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

The localizationnext term 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 previous termlocalizationnext term 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 Click to view the MathML source of spherical polynomials of degree at most n+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 Click to view the MathML source with minimal variance in space-domain, or in other words, to determine the optimally space-localized polynomial in Click to view the MathML source

Polynomial Interpolation on the Unit Sphere II.

The problem of interpolation at $(n+1)^2$ points on the unit sphere $mathbbS^2$ by spherical polynomials of degree at most $n$ is proved to have a unique solution for several sets of points. The points are located on a number of circles on the sphere with even number of points on each circle. The proof is based on a method of factorization of polynomials