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A real analytic approach to estimating oscillatory integrals.

紀錄類型:	書目-語言資料,手稿 : Monograph/item
正題名/作者:	A real analytic approach to estimating oscillatory integrals./
作者:	Gilula, Maxim.
面頁冊數:	1 online resource (118 pages)
附註:	Source: Dissertation Abstracts International, Volume: 77-12(E), Section: B.
Contained By:	Dissertation Abstracts International77-12B(E).
標題:	Mathematics. -
電子資源:	click for full text (PQDT)
ISBN:	9781339928715

A real analytic approach to estimating oscillatory integrals.
Gilula, Maxim.

A real analytic approach to estimating oscillatory integrals. - 1 online resource (118 pages)

Source: Dissertation Abstracts International, Volume: 77-12(E), Section: B.

Thesis (Ph.D.)

Includes bibliographical references

We develop an asymptotic expansion for oscillatory integrals with real analytic phases. We assume the phases satisfy a nondegeneracy condition originally considered by Varchenko, which is related to the Newton polyhedron. Analogous estimates for smooth and Ck phases are also proved. With algebraic techniques such as resolution of singularities, Varchenko was the first to obtain sharp estimates for oscillatory integrals with nondegenerate analytic phases, assuming the Newton distance of the phase is greater than 1. This condition has also been frequently used in modern literature; for example, Greenblatt and later Kamimoto-Nose obtained more general results by also using resolution of singularities. Using only real analytic methods that are very much in the spirit of van der Corput, we develop a full asymptotic expansion for analytic phases satisfying Varchenko's condition, and an asymptotic expansion with finitely many terms for Ck phases under the additional assumption that the Newton polyhedron intersects each coordinate axis. We demonstrate how the exponents in the asymptotic expansion of these integrals can be obtained completely geometrically via the Newton polyhedron. Important techniques include: dyadic decomposition; proving and then using a lower bound similar to that of Lojaciewicz for analytic functions, together with the method of stationary phase to integrate by parts; linear programming to get sharpest estimates (matching Varchenko's); and finally, repeated differentiation of the integral with respect to the oscillatory parameter in order to obtain higher order terms of the expansion.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2018

Mode of access: World Wide Web

ISBN: 9781339928715Subjects--Topical Terms:

527692
Mathematics.
Index Terms--Genre/Form:

554714
Electronic books.

A real analytic approach to estimating oscillatory integrals.
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100 1 $a Gilula, Maxim. $3 1180477
245 1 2 $a A real analytic approach to estimating oscillatory integrals.
264 0 $c 2016
300 $a 1 online resource (118 pages)
336 $a text $b txt $2 rdacontent
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500 $a Source: Dissertation Abstracts International, Volume: 77-12(E), Section: B.
500 $a Adviser: Philip T. Gressman.
502 $a Thesis (Ph.D.) $c University of Pennsylvania $d 2016.
504 $a Includes bibliographical references
520 $a We develop an asymptotic expansion for oscillatory integrals with real analytic phases. We assume the phases satisfy a nondegeneracy condition originally considered by Varchenko, which is related to the Newton polyhedron. Analogous estimates for smooth and Ck phases are also proved. With algebraic techniques such as resolution of singularities, Varchenko was the first to obtain sharp estimates for oscillatory integrals with nondegenerate analytic phases, assuming the Newton distance of the phase is greater than 1. This condition has also been frequently used in modern literature; for example, Greenblatt and later Kamimoto-Nose obtained more general results by also using resolution of singularities. Using only real analytic methods that are very much in the spirit of van der Corput, we develop a full asymptotic expansion for analytic phases satisfying Varchenko's condition, and an asymptotic expansion with finitely many terms for Ck phases under the additional assumption that the Newton polyhedron intersects each coordinate axis. We demonstrate how the exponents in the asymptotic expansion of these integrals can be obtained completely geometrically via the Newton polyhedron. Important techniques include: dyadic decomposition; proving and then using a lower bound similar to that of Lojaciewicz for analytic functions, together with the method of stationary phase to integrate by parts; linear programming to get sharpest estimates (matching Varchenko's); and finally, repeated differentiation of the integral with respect to the oscillatory parameter in order to obtain higher order terms of the expansion.
533 $a Electronic reproduction. $b Ann Arbor, Mich. : $c ProQuest, $d 2018
538 $a Mode of access: World Wide Web
650 4 $a Mathematics. $3 527692
650 4 $a Theoretical mathematics. $3 1180455
655 7 $a Electronic books. $2 local $3 554714
690 $a 0405
690 $a 0642
710 2 $a ProQuest Information and Learning Co. $3 1178819
710 2 $a University of Pennsylvania. $b Mathematics. $3 1180478
773 0 $t Dissertation Abstracts International $g 77-12B(E).
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10134938 $z click for full text (PQDT)