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Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature

Authors: H. N. Mhaskar, F. J. Narcowich and J. D. Ward
Journal: Math. Comp. 70 (2001), 1113-1130
MSC (2000): Primary 65D32; Secondary 41A17, 42C10
Published electronically: March 1, 2000
Corrigendum: Math. Comp. 71 (2002), 453-454
MathSciNet review: 1710640
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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 $\mathbb{R}^q$, 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.

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Additional Information

H. N. Mhaskar
Affiliation: Department of Mathematics, California State University, Los Angeles, CA 90032

F. J. Narcowich
Affiliation: Department of Mathematics, Texas A&M University, College Station, TX 77843-3368

J. D. Ward
Affiliation: Department of Mathematics, Texas A&M University, College Station, TX 77843-3368

Keywords: Marcinkiewicz-Zygmund inequalities, quadrature, scattered-data on spheres
Received by editor(s): January 26, 1999
Received by editor(s) in revised form: August 25, 1999
Published electronically: March 1, 2000
Additional Notes: Research of the authors was sponsored by the Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grant numbers F49620-97-1-0211 and F49620-98-1-0204. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the U.S. Government.
Article copyright: © Copyright 2000 American Mathematical Society