Abstract
This paper, divided into three parts (Part II-A, Part II-B and Part II-C), contains the detailed factorizational theory of asymptotic expansions of type (?) , , , where the asymptotic scale , , is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of . It follows two pre-viously published papers: the first, labelled as Part I, contains the complete (elementary but non-trivial) theory for ; the second is a survey highlighting only the main results without proofs. All the material appearing in §2 of the survey is here reproduced in an expanded form, as it contains all the preliminary formulas necessary to understand and prove the results. The remaining part of the survey—especially the heuristical considerations and consequent conjectures in §3—may serve as a good introduction to the complete theory.
Highlights
Following the line of thought in [1], case n = 2, we develop in this paper a general analytic theory of asymptotic expansions of type
For the reader’s convenience, the paper has been divided into three parts: the present Part II-A contains all the general results obtainable through two approaches based on different special factorizations of the nth-order differential operator whose kernel is spanned by (φ1,φn )
An iteration of the procedure gives all relations in (5.6) together with the representation formulas (5.11)-(5.12) for R0 ( x) and (5.15) for Rk ( x), k ≥ 1
Summary
The ordered n-tuple of real-valued functions (φ1, ,φn ) n ≥ 2, is termed a “Chebyshev asymptotic scale” (T.A.S. for short) on the half-open interval [T , x0[ , T ∈ » , x0 ≤ +∞ , provided the following properties are satisfied: φi ∈ Cn−1 [T , x0 [ , 1 ≤ i ≤ n;. 2) Choosing an half-open interval in this definition is a matter of convenience: the point x0 involved in the asymptotic relations is characterized as the endpoint not belonging to the interval, possibly x0 = +∞ , whereas the other endpoint marks off an interval whereon the inequalities involving the Wronskians are satisfied and these in turn allow certain integral representations valid on the whole given interval and essential to our theory These remarks make evident the analogous definition for an interval ]x0,T ] where: −∞ ≤ x0 , and T ∈ ». As far as C.F.’s of type (II) are concerned the present quick approach does not yield a C.F. of type (II) at b for each interval ]a + ,b[ , as asserted in Proposition 2.1-5
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