**"Marcinkiewicz averages of smooth orthogonal projections on sphere"**
#### Dziedziul, KarolWe 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 $\Gamma \subset {R}^d$, a smooth projection is localized in a neighborhood of an arbitrary precompact fundamental domain $R^d/\Gamma$. 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 $L^2({S}^{d-1})$. As an application we construct highly localized continuous Parseval frames on the sphere. |