In this thesis we study a generalization of the classical boundary value problem for the wave equation called the Cauchy problem on flat and compact symmetric spaces. Using methods similar to those used by S. Helgason in his study of the noncompact case, we derive a solution to this problem using the Fourier and Radon transforms on a flat symmetric space and establish that the solution satisfies Huygens' principle. In addition, we show that the solution is unique assuming a support condition on the function. We study a generalization of the classical energy form, establish that this form is positive definite and time-invariant and satisfies a Plancherel-type result.;In addition, we study the Cauchy problem for a compact symmetric space U/K associated to the compact Riemannian symmetric pair (U,K). Using the solution form derived by F. Gonzalez, we study the energy of a solution to the Cauchy problem in the compact case. We establish positivity and time-invariance of the energy form as well as a Plancherel-type result.
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