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 $\Gamma \subset {\mathbb{R}}^d$, a smooth projection is localized in a neighborhood of an arbitrary precompact fundamental domain ${\mathbb{R}}^d/\Gamma$. 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 $L^2({\mathbb{S}}^{d-1})$. As an application we construct highly localized continuous Parseval frames on the sphere.