Numerical integration over spheres of arbitrary dimension

Constructive Approximation

Volumn 25, Issue 1, 2007, Pages 41-71

  Brauchart, Johann S ^a   Hesse, Kerstin ^b  

^a VANDERBILT UNIVERSITY   (United States)

^b UNIVERSITY OF NEW SOUTH WALES   (Australia)

Author keywords

Cubature; Jacobi series; Numerical integration rules on the sphere; Reproducing kernel; Sobolev space; Sphere; Worst case error

Indexed keywords

EID: 33749644556     PISSN: 01764276     EISSN: 14320940     Source Type: Journal    
DOI: 10.1007/s00365-006-0629-4     Document Type: Article

References (16)

1. M. ABRAMOWITZ, I. A. STEGUN, editors (1970): Handbook of Mathematical Functions (with Formulas, Graphs, and Mathematical Tables). New York: Dover.
2. d-1 J. Complexity, 18:1001-1023.
3. A. ERDÉLYI, editor; W. MAGNUS, F. OBERHETTINGER, F. G. TRICOMI, research associates (1954): Tables of Integral Transforms. California Institute of Technology, Bateman Manuscript Project, Vol. II. New York: McGraw-Hill.
4. K. HESSE (to appear): A lower bound for the worst-case cubature error on spheres of arbitrary dimension. Numer. Math.
5. 2 in Sobolev spaces of arbitrary order. J. Approx. Theory.
6. 2. Bull. Austral. Math. Soc., 71:81-105.
7. 2. J. Complexity, 21:790-803.
8. S. LANG (1993): Real and Functional Analysis, 3rd ed. New York: Springer-Verlag.
9. Y. L. LUKE (1969): The Special Functions and Their Approximations, Vol. I. Mathematics in Science and Engineering, Vol. 53. New York: Academic Press.
10. H. N. MHASKAR, F. J. NARCOWICH, J. D. WARD (2001): Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature. Math. Comp., 70:1113-1130. Corrigendum: Math. Comp., 71:453-454.
11. C. MÜLLER (1966): Spherical Harmonics. Lecture Notes in Mathematics, Vol. 17. New York: Springer-Verlag.
12. P. P. PETRUSHEV (1998): Approximation by ridge functions and neural networks. SIAM J. Math. Anal., 30(1):155-189.
13. M. REIMER (1990): Constructive Theory of Multivariate Functions. Mannheim: BI Wissenschaftsverlag.
14. M. REIMER (2000): Hyperinterpolation on the sphere at the minimal projection order. J. Approx. Theory, 104(2):272-286.
15. A. H. STROUD (1971): Approximate Calculation of Multiple Integrals. Englewood Cliffs, NJ: Prentice-Hall.
16. G. SZEGO (1975): Orthogonal Polynomials, 4th ed. American Mathematical Society Colloquium Publications, Vol. 23. Providence, RI: American Mathematical Society.