The Geometry of Nodes in a Positive Quadrature on the Sphere

  • Manfred Reimer
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 137)

Abstract

Assume the integral I,
$$IF: = \int\limits_{{S^{r - 1}}} {F(x)d\omega (x),F \in C({S^{r - 1}}),} $$
(1)
ω standard measure on the unit sphere S r−1 in ℝ r , r≥3,is approximated by the quadrature Î,
$$\widehat IF: = \sum\limits_{j = 1}^M {{A_j}F\left( {{t_j}} \right),\quad F \in C\left( {{S^{r - 1}}} \right),} $$
(2)
with nodes t 1,…,t M ∈ S r-1 and weights A j >0 and satisfying
$$\widehat IF = IFforallF \in {\Bbb P}_\mu ^r,$$
(3)
µ r the space of all spherical polynomials of degree µ, µ ∈ ℕ. What can be said about the geometry of the nodes on the basis of this information?

Keywords

Unit Sphere Orthogonal Polynomial Quadrature Rule Positive Weight Golden Ratio 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Reimer: Spherical polynomial approximations: A survey. In: Advances in Multivariate Approximation, W. Haußmann, K. Jet-ter, M. Reimer (eds.), Wiley—VCH, Berlin 1999, pp. 231–252.Google Scholar
  2. [2]
    M. Reimer: Hyperinterpolation on the sphere at the minimal projection order, J. Approx. Theory 104 (2000), 272–286.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    G. N. Watson: A treatise on the theory of Bessel functions, 2nd ed. Cambridge Univ. Press, Cambridge 1966.MATHGoogle Scholar
  4. [4]
    V. Yudin: Covering a sphere and extremal properties of orthogonal polynomials, Discrete Math. Appl. 5 (1995), 371–379.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2001

Authors and Affiliations

  • Manfred Reimer
    • 1
  1. 1.Fachbereich MathematikUniversität DortmundDortmundGermany

Personalised recommendations