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Convolution Operators on Euclidean and Symmetric Spaces = = 欧氏空间与对称空间上的卷积算子.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Convolution Operators on Euclidean and Symmetric Spaces =/
其他題名:	欧氏空间与对称空间上的卷积算子.
作者:	Wang, Jue.
出版者:	Ann Arbor : ProQuest Dissertations & Theses, : 2022,
面頁冊數:	171 p.
附註:	Source: Dissertations Abstracts International, Volume: 83-09, Section: B.
Contained By:	Dissertations Abstracts International83-09B.
標題:	Mathematics. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28963531
ISBN:	9798206396652

Convolution Operators on Euclidean and Symmetric Spaces = = 欧氏空间与对称空间上的卷积算子.
Wang, Jue.

Convolution Operators on Euclidean and Symmetric Spaces =欧氏空间与对称空间上的卷积算子. - Ann Arbor : ProQuest Dissertations & Theses, 2022 - 171 p.

Source: Dissertations Abstracts International, Volume: 83-09, Section: B.

Thesis (Ph.D.)--Tufts University, 2022.

This item must not be sold to any third party vendors.

In the 1960s, L. Ehrenpreis established some classical results on the surjectivity of convolution operators on Euclidean spaces, including the existence of fundamental solutions. These results generalize the well-known Malgrange-Ehrenpreis Theorem. In this thesis, we first discuss some applications of Ehrenpreis' classical results. In particular, we obtain an explicit formula for the "snapshot" problem of the wave equation.After that, we extend Ehrenpreis' results from Euclidean spaces to noncompact symmetric spaces. We consider the right convolution operator cμ, where the convolution kernel μ is a compactly supported distribution. Precisely, we give a concrete condition for μ which holds if and only if cμ has a fundamental solution and is surjective on various spaces of functions or distributions. A corollary of our results is the Malgrange-Ehrenpreis Theorem on noncompact symmetric spaces, a classical result established by Helgason in 1964.Finally, we establish generalizations of Asgeirsson's Mean Value Theorem (AsMVT) on Euclidean spaces, noncompact rank-one symmetric spaces, and compact symmetric spaces, where the last one is our main result. The original AsMVT gives a mean value relation satisfied by smooth solutions of the ultrahyperbolic equation on Euclidean spaces. In our generalized AsMVT, the mean value relation is generalized to a convolution relation. Our results also extend Helgason's work about AsMVT on rank-one symmetric spaces.

ISBN: 9798206396652Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Asgeirsson's mean value theorem

Convolution Operators on Euclidean and Symmetric Spaces = = 欧氏空间与对称空间上的卷积算子.
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520 $a In the 1960s, L. Ehrenpreis established some classical results on the surjectivity of convolution operators on Euclidean spaces, including the existence of fundamental solutions. These results generalize the well-known Malgrange-Ehrenpreis Theorem. In this thesis, we first discuss some applications of Ehrenpreis' classical results. In particular, we obtain an explicit formula for the "snapshot" problem of the wave equation.After that, we extend Ehrenpreis' results from Euclidean spaces to noncompact symmetric spaces. We consider the right convolution operator cμ, where the convolution kernel μ is a compactly supported distribution. Precisely, we give a concrete condition for μ which holds if and only if cμ has a fundamental solution and is surjective on various spaces of functions or distributions. A corollary of our results is the Malgrange-Ehrenpreis Theorem on noncompact symmetric spaces, a classical result established by Helgason in 1964.Finally, we establish generalizations of Asgeirsson's Mean Value Theorem (AsMVT) on Euclidean spaces, noncompact rank-one symmetric spaces, and compact symmetric spaces, where the last one is our main result. The original AsMVT gives a mean value relation satisfied by smooth solutions of the ultrahyperbolic equation on Euclidean spaces. In our generalized AsMVT, the mean value relation is generalized to a convolution relation. Our results also extend Helgason's work about AsMVT on rank-one symmetric spaces.
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