Optimally space-localized band-limited wavelets on Sq-1Sq-1

The localization of a function can be analyzed with respect to different criteria. In this paper, we focus on the uncertainty relation on spheres introduced by Goh and Goodman [Uncertainty principles and asymptotic behavior, Appl. Comput. Harmon. Anal. 16 (2004) 69–89], where the localization of a function is measured in terms of the product of two variances, the variance in space domain and the variance in frequency   domain. After deriving an explicit formula for the variance in space domain of a function in the space Wn,qs of spherical polynomials of degree at most n+sn+s which are orthogonal to all spherical polynomials of degree at most n  , we are able to identify—up to rotation and multiplication by a constant—the polynomial in Wn,qs with minimal variance in space-domain, or in other words, to determine the optimally space-localized polynomial in Wn,qs.