Directional polynomial frames on spheres

TL;DR

This paper presents a flexible framework for constructing polynomial frames on spheres, enabling the design of systems with customizable localization and directional sensitivity for advanced position-frequency analysis.

Contribution

It introduces a systematic method to create polynomial frames on spheres, including classical and new systems, with verifiable properties and optimal localization features.

Findings

Framework includes spherical needlets and directional wavelets.

Conditions for optimal spatial localization are established.

Examples of well-localized, directional polynomial frames are provided.

Abstract

We introduce a general framework for the construction of polynomial frames in $L^2(\mathbb{S}^{d-1})$, $d \geq 3$, where the frame functions are obtained as rotated versions of an initial sequence of polynomials $\Psi^j$, $j\in \mathbb{N}_0$. The rotations involved are discretized using suitable quadrature rules. This framework includes classical constructions such as spherical needlets and directional wavelet systems, and at the same time permits the systematic design of new frames with adjustable spatial localization, directional sensitivity, and computational complexity. We show that a number of frame properties can be characterized in terms of simple, easily verifiable conditions on the Fourier coefficients of the functions $\Psi^j$. Extending an earlier result for zonal systems, we establish sufficient conditions under which the frame functions are optimally localized in space with respect to a spherical uncertainty principle, thus making the corresponding systems a viable tool for position-frequency analyses. To conclude this article, we explicitly discuss examples of well-localized and highly directional polynomial frames.