Cubature formula for spherical basis function networks

Abstract

Some mathematical models in geophysics and graphic processing need to compute integrals with scattered data on the sphere. Thus cubature formula plays an important role in computing these spherical integrals. This paper is devoted to establishing an exact positive cubature formula for spherical basis function networks. The authors give an existence proof of the exact positive cubature formula for spherical basis function networks, and prove that the cubature points needed in the cubature formula are not larger than the number of the scattered data.

Keywords

Cubature formula / Spherical basis function / Scattered data

References

[1] Lam P. M., Ho T. Y., Leung C. S. All-frequency lighting with multiscale spherical radial basis functions. IEEE Trans. Visual. Comput. Graph., 2010, 16: 43-56

[2] Tsai Y. T., Chang C. C., Jiang Q. Z. Importance sampling of products from illumination and BRDF using spherical radial basis functions. Visual. Comput., 2008, 24: 817-826

[3] Freeden W., Gervens T., Schreiner M.. Constructive Approximation on the Sphere, 1998, Oxford: Calderon Press

[4] Freeden W., Michel V., Nutz H.. Satellite-to-satellite tracking and satellite gravity gradiometry (Advanced techniques for high-resolution geopotential field determination). J. Engin. Math., 2002, 43: 19-56

[5] Hesse K., Sloan I. H.. Worst-case errors in a Sobolev space setting for cubature over the sphere S ². Bull. Austral. Math. Soc., 2005, 71: 81-105

[6] Hesse K., Sloan I. H.. Optimal lower bounds for cubature error on the sphere S ². J. Complex, 2005, 21: 790-803

[7] Hesse K.. A lower bound for the worst-case cubature error on spheres of arbitrary dimension. Numer. Math., 2006, 103: 413-433

[8] Mhaskar H. N., Narcowich F. J., Ward J. D.. Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature. Math. Comput., 2000, 70: 1113-1130

[9] Brown G., Dai F.. Approximation of smooth functions on compact two-point homogeneous spaces. J. Funct. Anal., 2005, 220: 401-423

[10] Dai F., Wang H. P.. Positive cubature formulas and Marcinkiewicz-Zygmund inequalities on spherical caps. Constr. Approx., 2010, 31: 1-36

[11] Dai F.. Multivariate polynomial inequalities with respect to doubling weights and A _∞ weights. J. Approx. Theory, 2006, 235: 137-170

[12] Brauchart J. S., Hesse K.. Numerical integration over spheres of arbitrary dimension. Constr. Approx., 2007, 25: 41-71

[13] Sun X., Chen Z.. Spherical basis functions and uniform distribution of points on spheres. J. Approx. Theory, 2008, 151: 186-207

[14] Schoenberg I. J.. Positive definite functions on spheres. Duke Math. J., 1942, 9: 96-108

[15] Jetter K., Stöckler J., Ward J. D.. Error estimates for scattered data interpolation. Math. Comput., 1999, 68: 743-747

[16] Levesley J., Sun X.. Approximation in rough native spaces by shifts of smooth kernels on spheres. J. Approx. Theory, 2005, 133: 269-283

[17] Narcowich F. J., Ward J. D.. Scattered data interpolation on spheres: Error estimates and locally supported basis functions. SIAM J. Math. Anal., 2002, 33: 1393-1410

[18] Narcowich F. J., Sun X., Ward J. D. Direct and inverse Sobolev error estimates for scattered data interpolation via spherical basis functions. Found. Comput. Math., 2007, 7: 369-370

[19] Rudin W.. Functional Analysis, 1973, New York: McGraw-Hill

[20] Borwein P. B., Erdélyi T.. Polynomials and Polynomial Inequalities, Graduate Texts in Mathematics, 1995, New York: Springer-Verlag