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

DOI:
https://doi.org/10.1090/S0025-5718-00-01240-0

Published electronically:
March 1, 2000

Corrigendum:
Math. Comp. 71 (2002), 453-454

MathSciNet review:
1710640

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Abstract | References | Similar Articles | Additional Information

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 , 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*, 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 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.

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Additional Information

**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:
https://doi.org/10.1090/S0025-5718-00-01240-0

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

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