Spherical designs, discrepancy and numerical integration
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- by Peter J. Grabner and Robert F. Tichy PDF
- Math. Comp. 60 (1993), 327-336 Request permission
Abstract:
A spherical design is a point configuration on the sphere, which yields exact equal-weight quadrature formulae for polynomials up to a given degree. Until now only very specific constructions for spherical designs are known. We establish connections to spherical cap discrepancy and show some general discrepancy bounds. Furthermore, we reformulate the problem of constructing designs as an optimization problem and develop an algorithm for finding ’practical designs’.References
- József Beck and William W. L. Chen, Irregularities of distribution, Cambridge Tracts in Mathematics, vol. 89, Cambridge University Press, Cambridge, 1987. MR 903025, DOI 10.1017/CBO9780511565984
- Bruno Buchberger, Applications of Gröbner bases in nonlinear computational geometry, Mathematical aspects of scientific software (Minneapolis, Minn., 1986/87) IMA Vol. Math. Appl., vol. 14, Springer, New York, 1988, pp. 59–87. MR 938107, DOI 10.1007/978-1-4684-7074-1_{3}
- P. Delsarte, J. M. Goethals, and J. J. Seidel, Spherical codes and designs, Geometriae Dedicata 6 (1977), no. 3, 363–388. MR 485471, DOI 10.1007/bf03187604 H. Fischer, Beiträge zur Computerzahlentheorie: Lineare Rekursionen, Designs und Diophantische Gleichungen, Thesis, Techn. Univ. Vienna, 1992.
- Walter Gautschi, Advances in Chebyshev quadrature, Numerical analysis (Proc. 6th Biennial Dundee Conf., Univ. Dundee, Dundee, 1975) Lecture Notes in Math., Vol. 506, Springer, Berlin, 1976, pp. 100–121. MR 0468117
- C. D. Godsil, Polynomial spaces, Proceedings of the Oberwolfach Meeting “Kombinatorik” (1986), 1989, pp. 71–88. MR 974814, DOI 10.1016/0012-365X(88)90134-3
- Peter J. Grabner, Erdős-Turán type discrepancy bounds, Monatsh. Math. 111 (1991), no. 2, 127–135. MR 1100852, DOI 10.1007/BF01332351
- Edmund Hlawka, Beiträge zur Theorie der Gleichverteilung und ihren Anwendungen. I. Einleitung, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 197 (1988), no. 1-3, 1–94 (German). MR 1004374
- Loo Keng Hua and Yuan Wang, Applications of number theory to numerical analysis, Springer-Verlag, Berlin-New York; Kexue Chubanshe (Science Press), Beijing, 1981. Translated from the Chinese. MR 617192
- L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0419394
- A. Lubotzky, R. Phillips, and P. Sarnak, Hecke operators and distributing points on the sphere. I, Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S149–S186. Frontiers of the mathematical sciences: 1985 (New York, 1985). MR 861487, DOI 10.1002/cpa.3160390710
- A. D. McLaren, Optimal numerical integration on a sphere, Math. Comp. 17 (1963), 361–383. MR 159418, DOI 10.1090/S0025-5718-1963-0159418-2
- Claus Müller, Spherical harmonics, Lecture Notes in Mathematics, vol. 17, Springer-Verlag, Berlin-New York, 1966. MR 0199449
- I. P. Natanson, Konstruktive Funktionentheorie, Akademie-Verlag, Berlin, 1955 (German). MR 0069915
- D. J. Newman and H. S. Shapiro, Jackson’s theorem in higher dimensions, On Approximation Theory (Proceedings of Conference in Oberwolfach, 1963) Birkhäuser, Basel, 1964, pp. 208–219. MR 0182828 J. J. Seidel, Integration over spheres, Discrete Geometry (A. Florian, ed.), Salzburg, 1985, pp. 233-242.
- P. D. Seymour and Thomas Zaslavsky, Averaging sets: a generalization of mean values and spherical designs, Adv. in Math. 52 (1984), no. 3, 213–240. MR 744857, DOI 10.1016/0001-8708(84)90022-7 G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1939.
- Gerold Wagner, On averaging sets, Monatsh. Math. 111 (1991), no. 1, 69–78. MR 1089385, DOI 10.1007/BF01299278
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Math. Comp. 60 (1993), 327-336
- MSC: Primary 11K45; Secondary 65C05
- DOI: https://doi.org/10.1090/S0025-5718-1993-1155573-5
- MathSciNet review: 1155573