Mathematics of Computation

Author(s): H. N. Mhaskar; F. J. Narcowich; J. D. Ward.
Journal: Math. Comp. 70 (2001), 1113-1130.
MSC (2000): Primary 65D32; Secondary 41A17, 42C10
Posted: March 1, 2000
Corrigenda: Math. Comp. 71 (2002), 453-454
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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

-Marcinkiewicz-Zygmund inequalities for such sites.

1.: L. Bos, N. Levenberg, P. Milman, and B. A. Taylor, Tangential Markov inequalities characterize algebraic submanifolds of $\mathbb R^N$ , Ind. Univ. Math. J., 44 (1995), 115-138. MR 96i:41009
2.: J. R. Driscoll and D. M. Healy, Computing Fourier Transforms and Convolutions for the 2-Sphere, Adv. in Appl. Math., 15 (1994), 202-250. MR 95h:65108
3.: N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1958. MR 22:8302
4.: T. Erdélyi, Notes on inequalities with doubling weights, J. Approx. Theory 100 (1999), 60-72. CMP 2000:01
5.: H. Flanders, Differential Forms, Academic Press, New York, 1963. MR 28:5397
6.: W. Freeden, O. Glockner, and M. Schreiner, Spherical Panel Clustering and Its Numerical Aspects, AGTM Report No. 183, University of Kaiserlautern, Geomathematics Group, 1997.
7.: W. Freeden and U. Windheuser, Spherical Wavelet Transform and Its Discretization, AGTM Report No. 125, University of Kaiserlautern, Geomathematics Group, 1995.
8.: J. M. Goethals and J. J. Seidel, Cubature formulae, polytopes, and spherical designs, in The Geometric Vein, Coexeter Festschrift, (C. Davis et al., eds.), Springer, New York, 1981, pp. 203-218. MR 83k:05033
9.: R. B. Holmes, Geometric functional analysis and its applications, Springer-Verlag, New York, 1975. MR 53:14085
10.: K. Jetter, J. Stöckler, and J. D. Ward, Error estimates for scattered data interpolation, Math. Comp., 68 (1999), 743-747. MR 99i:41032
11.: K. Jetter, J. Stöckler, and J. D. Ward, Norming sets and spherical cubature formulas, in Computational Mathematics, (Chen, Li, C. Micchelli, Y. Xu, eds.), Marcel Decker, New York, 1998, pp. 237-245. MR 99i:65022
12.: G. Mastoianni and V. Totik, Weighted polynomial inequalities with doubling and $A_\infty$ weights, to appear in J. London Math. Soc.
13.: H. N. Mhaskar, F. J. Narcowich and J. D. Ward, Approximation Properties of Zonal Function Networks Using Scattered Data on the Sphere, Adv. in Comp. Math., to appear.
14.: H. N. Mhaskar and J. Prestin, Marcinkiewicz-Zygmund Inequalities, in Approximation Theory: In Memory of A. K. Varma, (N. K. Govil, R. N. Mohapatra, Z. Nashed, A. Sharma, and J. Szabados Eds.), Marcel Dekker, to appear.
15.: C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, Vol. 17, Springer Verlag, Berlin, 1966. MR 33:7593
16.: P. Petrushev, Approximation by Ridge Functions and Neural Networks, SIAM J. Math. Anal., 30 (1999), 155-189. MR 99g:41031
17.: D. Potts, G. Steidl, and M. Tasche, Fast algorithms for discrete polynomial transforms, Math. Comp., 67 (1998), 1577-1590. MR 99b:65183
18.: E. M. Stein, Interpolation in polynomial classes and Markoff's inequality, Duke Math. J., 24 (1957), 467-476. MR 19:956b
19.: E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, New Jersey, 1971. MR 46:4102
20.: G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., Providence, 1975. MR 51:8724
21.: A. Zygmund, A remark on conjugate series, Proc. London Math. Soc., 34 (1932), 392-400.

Retrieve articles in Mathematics of Computation with MSC (2000): 65D32, 41A17, 42C10

H. N. Mhaskar
Affiliation: Department of Mathematics, California State University, Los Angeles, CA 90032
Email: hmhaskar@calstatela.edu

F. J. Narcowich
Affiliation: Department of Mathematics, Texas A&M University, College Station, TX 77843-3368
Email: fnarc@math.tamu.edu

J. D. Ward
Affiliation: Department of Mathematics, Texas A&M University, College Station, TX 77843-3368
Email: jward@math.tamu.edu

DOI: 10.1090/S0025-5718-00-01240-0
PII: S 0025-5718(00)01240-0
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
Posted: 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.
Copyright of article: Copyright 2000, American Mathematical Society

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