Plane Waves and Spherical Means Applied to Partial Differential Equations

Elementary and self-contained, this heterogeneous collection of results on partial differential equations employs certain elementary identities for plane and spherical integrals of an arbitrary function, showing how a variety of results on fairly general differential equations follow from those identities. The first chapter deals with the decomposition of arbitrary functions into functions of the type of plane waves. Succeeding chapters introduce the first application of the Radon transformation and examine the solution of the initial value problem for homogeneous hyperbolic equations with constant coefficients and the problem of determining a function from its integrals over spheres of radius 1. 1955 edition.

I. Decomposition of an Arbitrary Function into Plane Waves
II. The Initial Value Problem for Hyperbolic Homogeneous Equations with Constant Coefficients
III. The Fundamental Solution of a Linear Elliptic Differential Equation with Analytic Coefficients
IV. Identities for Spherical Means
V. The Theorems of Asgeirsson and Howard
VI. Determination of a Function from its Integrals over Spheres of a Fixed Radius
VII. Differentiability Properties of Solutions of Elliptic Systems
VIII. Regularity Properties of Integrals of Solutions over Time-like Lines
Bibliography
Index