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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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