Geodesic
One Approach to the Computation of Asymptotics of Integrals of Rapidly Varying Functions
Matematičeskie zametki, Tome 103 (2018) no. 5, pp. 680-692.

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We consider integrals of the form $$ I(x,h)=\frac{1}{(2\pi h)^{k/2}}\int_{\mathbb{R}^k} f\biggl(\frac{S(x,\theta)}{h}\,,x,\theta\biggr)\,d\theta, $$ where $h$ is a small positive parameter and $S(x,\theta)$ and $f(\tau,x,\theta)$ are smooth functions of variables $\tau\in\mathbb{R}$, $x\in\mathbb{R}^n$, and $\theta\in\mathbb{R}^k$; moreover, $S(x,\theta)$ is real-valued and $f(\tau,x,\theta)$ rapidly decays as $|\tau|\to\infty$. We suggest an approach to the computation of the asymptotics of such integrals as $h\to0$ with the use of the abstract stationary phase method.
Keywords: rapidly decaying function, integral, asymptotics, abstract stationary phase method. 
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S. Yu. Dobrokhotov; V. E. Nazaikinskii; A. V. Tsvetkova. One Approach to the Computation of Asymptotics of Integrals of Rapidly Varying Functions. Matematičeskie zametki, Tome 103 (2018) no. 5, pp. 680-692. http://geodesic.mathdoc.fr/item/MZM_2018_103_5_a3/

[1] V. P. Maslov, Teoriya vozmuschenii i asimptoticheskie metody, Izd-vo Mosk. un-ta, M., 1965

[2] V. P. Maslov, M. V. Fedoryuk, Kvaziklassicheskoe priblizhenie dlya uravnenii kvantovoi mekhaniki, Nauka, M., 1976 | MR | Zbl

[3] M. V. Fedoryuk, Metod perevala, Nauka, M., 1977 | MR | Zbl

[4] V. I. Arnold, Teoriya katastrof, Nauka, M., 1990 | MR | Zbl

[5] V. I. Arnold, A. N. Varchenko, S. M. Gusein-Zade, Osobennosti differentsiruemykh otobrazhenii. Tom 2. Monodromiya i asimptotiki integralov, Nauka, M., 1984 | MR

[6] A. I. Allilueva, S. Yu. Dobrokhotov, S. A. Sergeev, A. I. Shafarevich, “Novye predstavleniya kanonicheskogo operatora Maslova i lokalizovannye asimptoticheskie resheniya strogo giperbolicheskikh sistem”, Dokl. AN, 464:3 (2015), 261–266 | DOI | MR | Zbl

[7] I. Bryuning, V. V. Grushin, S. Yu. Dobrokhotov, “Osrednenie lineinykh operatorov, adiabaticheskoe priblizhenie i psevdodifferentsialnye operatory”, Matem. zametki, 92:2 (2012), 163–180 | DOI | MR | Zbl

[8] S. Yu. Dobrokhotov, B. Tirotstsi, A. I. Shafarevich, “Predstavleniya bystroubyvayuschikh funktsii kanonicheskim operatorom Maslova”, Matem. zametki, 82:5 (2007), 792–796 | DOI | MR | Zbl

[9] S. Yu. Dobrokhotov, P. N. Zhevandrov, V. P. Maslov, A. I. Shafarevich, “Asimptoticheskie bystroubyvayuschie resheniya lineinykh strogo giperbolicheskikh sistem s peremennymi koeffitsientami”, Matem. zametki, 49:4 (1991), 31–46 | MR | Zbl

[10] V. G. Danilov, Le Vu An, “Ob integralnykh operatorakh Fure”, Matem. sb., 110 (152):3 (11) (1979), 323–368 | MR | Zbl

[11] S. Kuksin, “Asymptotic expansions for some integrals of quotients with degenerated divisors”, Russ. J. Math. Phys., 24:4 (2017), 476–487 | MR | Zbl

[12] V. P. Maslov, Operatornye metody, Nauka, M., 1973 | MR | Zbl