The Spherical Mean Value Operators on Euclidean and Hyperbolic Spaces.
Abstract: 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 o... read moref 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.
Thesis (Ph.D.)--Tufts University, 2012.
Submitted to the Dept. of Mathematics.
Advisor: Fulton Gonzalez.
Committee: Sigurdur Helgason, Eric Todd Quinto, and Loring Tu.
Keyword: Mathematics.read less