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Oscillatory integrals and volumes with semiquasihomogeneous phase. (English. Russian original) Zbl 0778.58065
Funct. Anal. Appl. 26, No. 1, 46-48 (1992); translation from Funkts. Anal. Prilozh. 26, No. 1, 59-61 (1992).
Let $$I(\tau,F,\varphi)=\int_{\mathbb{R}^ n}\varphi\exp(i\tau F)$$, where $$\varphi:\mathbb{R}^ n\to\mathbb{R}$$ is $$C^ \infty$$ with support in a small neighborhood of the origin, and $$F$$ is defined in the support of $$\varphi$$. The author obtains asymptotic expansions for this integral as $$\tau\to\infty$$ when $$F$$ is semiquasihomogeneous, that is, $$F=f+f_ 1$$, where $$f$$ is a quasihomogeneous polynomial of degree 1 with weights $$\alpha=(\alpha_ 1,\ldots,\alpha_ n)$$ and $$f_ 1=\sum a_ mx^ m$$, where $$a_ m\geq 0$$ if $$(m,\alpha)\leq 1$$, is real-analytic. The principal term of these asymptotics, for $$|\alpha|<1$$, has the form $$a(f)\tau^{-|\alpha|}\varphi(0)$$, where $$a(f)>0$$. An analogous expansion is obtained for the Lebesgue integral of $$\varphi$$ over the set $$\{x\in\mathbb{R}^ n:0\leq F(x)\leq\varepsilon\}$$, with $$\varepsilon>0$$. This Lebesgue integral is the “volume” of $$F=f+f_ 1$$. Finally, the author obtains a result concerning the relationship between the semiquasihomogeneity of $$F$$ and the $$\mathbb{R}$$-nondegeneracy of the principal part of the Maclaurin series of $$f$$.

##### MSC:
 58J37 Perturbations of PDEs on manifolds; asymptotics 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 58C25 Differentiable maps on manifolds 58K99 Theory of singularities and catastrophe theory
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##### References:
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