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

Full-text PDF Free Access

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 𝑅^{𝑁}*, Indiana Univ. Math. J.**44**(1995), no. 1, 115–138. MR**1336434**, 10.1512/iumj.1995.44.1980**2.**James R. Driscoll and Dennis M. Healy Jr.,*Computing Fourier transforms and convolutions on the 2-sphere*, Adv. in Appl. Math.**15**(1994), no. 2, 202–250. MR**1277214**, 10.1006/aama.1994.1008**3.**Nelson Dunford and Jacob T. Schwartz,*Linear Operators. I. General Theory*, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR**0117523****4.**T. Erdélyi,*Notes on inequalities with doubling weights*, J. Approx. Theory**100**(1999), 60-72. CMP**2000:01****5.**Harley Flanders,*Differential forms with applications to the physical sciences*, Academic Press, New York-London, 1963. MR**0162198****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*, The geometric vein, Springer, New York-Berlin, 1981, pp. 203–218. MR**661779****9.**Richard B. Holmes,*Geometric functional analysis and its applications*, Springer-Verlag, New York-Heidelberg, 1975. Graduate Texts in Mathematics, No. 24. MR**0410335****10.**Kurt Jetter, Joachim Stöckler, and Joseph D. Ward,*Error estimates for scattered data interpolation on spheres*, Math. Comp.**68**(1999), no. 226, 733–747. MR**1642746**, 10.1090/S0025-5718-99-01080-7**11.**K. Jetter, J. Stöckler, and J. D. Ward,*Norming sets and spherical cubature formulas*, Advances in computational mathematics (Guangzhou, 1997) Lecture Notes in Pure and Appl. Math., vol. 202, Dekker, New York, 1999, pp. 237–244. MR**1661538****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.**Claus Müller,*Spherical harmonics*, Lecture Notes in Mathematics, vol. 17, Springer-Verlag, Berlin-New York, 1966. MR**0199449****16.**Pencho P. Petrushev,*Approximation by ridge functions and neural networks*, SIAM J. Math. Anal.**30**(1999), no. 1, 155–189 (electronic). MR**1646689**, 10.1137/S0036141097322959**17.**Daniel Potts, Gabriele Steidl, and Manfred Tasche,*Fast algorithms for discrete polynomial transforms*, Math. Comp.**67**(1998), no. 224, 1577–1590. MR**1474655**, 10.1090/S0025-5718-98-00975-2**18.**Elias M. Stein,*Interpolation in polynomial classes and Markoff’s inequality*, Duke Math. J.**24**(1957), 467–476. MR**0091368****19.**Elias M. Stein and Guido Weiss,*Introduction to Fourier analysis on Euclidean spaces*, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR**0304972****20.**Gábor Szegő,*Orthogonal polynomials*, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. MR**0372517****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