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Abstract  

We establish a general analytic theory of asymptotic expansions of type

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$f(x) = a_1 \varphi _1 (x) + \cdots + a_n \varphi _n (x) + o(\varphi _n (x)) x \to x_0 ,$$ \end{document}
(1), where the given ordered n-tuple of real-valued functions (ϕ 1, ..., ϕ n) forms an asymptotic scale at x 0. 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, ϕ 1, ..., ϕ n. 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 starts from simple geometric considerations and gives conditions expressed as the existence of finite limits, as xx 0, of certain Wronskian determinants constructed with f, ϕ 1, ..., ϕ n. There is a link between the two approaches and it turns out that some of the integral conditions found via the factorizational approach have geometric meanings. Our theory extends to more general expansions the theory of real-power asymptotic expansions thoroughly investigated in previous papers. In the first part of our work we study the case of two comparison functions ϕ 1, ϕ 2 because the pertinent theory requires a very limited theoretical background and completely parallels the theory of polynomial expansions.

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Abstract  

In two papers, the problem of formal differentiation of an asymptotic expansion in the real domain of type

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$f(x) - a_1 x^{\alpha _1 } + \cdots + a_n x^{\alpha _n } + o(x^{\alpha _n } ),x \to + \infty ,$$ \end{document}
is amply studied. In Part I, we show that the classical viewpoints and techniques concerning formal differentiation of an asymptotic relation
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$f(x) - ax^\alpha + o(x^\alpha ),x \to + \infty ,$$ \end{document}
give either unsatisfactory or partial results when applied to an asymptotic expansion with at least two meaningful terms. Simple examples show that some of these results are the best possible in the classical context. Hence a change of viewpoint is necessary to arrive at useful results.

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Abstract  

In Part II of our work we approach the problem discussed in Part I from the new viewpoint of canonical factorizations of a certain nth order differential operator L. The main results include: (i)  characterizations of the set of relations

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$f^{(k)} (x) = P^{(k)} (x) + o^{(k)} (x^{\alpha _n - k} ),x \to + \infty ,0 \leqslant k \leqslant n - 1,$$ \end{document}
where
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$P(x) = a_1 x^{\alpha _1 } + \cdots + a_n x^{\alpha _n } and \alpha _1 > \alpha _2 > \cdots > \alpha _n ,$$ \end{document}
by means of suitable integral conditions (ii)  formal differentiation of a real-power asymptotic expansion under a Tauberian condition involving the order of growth of L(iii)  remarkable properties of asymptotic expansions of generalized convex functions.

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Abel, U. and Heilmann, M. , The complete asymptotic expansion for Bernstein-Durrmeyer operators with Jacobi weights, Mediterr. J. Math. , 1 (2004), no. 4, 487–499. MR 2005k :41075

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. , 2001 , Approximation Theory, Asymptotical expansions, suppl. 1, S30–S47. [2] Babenko , A. G. , Chernykh , N. I. , Shevaldin , V. T. 1999 The Jackson–Stechkin inequality in L 2 with

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References [1] Erdélyi , A. 1956 Asymptotic Expansions Dover New York

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1 9 Abel, U. and Heilmann, M. , The complete asymptotic expansion for Bernstein-Durrmeyer operators with Jacobi weights, Mediterr. J. Math. , 1 (2004), 487

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References [1] Accetta , G. and Orsingher , E. , Asymptotic expansion of fundamental solutions of higher order heat equations

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multiplication parameter, which by itself well indicates a certain extent of inherent inexactness of such a kind of kinetic evaluation. A better insight of this inquisitiveness was provided by the use of an asymptotic expansion [ 88 ] of a series with a

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