Harmonic analysis of scalar and vector fields inn
J. Denmead Smith · Mathematical Proceedings of the Cambridge Philosophical Society · 1972
Abstract It is shown that a real scalar function in nwhich is of classCnand either has zero mean on all spheres of unit radius, or has zero mean in all balls of unit radius admits a unique expansion in terms of eigenfunctions of the Laplacian operator. In a similar manner, a suitably smooth vector-valued function in nwhich has zero flux through all spheres cf unit radius is shown to be decomposable as the sum of a solenoidal part and a series of conservative parts that are eigenfunctions of the Laplacian. Applications are given, including some in complex analysis.