Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature

Abstract

Abstract. Geodetic and meteorological data, collected via satellites for example, are genuinely scattered and not confined to any special set of points. Even so, known quadrature formulas used in numerically computing integrals involving such data have had restrictions either on the sites (points) used or, more significantly, on the number of sites required. Here, for the unit sphere embedded in R q , we obtain quadrature formulas that are exact for spherical harmonics of a fixed order, have nonnegative weights, and are based on function values at scattered sites. To be exact, these formulas require only a number of sites comparable to the dimension of the space. As a part of the proof, we derive L 1 -Marcinkiewicz-Zygmund inequalities for such sites.

Localized Tight Frames on Spheres

“…Using the expression on the right in (15) in the series definition of K ε,n , we get this representation:…”

“…To do the discretizations required to construct tight spherical frames in section 5, we need a strengthened version of the quadrature formula given in [14,15]. There are two reasons for this.…”

“…At this point, the order of the spherical harmonics is such that we can compute the integral exactly using a quadrature formula introduced in [14,15] and, in section 4, developed into the tool we need here. The point is that the frame functions have the form…”

Fast evaluation of quadrature formulae on the sphere

“…, k. Note that this usually does not yet recover the coefficients a n k from (1.1) for which we denote the computed coefficientsã n k . The coefficients a n k can be obtained from values of the function f on a set of arbitrary nodes (ϑ d , ϕ d ) provided that a quadrature rule with weights w d and sufficient high degree of exactness is available (see also [5,10]). Then the sum in (1.3) changes to…”

Corrigendum to ``Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature''

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Professor Dr. Joachim Stöckler has kindly pointed out to us that our proofs [2] of the estimates (4.5) and (4.9) are valid only in the case when we do not require the quadrature weights a ξ to be nonnegative, since the Krein-Rutman extensions of nonnegative functionals are not guaranteed to be norm-preserving. The purpose of this note is to point out that the existence of nonnegative weights a ξ satisfying (4.4) necessarily implies the following analogue (0.1) of (4.5).

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