Marcinkiewicz Averages of Smooth Orthogonal Projections on Sphere

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 Γ⊂ℝ^d , a smooth projection is localized in a neighborhood of an arbitrary precompact fundamental domain 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(𝕊^d-1) . As an application we construct highly localized continuous Parseval frames on the sphere.