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

This note deals with integrals of the form

$I\left(s,f\right):={\int }_{{ℝ}^{n}}g\left({x}^{\alpha }/s\right){x}^{\beta }{log}^{\gamma }xf\left(x\right)dx$

where $g\in S\left({ℝ}^{n}\right)$ (the Schwartz space), $f\in {C}_{0}^{\infty }\left({ℝ}^{n}\right)$, $s>0$, and $\alpha ,\beta \in {ℝ}^{n}$ with ${\alpha }_{i}>0$, ${\beta }_{i}\ge 0$, 1$\le i\le n$, $\gamma \in {ℤ}_{+}^{n}$. The asymptotic expansion of I(s,f) as $s\to 0$ 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)].

##### MSC:
 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
##### Keywords:
asymptotic expansions for integrals
##### References:
 [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.