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Zbl 0855.34062
DurĂ¡n, A.L.; Estrada, R.; Kanwal, R.P.
Pre-asymptotic expansions.
(English)
[J] J. Math. Anal. Appl. 202, No.2, 470-484 (1996). ISSN 0022-247X

Let $\{V_n\}$ be a sequence of vector spaces with $V_{n + 1} \subseteq V_n$ for $n = 0,1,2, \dots$. A series $\sum^\infty_{n = 0} v_n$ with $v_n \in V_n$ is called a pre-asymptotic expansion of $v \in V_0$, written $v \sim \sum^\infty_{n = 0} v_n$ with respect to $\{V_n\}$, if $v - \sum^N_{n = 0} v_n \in V_{N + 1}$ for all $N$. Under suitable assumptions it is shown that the solution $v$ of a linear operator equation $Tv = w$ with $v \in V_0$, $w \in \cap^\infty_{n = 0} V_n$ possesses a pre-asymptotic expansion. In particular, for linear differential operators with polynomial coefficients, formal solutions of the form $\sum_{n = 0}^\infty a_n \delta^{(n)} (x)$ can be interpreted as pre-asymptotic expansions of distributional solutions. Sometimes, pre-asymptotic expansions $\psi (x) \sim \sum^\infty_{n = 0} \psi_n (x)$ are connected with ordinary asymptotic expansions $\psi (Ex) \sim \sum^\infty_{n = 0} \psi_n (Ex)$ in the sense of $\psi (Ex) = \sum^N_{n = 0} \psi_n (Ex) + o(E^N)$ for $E \to 0$.
[L.Berg (Rostock)]
MSC 2000:
*34E05 Asymptotic expansions (ODE)
41A60 Asymptotic problems in approximation
46F10 Operations with distributions (generalized functions)

Keywords: pre-asymptotic expansion; linear operator equation; linear differential operators with polynomial coefficients; formal solutions; asymptotic expansions

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