  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.
