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On the asymptotic expansion of some integrals. (English) Zbl 0531.41027

This note deals with integrals of the form

I(s,f):= n g(x α /s)x β log γ xf(x)dx

where gS( n ) (the Schwartz space), fC 0 ( n ), s>0, and α,β n with α i >0, β i 0, 1in, γ + n . The asymptotic expansion of I(s,f) as s0 is derived by induction on n using a two-variable asymptotic expansion in the induction step. As a special case one obtains a recent result of D. Barlet [Invent. Math. 68, 129-174 (1982; Zbl 0508.32003)].

41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
[1]D. Barlet, Développement asymptotique des fonctions obtenue par intégration sur les fibres. Inventiones math.68, 129-174 (1982). · Zbl 0508.32003 · doi:10.1007/BF01394271
[2]J.Brüning and E.Heintze, The Minakshisundaram-Pleijel expansion in the equivariant case. To appear.
[3]I. M.Gelfand and G. E.Shilov, Generalized Functions I, Properties and Operations. New York 1964.
[4]P. Jeanquartier, Développement asymptotique de la distribution de Dirac attachée à une fonction analytique. C.R. Acad. Sci. Paris271, 1159-1161 (1970).
[5]B.Malgrange, Intégrales asymptotiques et monodromie. Ann. scient. Ec. Norm. Sup. 7 (4e série), 405-430 (1974).
[6]E.Ya. Riekstyn’sh, Asymptotic expansions of integrals (Asimptoticheskie razloshe integralov). Vols. 1, 2, 3. Riga 1974, 1977, 1981.