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Monatshefte für Mathematik

, Volume 111, Issue 1, pp 69–78 | Cite as

On averaging sets

  • Gerold Wagner
  • Bodo Volkmann
Article

Abstract

For any set Φ={f1,f2,...,fs} ofC3-functions on the interval [−1, 1], and for any weight functionw(x) satisfyingL1w(x)L2(1−|x|)β(L1,L2>0, β≥0) and\(\int_{ - 1}^1 {w(x)dx = 1} \), we give a constructive proof for the existence of quadrature formulas of the type
$$\frac{1}{n}\sum\limits_{j = 1}^n {f_\mu (x_j )} \int_{ - 1}^1 {f_\mu (x)w(x)dx} {\text{ }}(\mu = 1,2,{\text{ }} \ldots ,s)$$
for sufficiently largen, −1<x1<x2<...<xn<1. Assuming the orthonormality of the derivativesf′1,f′2,...,f′s with respect to the weight functionw(x), we obtain explicit bounds for the numbern of interpolation points for which such formulas exist. As an application to combinatorics, we prove the existence ofd-dimensional sphericalt-designs of sizen for eachn>cd·t12d4,cd>0 a constant.

Keywords

Quadrature Formula Interpolation Point Constructive Proof Explicit Bound 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    Conway, J. H., Sloane, N. J. A.: Sphere Packings, Lattices and Groups. New York-Heidelberg-Berlin: Springer. 1988.Google Scholar
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    Delsarte, P., Goethals, J. M., Seidel, J. J.: Spherical codes and designs. Geom. Dedic.6, 363–388 (1977).Google Scholar
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    Erdelyi, A. et al.: Higher Transcendental Functions. New York-Toronto-London: McGraw-Hill. 1953.Google Scholar
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    Natanson, I. P.: Constructive Function Theory. New York: Frederick Ungar Publ. Company. 1965.Google Scholar
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    Seymour, P. D., Zaslavsky, T.: Averaging Sets. Adv. of Math.52, 213–240 (1984).Google Scholar
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    Wagner, G.: On a new method for constructing good point sets on spheres. To appear in Discrete and Computational Geometry.Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Gerold Wagner
    • 1
    • 2
  • Bodo Volkmann
    • 1
  1. 1.Mathematisches Institut AUniversität StuttgartStuttgartFederal Republic of Germany
  2. 2.Faculty of Science and TechnologyKeio UniversityYokohamaJapan

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