Sufficient Conditions for Sampling and Interpolation on the Sphere

Abstract

ABSTRACT. We obtain sufficient conditions for arrays of points, Z = {Z(L)} L≥1 , on the unit sphere Z(L) ⊂ S d , to be Marcinkiewicz-Zygmund and interpolating arrays for spaces of spherical harmonics. The conditions are in terms of the mesh norm and the separation radius of Z(L).

A Density Theorem for Weighted Fekete Sets

“…[30], Section III.1. The convergence of the counting measures has been generalized in various directions, see for instance [5,8,9,12,13,16,17,18,21,24,26,28,29,33] and the references there. One of those directions concerns the case of a complex line bundle over a compact manifold [9,21,29].…”

Approximation and quadrature by weighted least squares polynomials on the sphere

“…It follows from [19,2,7] that such MZ families exist if the families are dense enough. We remark that Fekete points of degree ⌊n(1+ε)⌋ (ε > 0) on the sphere are MZ families with the equal weights τ n,k = 1 dn , see [17], where ⌊a⌋ is the largest integer not exceeding a ∈ R. Also, sufficient conditions and necessary density conditions for MZ families with the equal weights τ n,k = 1 dn on the sphere are obtained in [18] and [16], respectively.…”

“…It follows from [31], [1] and [13] that It follows from [16,17,18] that there exist MZ families with…”

Weighted $\ell_q$ approximation problems on the ball and on the sphere

“…Remark 4.7. It follows from [14,15,16] that there exist L q -MZ families on S d with l n ≍ N ≍ n d . For such L q -MZ family, combining (4.4) with [22, Theorem 1.2], we obtain for 1 ≤ p, q ≤ ∞, r > d max{1/p, 1/q}, sup f ∈BW r p (S d )…”

“…Necessary density conditions for L 2,µ -MZ families on B d were obtained in [1,4]. There are many papers devoted to studying MZ families on the sphere and compact manifold (see [7,14,16,18,20]).…”