Frame and wavelet systems on the sphere

TL;DR

This paper develops a new frame and wavelet system on the sphere by solving a weighted minimum problem, leading to efficient localization and orthogonality properties useful for spherical analysis.

Contribution

It introduces a novel approach to constructing frame and wavelet systems on the sphere using solutions to a weighted minimum problem, improving localization and orthogonality.

Findings

Constructed a frame system on the sphere with optimal localization.

Connected the system to discrete orthogonality of spherical functions.

Compared the efficiency with previous methods, showing improvements.

Abstract

In this paper we formulate a weighted version of minimum problem (1.4) on the sphere and we show that, for $K\le L$, if $\set{\phi_k}^K_{k=1}$ consists of the spherical functions with degree less than $N$ we can localize the points $(\xi_1,...,\xi_L)$ on the sphere so that the solution of this problem is the simplest possible. This localization is connected to the discrete orthogonality of the spherical functions which was proved in [3]. Using these points we construct a frame system and a wavelet system on the sphere and we study the properties of these systems. For $K>L$ a similar construction was made in paper [4], but in that case the solution of the minimum problem (1.4) is not as efficient as it is in our case. The analogue of Fej\'er and de la Val\'ee-Poussin summation methods introduced in [3] can be expressed by the frame system introduced in this paper.