Decomposition of Besov and Triebel–Lizorkin spaces on the sphere

Abstract

A discrete system of almost exponentially localized elements (needlets) on the n-dimensional unit sphere S n is constructed. It shown that the needlet system can be used for decomposition of Besov and Triebel-Lizorkin spaces on the sphere. As an application of Besov spaces on S n , a Jackson estimate for nonlinear m-term approximation from the needlet system is obtained.

Sub-exponentially localized kernels and frames induced by orthogonal expansions

“…The localization principle put forward in [18] says that for all "natural orthogonal systems" the kernels {L n (x, y)} decay at rates faster than any inverse polynomial rate away from the main diagonal y = x in E × E with respect to the distance in E. This principle is very well-known in the case of the trigonometric system (and the Fourier transform) and not so long ago was established for spherical harmonics [15,16], Jacobi polynomials [1,17], orthogonal polynomials on the ball [18], and Hermite and Laguerre functions [2,4,19,9]. Surprisingly, however, the localization principle as formulated above fails to be true for tensor product Jacobi polynomials and, in particular, for tensor product Legendre or Chebyshev polynomials, as will be shown in §10.…”

Fast memory efficient evaluation of spherical polynomials at scattered points

“…Spherical polynomial evaluation via "father needlets". "Mother needlets" on the sphere have been developed and used in [11,12]. For the purposes of evaluation of spherical polynomials it is convenient to use spherical "father needlets", defined as kernels of the form…”

“…For other purposes we have also developed algorithms for approximation and representation of functions on the sphere based on the "mother needlets" introduced in [11,12], which are similar in nature to the widely used wavelets. We shall not present the details of that development here.…”

“…In this paper we present an alternative method for solving problem (ii), based on the spherical "needlets" introduced in [11,12]. We propose a new approximation scheme based on "father needlets".…”

Exact solutions of the vorticity equation on the sphere as a manifold

“…After introducing the manifold S 2 and the Riemannian manifolds (S 2 , g), a general type of spaces (Besov and Triebel-Lizorkin spaces) on the sphere may also be introduced (Narcowich et al, 2006). Using the power of a Laplace operator, the Sobolev space on Riemannian manifolds can also be incorporated as a field currently undergoing great development (Aubin, 1998;Hebey, 2000).…”