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)


Assume the integral I,
$$IF: = \int\limits_{{S^{r - 1}}} {F(x)d\omega (x),F \in C({S^{r - 1}}),} $$
ω standard measure on the unit sphere S r−1 in ℝ r , r≥3,is approximated by the quadrature Î,
$$\widehat IF: = \sum\limits_{j = 1}^M {{A_j}F\left( {{t_j}} \right),\quad F \in C\left( {{S^{r - 1}}} \right),} $$
with nodes t 1,…,t M ∈ S r-1 and weights A j >0 and satisfying
$$\widehat IF = IFforallF \in {\Bbb P}_\mu ^r,$$
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?


Unit Sphere Orthogonal Polynomial Quadrature Rule Positive Weight Golden Ratio 
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Copyright information

© Springer Basel AG 2001

Authors and Affiliations

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

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