Marcinkiewicz Averages of Smooth Orthogonal Projections on Sphere

Abstract

We construct a single smooth orthogonal projection with desired localization whose average under a group action yields the decomposition of the identity operator. For any full rank lattice $$\Gamma \subset \mathbb {R}^d$$ Γ ⊂ R d , a smooth projection is localized in a neighborhood of an arbitrary precompact fundamental domain $$\mathbb {R}^d/\Gamma $$ R d / Γ . We also show the existence of a highly localized smooth orthogonal projection, whose Marcinkiewicz average under the action of SO(d), is a multiple of the identity on $$L^2(\mathbb {S}^{d-1})$$ L 2 ( S d - 1 ) . As an application we construct highly localized continuous Parseval frames on the sphere.