Marcinkiewicz–Zygmund inequalities and interpolation by spherical harmonics

Abstract

We find necessary density conditions for Marcinkiewicz-Zygmund inequalities and interpolation for spaces of spherical harmonics in S d with respect to the L p norm. Moreover, we prove that there are no complete interpolation families for p = 2.

Equidistribution of Fekete Points on the Sphere

“…We need a polynomial p that peaks at one point, has degree [εL] and decays fast far away from the picking point. For this purpose we will use powers of the Jacobi polynomials which are natural in this context because they are the reproducing kernels in Π L , see [Mar07]. The Jacobi polynomials P (α,β) L of degree L and index (α, β) are the orthogonal polynomials on [−1, 1] with respect to the weight function (1 − x) α (1 + x) β with α, β > −1.…”

Beurling–Landauʼs density on compact manifolds

“…The proofs of these results follow from standard techniques and the ideas in [11,Theorem 4.7], replacing the corresponding gradient estimates obtained in [14].…”

“…Pesenson and his co-authors (see [6] for a detailed discussion). The goal of this work is to extend the theory of Beurling-Landau on the discretization of functions in the Paley-Wiener space on R n to functions in M. This should be possible because there is already a literature on the subject in the case M = S m (see [11] for more details). In the present work, we study the interpolating and Marcinkiewicz-Zygmund families for the spaces E L .…”

Marcinkiewicz–Zygmund inequalities and interpolation by spherical polynomials with respect to doubling weights

“…Usually, in the definition of an L p,w -interpolating family for 1 ≤ p ≤ ∞, the above condition is replaced by the one that there exists a sequence of polynomials {Q L } L≥0 , Q L ∈ Π L , uniformly bounded in L p,w such that Q L (z Lj ) = c Lj , 1 ≤ j ≤ m L (see [11]). However, the two conditions are equivalent.…”