Local harmonic analysis on spheres
Robert S. Strichartz · Journal of Functional Analysis · 1988
A continuous expansion ƒ(x) = ∝∞∞ ƒλ(x)h(λ)dλ is established for functions or distributions supported on a sufficiently small set K in the sphere Sn, valid in a neighborhood Ω of K, where ƒλ(x) are eigenfunctions of the Laplacian on Ω with eigenvalue ((n − 1)2)2 − λ2. The function h(λ) and the operators Pλ that give ƒλ from ƒ are given by explicit formulas. When n is odd, the function ƒλ(x) can be uniquely characterized by analyticity and growth conditions in λ of Paley-Wiener type. The expansion is applied to study functions of the Laplacian and operators that commute with the Laplacian.