for polynomial expansions [2] and for real-power expansions [5] in an indirect way by expressing the two sets of differentiated expansions as suitable sets of expansions involving the standard operators d d k k…”

“…The reader must not think that we are now filling a few pages with trivialities about Taylor's formula; as a matter of fact if we apply our theory to the operator n D φ in the case 0 x = +∞ we obtain the results about asymptotic parabolas for the function f φ whose theory is thoroughly studied in [2]. But for 0 x ∈  the first factorizational appproach characterizes a set of asymptotic expansions wherein (quite surprisingly) the estimates of the remainders in the differentiated expansions may follow algebraic rules different from those valid both in the case 0 x = +∞ and in the case of the standard Taylor's formula; and the second factorizational appproach gives Taylor's formula as a "limit" of Taylor's formulas which is a classical elementary result to be commented on in our context.…”

“…235-236) valid under weaker regularity assumptions involving only the existence of the highestorder left derivative of the given function and its limit as x . This fact historically is the idea underlying the geometric theory of limit parabolas at +∞ , see ( [2], §1) where the two main results characterize expansions involving remainder-estimates at +∞ either of the form…”

“…Under the regularity assumptions of the factorizational theory, this class is characterized by an nth-order differential inequality whereas in the yet-to-be-developed geometric theory it will be the class of generalized convex functions as in the special case of polynomial expansions ( [2], §4).…”

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

“…φ i (x) = x α i . In [5] the author collected and systematized various scattered results concerning polynomial expansions (1.3) f (x) = a n x n + · · · + a 1 x + a 0 + o(1) , x → +∞ , with an eye to highlight the geometric approach and to link different approaches by a unique thread. In [6; 7] the author developed a theory for expansions in real powers (1.4) f (x) = a 1 x α 1 + · · · + a n x αn + o(x αn ) , x → +∞ ; α 1 > · · · > α n , with the aim of obtaining complete and applicable results about the formal differentiation of (1.4), results not obtainable by any of the classical approaches used for formal differentiation of the asymptotic relations f (x) = O(x γ ) or f (x) = o(x γ ).…”