Explicit spherical designs. (English) Zbl 1487.05044
Summary: Since the introduction of the notion of spherical designs by P. Delsarte et al. [Geom. Dedicata 6, 363–388 (1977; Zbl 0376.05015)], finding explicit constructions of spherical designs had been an open problem. Most existence proofs of spherical designs rely on the topology of the spheres, hence their constructive versions are only computable, but not explicit. That is to say that these constructions can only give algorithms that produce approximations of spherical designs up to arbitrary given precision, while they are not able to give any spherical designs explicitly. Inspired by recent work on rational designs, i.e. designs consisting of rational points, we generalize the known construction of spherical designs that uses interval designs with Gegenbauer weights, and give an explicit formula of spherical designs of arbitrary given strength on the real unit sphere of arbitrary given dimension.
MSC:
| 05B30 | Other designs, configurations |
Citations:
Zbl 0376.05015References:
| [1] | Abramowitz, Milton; Stegun, Irene A., Handbook of mathematical functions: with formulas, graphs, and mathematical tables, 55 (1964), Courier Corporation · Zbl 0171.38503 |
| [2] | Bajnok, Bela, Construction of spherical \(t\)-designs, Geom. Dedicata, 43, 2, 167-179 (1992) · Zbl 0765.05032 · doi:10.1007/BF00147866 |
| [3] | Bondarenko, Andriy; Radchenko, Danylo; Viazovska, Maryna, Optimal asymptotic bounds for spherical designs, Ann. of Math. (2), 178, 2, 443-452 (2013) · Zbl 1270.05026 · doi:10.4007/annals.2013.178.2.2 |
| [4] | Bondarenko, Andriy; Radchenko, Danylo; Viazovska, Maryna, Well-separated spherical designs, Constr. Approx., 41, 1, 93-112 (2015) · Zbl 1314.52020 · doi:10.1007/s00365-014-9238-2 |
| [5] | Bondarenko, Andriy V.; Viazovska, Maryna S., Spherical designs via Brouwer fixed point theorem, SIAM J. Discrete Math., 24, 1, 207-217 (2010) · Zbl 1229.05057 · doi:10.1137/080738313 |
| [6] | Chen, Xiaojun; Frommer, Andreas; Lang, Bruno, Computational existence proofs for spherical \(t\)-designs, Numer. Math., 117, 2, 289-305 (2011) · Zbl 1208.65032 · doi:10.1007/s00211-010-0332-5 |
| [7] | Chen, Xiaojun; Womersley, Robert S., Existence of solutions to systems of underdetermined equations and spherical designs, SIAM J. Numer. Anal., 44, 6, 2326-2341 (2006) · Zbl 1129.65035 · doi:10.1137/050626636 |
| [8] | Cui, Zhen; Xia, Jiacheng; Xiang, Ziqing, Rational designs, Adv. Math., 352, 541-571 (2019) · Zbl 1416.05062 · doi:10.1016/j.aim.2019.06.012 |
| [9] | Delsarte, P.; Goethals, J. M.; Seidel, J. J., Spherical codes and designs, Geometriae Dedicata, 6, 3, 363-388 (1977) · Zbl 0376.05015 · doi:10.1007/bf03187604 |
| [10] | Folland, Gerald B., How to integrate a polynomial over a sphere, Amer. Math. Monthly, 108, 5, 446-448 (2001) · Zbl 1046.26503 · doi:10.2307/2695802 |
| [11] | Gautschi, Walter, On inverses of Vandermonde and confluent Vandermonde matrices, Numer. Math., 4, 117-123 (1962) · Zbl 0108.12501 · doi:10.1007/BF01386302 |
| [12] | Hardin, Ronald H.; Sloane, Neil J. A., McLaren’s improved snub cube and other new spherical designs in three dimensions, Discrete Comput. Geom., 15, 4, 429-441 (1996) · Zbl 0858.05024 · doi:10.1007/BF02711518 |
| [13] | Korevaar, Jacob; Meyers, J. L. H., Spherical Faraday cage for the case of equal point charges and Chebyshev-type quadrature on the sphere, Integral Transform. Spec. Funct., 1, 2, 105-117 (1993) · Zbl 0823.41026 · doi:10.1080/10652469308819013 |
| [14] | Kuperberg, Greg, Special moments, Adv. in Appl. Math., 34, 4, 853-870 (2005) · Zbl 1077.62007 · doi:10.1016/j.aam.2004.11.005 |
| [15] | Rabau, Patrick; Bajnok, Bela, Bounds for the number of nodes in Chebyshev type quadrature formulas, J. Approx. Theory, 67, 2, 199-214 (1991) · Zbl 0751.41026 · doi:10.1016/0021-9045(91)90018-6 |
| [16] | Seymour, Paul D.; Zaslavsky, Thomas, Averaging sets: a generalization of mean values and spherical designs, Adv. in Math., 52, 3, 213-240 (1984) · Zbl 0596.05012 · doi:10.1016/0001-8708(84)90022-7 |
| [17] | Stein, Elias M.; Shakarchi, Rami, Real analysis: measure theory, integration, and Hilbert spaces (2005), Princeton University Press · Zbl 1081.28001 · doi:10.1515/9781400835560 |
| [18] | Venkov, Boris B., Even unimodular extremal lattices, Trudy Mat. Inst. Steklov., 165, 43-48 (1984) · Zbl 0544.10017 |
| [19] | Wagner, Gerold, On averaging sets, Monatsh. Math., 111, 1, 69-78 (1991) · Zbl 0721.65011 · doi:10.1007/BF01299278 |
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