On Euclidean space, we prove the surjectivity of the spherical mean value operator from the class of smooth functions to itself, and from the class of distributions to itself, and obtain range characterizations of this operator on the class of compactly supported distributions and functions, respectively. On the three dimensional hyperbolic space, we obtain a range characterization of the spherical mean value operator on the class of compactly supported distributions. From this we show that this operator is surjective from the space of smooth functions onto itself.
We extend some results on the spherical mean obtained by Fritz John. We derive a formula for the iterated spherical mean in hyperbolic spaces. We also show that a smooth function f on the three dimensional hyperbolic space with known averages over all spheres of a fixed radius r > 0 is uniquely determined, if the values of f are known on certain split annuli where the sum of the thicknesses of the annuli is r. Finally we obtain an explicit solution to the inhomogeneous spherical mean value equation in three dimensional hyperbolic space.
We provide supplementary support theorems for the single radius spherical mean in Euclidean space of dimension 3 and 5, and a support theorem for the single radius spherical mean in Hyperbolic space of dimension 3.
|Advisor:||Gonzalez, Fulton B.|
|Commitee:||Helgason, Sigurdur, Quinto, Eric Todd, Tu, Loring|
|School Location:||United States -- Massachusetts|
|Source:||DAI-B 74/04(E), Dissertation Abstracts International|
|Subjects:||Applied Mathematics, Mathematics|
|Keywords:||Delay differential equations, Iterated mean values, Range characterization, Spherical mean, Support theorem, Surjectivity|
Copyright in each Dissertation and Thesis is retained by the author. All Rights Reserved
The supplemental file or files you are about to download were provided to ProQuest by the author as part of a
dissertation or thesis. The supplemental files are provided "AS IS" without warranty. ProQuest is not responsible for the
content, format or impact on the supplemental file(s) on our system. in some cases, the file type may be unknown or
may be a .exe file. We recommend caution as you open such files.
Copyright of the original materials contained in the supplemental file is retained by the author and your access to the
supplemental files is subject to the ProQuest Terms and Conditions of use.
Depending on the size of the file(s) you are downloading, the system may take some time to download them. Please be