Cubature Formulas on Spheres

In problems that deal with data, as frequently encountered in applied mathematics, it is often necessary to discretize integrals to obtain discrete processes of approximation. Cubature formulas, a synonym for numerical integration formulas, are essential tools for discretizing integrals. In contrast to the one-variable case, fundamental problems of cubature formulas in several variables are still open, including those on the sphere. In this chapter, we discuss several aspects of cubature formulas on the sphere.

[1]  E. Stein.  Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. , 1970 .

[2]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[3]  V. I. Lebedev,et al.  Spherical quadrature formulas exact to orders 25–29 , 1977 .

[4]  J. Seidel,et al.  Spherical codes and designs , 1977 .

[5]  S. G. Hoggar,et al.  t-Designs in Projective Spaces , 1982, Eur. J. Comb..

[6]  Kendall Atkinson,et al.  Numerical integration on the sphere , 1982, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[7]  P. Seymour,et al.  Averaging sets: A generalization of mean values and spherical designs , 1984 .

[8]  M. Taibleson.  Estimates for Finite Expansions of Gegenbauer and Jacobi Polynomials , 1985 .

[9]  D. Hajela.  Construction techniques for some thin sets in duals of compact abelian groups , 1986 .

[10]  J. Lindenstrauss,et al.  Distribution of points on spheres and approximation by zonotopes , 1988 .

[11]  N. J. A. Sloane,et al.  Sphere Packings, Lattices and Groups , 1987, Grundlehren der mathematischen Wissenschaften.

[12]  B. Reznick.  Sums of Even Powers of Real Linear Forms , 1992 .

[13]  Yu. I. Lyubich,et al.  Isometric embed-dings between classical Banach spaces, cubature formulas, and spherical designs , 1993 .

[14]  J. Korevaar,et al.  Chebyshev-type quadrature on multidimensional domains , 1994 .

[15]  J. J. Seidel,et al.  Isometric embeddings and geometric designs , 1994, Discret. Math..

[16]  N. J. A. Sloane,et al.  McLaren’s improved snub cube and other new spherical designs in three dimensions , 1996, Discret. Comput. Geom..

[17]  E. Saff,et al.  Distributing many points on a sphere , 1997 .

[18]  Vladimir A. Yudin.  Lower bounds for spherical designs , 1997 .

[19]  R. Caflisch.  Monte Carlo and quasi-Monte Carlo methods , 1998, Acta Numerica.

[20]  Yuan Xu,et al.  Invariant Cubature Formulae for Spheres and Balls by Combinatorial Methods , 2000, SIAM J. Numer. Anal..

[21]  Hrushikesh Narhar Mhaskar,et al.  Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature , 2001, Math. Comput..

[22]  Yuan Xu,et al.  Constructing fully symmetric cubature formulae for the sphere , 2001, Math. Comput..

[23]  Ian H. Sloan,et al.  Extremal Systems of Points and Numerical Integration on the Sphere , 2004, Adv. Comput. Math..

[24]  Hrushikesh Narhar Mhaskar,et al.  Local quadrature formulas on the sphere , 2004, J. Complex..

[25]  Feng Dai,et al.  Approximation of smooth functions on compact two-point homogeneous spaces , 2005, math/0510007.

[26]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[27]  Yeol Je Cho,et al.  Topological Degree Theory and Applications , 2006 .

[28]  Paul C. Leopardi.  A PARTITION OF THE UNIT SPHERE INTO REGIONS OF EQUAL AREA AND SMALL DIAMETER , 2006 .

[29]  Pencho Petrushev,et al.  Localized Tight Frames on Spheres , 2006, SIAM J. Math. Anal..

[30]  F. Dai,et al.  Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps , 2007, math/0703768.

[31]  Eiichi Bannai,et al.  A survey on spherical designs and algebraic combinatorics on spheres , 2009, Eur. J. Comb..

[32]  Andriy Bondarenko,et al.  Optimal asymptotic bounds for spherical designs , 2010, 1009.4407.

[33]  Xiaojun Chen,et al.  Well Conditioned Spherical Designs for Integration and Interpolation on the Two-Sphere , 2010, SIAM J. Numer. Anal..