  This paper examines filtering on a sphere, by first examining the roles of
spherical harmonic magnitude and phase. We show that phase is more important
than magnitude in determining the structure of a spherical function. We examine
the properties of linear phase shifts in the spherical harmonic domain, which
suggest a mechanism for constructing finite-impulse-response (FIR) filters. We
show that those filters have desirable properties, such as being associative,
mapping spherical functions to spherical functions, allowing directional
filtering, and being defined by relatively simple equations. We provide
examples of the filters for both spherical and manifold data.
