Marcinkiewicz averages of smooth orthogonal projections on sphere

TL;DR

This paper constructs smooth orthogonal projections on spheres with specific localization properties, enabling the decomposition of identity operators and the creation of highly localized Parseval frames on the sphere.

Contribution

It introduces a method to produce smooth localized orthogonal projections on spheres and their averages, leading to new Parseval frames with desirable localization properties.

Findings

Existence of smooth localized orthogonal projections on spheres.

Construction of Parseval frames with strong localization.

Averaging projections yields the identity operator.

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.