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Pre-asymptotic expansions. (English) Zbl 0855.34062
Let {V n } be a sequence of vector spaces with V n+1 V n for n=0,1,2,. A series n=0 v n with v n V n is called a pre-asymptotic expansion of vV 0 , written v n=0 v n with respect to {V n }, if v- n=0 N v n V N+1 for all N. Under suitable assumptions it is shown that the solution v of a linear operator equation Tv=w with vV 0 , w n=0 V n possesses a pre-asymptotic expansion. In particular, for linear differential operators with polynomial coefficients, formal solutions of the form n=0 a n δ (n) (x) can be interpreted as pre-asymptotic expansions of distributional solutions. Sometimes, pre-asymptotic expansions ψ(x) n=0 ψ n (x) are connected with ordinary asymptotic expansions ψ(Ex) n=0 ψ n (Ex) in the sense of ψ(Ex)= n=0 N ψ n (Ex)+o(E N ) for E0.
Reviewer: L.Berg (Rostock)
MSC:
34E05Asymptotic expansions (ODE)
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
46F10Operations with distributions (generalized functions)