Pointwise Fourier Inversion: A Wave Equation Approach

Functions of the Laplace operator F(- δ) can be synthesized from the solution operator to the wave equation. When F is the characteristic function of [0, R₂], this gives a representation for radial Fourier inversion. A number of topics related to pointwise convergence or divergence of such inversion, as R → ∞, are studied in this article. In some cases, including analysis on Euclidean space, spheres, hyperbolic space, and certain other symmetric spaces, exact formulas for fundamental solutions to wave equations are available. In other cases, parametrices and other tools of microlocal analysis are effective.