On the Construction of Spherical Designs

Authors

  • L.C. Leal Junior
  • V.A. Menegatto

DOI:

https://doi.org/10.5540/tema.2007.08.03.0423

Abstract

We study special subsets of the unit sphere in Rm, m 2, the socalled spherical designs in the literature. Among other things we introduce a new equivalence for the concept and investigate the construction of designs through rotations of Rm and projections over the equator of the sphere.

References

[1] R. Askey, “Orthogonal Polynomials and Special Functions”, Philadelphia: Society for Industrial and Applied Mathematics, 1975, Regional Conference Series in Applied Mathematics, 21, 1974.

S. Axler, P. Bourdon, W. Ramey, “Harmonic Function Theory”, Graduate Texts in Mathematics, 137. Springer-Verlag, New York, 1992.

L. Carlitz, The product of two ultraspherical polynomials, Proc. Glasgow Math. Assoc. 5 (1961), 76-79.

P. Delsarte, J.M. Goethals, J.J. Seidel, Spherical codes and designs, Geometriae Dedicata, 6, No. 3 (1977), 363-388.

J.M. Goethals, J.J., Seidel, Spherical designs. Relations between combinatorics and other parts of mathematics (Proc. Sympos. Pure Math., Ohio State Univ., Columbus, Ohio, 1978), pp. 255–272, Proc. Sympos. Pure Math., XXXIV, Amer. Math. Soc., Providence, R.I., 1979.

J.M. Goethals, J.J. Seidel, The football, Nieuw Arch. Wisk. (3) 29, No. 1 (1981), 50-58.

H. Groemer, “Geometric Applications of Fourier Series and Spherical Harmonics”. Encyclopedia of Mathematics and its Applications, vol. 61, Cambridge University Press, Cambridge, 1996.

E. Hylleraas, Linearization of products of Jacobi polinomials, Math. Scand., 10 (1962), 189-200.

C. M¨uller, “Analysis of Spherical Symmetries in Euclidean Spaces”, Applied Mathematical Sciences, 129. Springer-Verlag, New York, 1998.

E.M. Stein, G. Weiss, “Introduction to Fourier Analysis on Euclidean Spaces”, Princeton Mathematical Series, No. 32. Princeton University Press, Princeton, N.J., 1971.

G. Szeg˝o, “Orthogonal Polynomials”, American Mathematical Society Colloquium Publications, v. 23. American Mathematical Society, New York, 1939.

V.A. Yudin, Rotation of spherical designs. (Russian) Problemy Peredachi Informatsii 36, No. 3 (2000), 39-45; translation in Probl. Inf. Transm. 36, No. 3 (2000), 230-236.

Published

2007-06-01

How to Cite

Leal Junior, L., & Menegatto, V. (2007). On the Construction of Spherical Designs. Trends in Computational and Applied Mathematics, 8(3), 423–432. https://doi.org/10.5540/tema.2007.08.03.0423

Issue

Section

Original Article