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2013, 2013(special): 499-514. doi: 10.3934/proc.2013.2013.499

## Discretizing spherical integrals and its applications

 1 Institute for Information and System Sciences, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China, China 2 Department of Mathematics, Missouri State University, Spring eld, MO 65810, United States

Received  October 2012 Revised  February 2013 Published  November 2013

Efficient discretization of spherical integrals is required in many numerical methods associated with solving differential and integral equations on spherical domains. In this paper, we discuss a discretization method that works particularly well with convolutions of spherical integrals. We utilize this method to construct spherical basis function networks, which are subsequently employed to approximate the solutions of a variety of differential and integral equations on spherical domains. We show that, to a large extend, the approximation errors depend only on the smoothness of the spherical basis function. We also derive error estimates of the pertinent approximation schemes. As an application, we discuss a Galerkin type solutions for spherical Fredholm integral equations of the first kind, and obtain rates of convergence of the spherical basis function networks to the solutions of these equations.
Citation: Shaobo Lin, Xingping Sun, Zongben Xu. Discretizing spherical integrals and its applications. Conference Publications, 2013, 2013 (special) : 499-514. doi: 10.3934/proc.2013.2013.499
##### References:
 [1] J. Bourgain and J. Lindenstrauss, Distribution of points on spheres and approximation by zonotopes, Israel J. Math., 64 (1988), 25-31. [2] J. Bourgain and J. Lindenstrauss, Approximating the ball by a Minkowski sum of segments with equal length, Discrete Comput. Geom., 9 (1993), 131-144. [3] G. Brown and F. Dai, Approximation of smooth functions on compact two-point homogeneous spaces, J. Funct. Anal., 220 (2005), 401-423. [4] F. Dai, Jackson-type inequality for doubling weights on the sphere, Constr. Approx., 24 (2006), 91-112. [5] F. Dai, On generalized hyperinterpolation on the sphere, Proc. Amer. Math. Soc., 134 (2006), 2931-2941. [6] F. Dai and Y. Xu, Moduli of smoothness and approximation on the unit sphere and the unit ball, Adv. Math., 224 (2010), 1233-1310. [7] R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin, 1993. [8] Z. Ditzian, Jackson-type inequality on the sphere, Acta Math. Hungar., 102 (2004), 1-35. [9] G. E. Fasshauer and L. L. Schumaker, Scattered data fitting on the sphere, in Mathematical Methods for Curves and Surfaces II (M. Dælen, T. Lyche, and L. L. Schumaker, eds.), Vanderbilt University Press, Nashville, TN, 1998, pp. 117-166. [10] W. Freeden, T. Gervens and M. Schreiner, Constructive Approximation on The Sphere, Calderon Press, Oxford, 1998. [11] W. Freeden and V. Michel, Constructive approximation and numerical methods in geodetic research today - an attempt at a categorization based on an uncertainty principle, J. Geod., 73 (1999), 452-465. [12] W. Freeden, V. Michel and M. Stenger, Multiscale signal-to-noise thresholding, Berichte der Arbeitsgruppe Technomathematik, Rep 224, Universität Kaiserslautern. [13] W. Freeden and S. Perevrzev, Spherical Tikhonov regularization wavelets in satellite gravity gradiometry with random noise, J. Geod., 74 (2001), 730-736. [14] W. Freeden, V. Michel and H. Nutz, Satellite-to-satellite tracking and satellite gravity gradiometry (Advanced techniques for high-resolution geopotential field determination), J. Engin. Math., 43 (2002), 19-56. [15] Q. T. Le Gia, F. J. Narcowich, J. D. Ward and H. Wendland, Continuous and discrete least-squares approximation by radial basis functions on spheres, J. Approx. Theory, 143 (2006), 124-133. [16] S. M. Gomes, A. K. Kushpel and J. Levesley, Approximation in $L_2$ Sobolev spaces on the 2-sphere by quasi-interpolation, J. Four. Anal. Appl., 7 (2001), 283-295. [17] S. Hubbert and T. M. Morton, A Duchon framework for the sphere, J. Approx. Theory, 129 (2004), 28-57. [18] S. Hubbert and T. M. Morton, $L_p$-error estimates for radial basis function interpolation on the sphere, J. Approx. Theory, 129 (2004), 58-77. [19] K. Jetter, J. Stöckler and J. D. Ward, Error estimates for scattered data interpolation on spheres, Math. Comp., 68 (1999), 743-747. [20] A. K. Kushpel and J. Levesley, Quasi-Interpolation on the 2-sphere using radial polynomials, J. Approx. Theory, 102 (2000), 141-154. [21] P. Leopardi, Diameter bounds for equal area partitions of the unit sphere, Electron. Trans. Numer. Anal., 35 (2009), 1-16. [22] J. Levesley and X. Sun, Approximation in rough native spaces by shifts of smooth kernels on spheres, J. Approx. Theory, 133 (2005), 269-283. [23] S. B. Lin, F. L. Cao and Z. B. Xu, A convergence rate for approximate solutions of Fredholm integral equations of the first kind, Positivity, 16 (2012), 641-652. [24] S. B. Lin, F. L. Cao, X. Y. Chang and Z. B. Xu, A general radial quasi-interpolation operator on the sphere, J. Approx. Theory, 164 (2012), 1402-1414. [25] H. N. Mhaskar, F. J. Narcowich and J. D. Ward, Approximation properties of zonal function networks using scattered data on the sphere, Adv. Comp. Math., 11 (1999), 121-137. [26] H. N. Mhaskar, F. J. Narcowich and J. D. Ward, Spherical Marcinkiewicz-Zymund inequalities and positive quadrature, Math. Comp., 70 (2001), 1113-1130. [27] H. Q. Minh, Reproducing kernel Hilbert spaces in learning theory, Ph. D. Thesis in Mathematics, Brown University, 2006. [28] H. Q. Minh, Some Properties of Gaussian reproducing kernel Hilbert spaces and their implications for function approximation and learning theory, Constr. Approx., 32 (2010), 307-338. [29] T. M. Morton and M. Neamtu, Error bounds for solving pseudo-differential equations on spheres by colloctation with zonal kernels, J. Approx. Theory, 114 (2002), 242-268. [30] C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, Vol. 17, Springer, Berlin, 1966. [31] F. J. Narcowich and J. D. Ward, Scattered data interpolation on spheres: Error estimates and locally supported basis functions, SIAM J. Math. Anal., 33 (2002), 1393-1410. [32] F. J. Narcowich, X. P. Sun, J. D. Ward and H. Wendland, Direct and inverse sobolev error estimates forscattered data interpolation via spherical basis functions, Found. Comp. Math., 7 (2007), 369-370. [33] M. Neamtu and L. L. Schumaker, On the approximation order of splines on spherical triangulations, Adv. Comp. Math., 21 (2004), 3-20. [34] E. A. Rakhmanov, E. B. Saff and Y. M. Zhou, Electrons on the sphere, Computational methods and function theory 1994 (Penang), 293-309, Ser. Approx. Decompos. 5, World Sci. Publ., River Edge, NJ, 1995. [35] X. Sun and Z. Chen, Spherical basis functions and uniform distribution of points on spheres, J. Approx. Theory, 151 (2008), 186-207. [36] Y. T. Tsai and Z. C. Shih, All-frequency precomputed radiance transfer using spherical radial basis functions and clustered tensor approximation, ACM Trans. Graph., 25 (2006), 967-976. [37] Y. T. Tsai, C. C. Chang, Q. Z. Jiang and S. C. Weng, Importance sampling of products from illumination and BRDF using spherical radial basis functions, Visual Comp., 24 (2008), 817-826. [38] G. Wagner, On a new method for constructing good point sets on spheres, Discrete Comput. Geom., 9 (1993), 111-129. [39] G. Wahba, Convergence rates of certain approximate solutions to Fredholm integral equations of the first kind, J. Approx. Theory, 7 (1973), 167-185. [40] K. Wang and L. Li, Harmonic Analysis and Approximation on The Unit Sphere, Science Press, Beijing, 2000.

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##### References:
 [1] J. Bourgain and J. Lindenstrauss, Distribution of points on spheres and approximation by zonotopes, Israel J. Math., 64 (1988), 25-31. [2] J. Bourgain and J. Lindenstrauss, Approximating the ball by a Minkowski sum of segments with equal length, Discrete Comput. Geom., 9 (1993), 131-144. [3] G. Brown and F. Dai, Approximation of smooth functions on compact two-point homogeneous spaces, J. Funct. Anal., 220 (2005), 401-423. [4] F. Dai, Jackson-type inequality for doubling weights on the sphere, Constr. Approx., 24 (2006), 91-112. [5] F. Dai, On generalized hyperinterpolation on the sphere, Proc. Amer. Math. Soc., 134 (2006), 2931-2941. [6] F. Dai and Y. Xu, Moduli of smoothness and approximation on the unit sphere and the unit ball, Adv. Math., 224 (2010), 1233-1310. [7] R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin, 1993. [8] Z. Ditzian, Jackson-type inequality on the sphere, Acta Math. Hungar., 102 (2004), 1-35. [9] G. E. Fasshauer and L. L. Schumaker, Scattered data fitting on the sphere, in Mathematical Methods for Curves and Surfaces II (M. Dælen, T. Lyche, and L. L. Schumaker, eds.), Vanderbilt University Press, Nashville, TN, 1998, pp. 117-166. [10] W. Freeden, T. Gervens and M. Schreiner, Constructive Approximation on The Sphere, Calderon Press, Oxford, 1998. [11] W. Freeden and V. Michel, Constructive approximation and numerical methods in geodetic research today - an attempt at a categorization based on an uncertainty principle, J. Geod., 73 (1999), 452-465. [12] W. Freeden, V. Michel and M. Stenger, Multiscale signal-to-noise thresholding, Berichte der Arbeitsgruppe Technomathematik, Rep 224, Universität Kaiserslautern. [13] W. Freeden and S. Perevrzev, Spherical Tikhonov regularization wavelets in satellite gravity gradiometry with random noise, J. Geod., 74 (2001), 730-736. [14] W. Freeden, V. Michel and H. Nutz, Satellite-to-satellite tracking and satellite gravity gradiometry (Advanced techniques for high-resolution geopotential field determination), J. Engin. Math., 43 (2002), 19-56. [15] Q. T. Le Gia, F. J. Narcowich, J. D. Ward and H. Wendland, Continuous and discrete least-squares approximation by radial basis functions on spheres, J. Approx. Theory, 143 (2006), 124-133. [16] S. M. Gomes, A. K. Kushpel and J. Levesley, Approximation in $L_2$ Sobolev spaces on the 2-sphere by quasi-interpolation, J. Four. Anal. Appl., 7 (2001), 283-295. [17] S. Hubbert and T. M. Morton, A Duchon framework for the sphere, J. Approx. Theory, 129 (2004), 28-57. [18] S. Hubbert and T. M. Morton, $L_p$-error estimates for radial basis function interpolation on the sphere, J. Approx. Theory, 129 (2004), 58-77. [19] K. Jetter, J. Stöckler and J. D. Ward, Error estimates for scattered data interpolation on spheres, Math. Comp., 68 (1999), 743-747. [20] A. K. Kushpel and J. Levesley, Quasi-Interpolation on the 2-sphere using radial polynomials, J. Approx. Theory, 102 (2000), 141-154. [21] P. Leopardi, Diameter bounds for equal area partitions of the unit sphere, Electron. Trans. Numer. Anal., 35 (2009), 1-16. [22] J. Levesley and X. Sun, Approximation in rough native spaces by shifts of smooth kernels on spheres, J. Approx. Theory, 133 (2005), 269-283. [23] S. B. Lin, F. L. Cao and Z. B. Xu, A convergence rate for approximate solutions of Fredholm integral equations of the first kind, Positivity, 16 (2012), 641-652. [24] S. B. Lin, F. L. Cao, X. Y. Chang and Z. B. Xu, A general radial quasi-interpolation operator on the sphere, J. Approx. Theory, 164 (2012), 1402-1414. [25] H. N. Mhaskar, F. J. Narcowich and J. D. Ward, Approximation properties of zonal function networks using scattered data on the sphere, Adv. Comp. Math., 11 (1999), 121-137. [26] H. N. Mhaskar, F. J. Narcowich and J. D. Ward, Spherical Marcinkiewicz-Zymund inequalities and positive quadrature, Math. Comp., 70 (2001), 1113-1130. [27] H. Q. Minh, Reproducing kernel Hilbert spaces in learning theory, Ph. D. Thesis in Mathematics, Brown University, 2006. [28] H. Q. Minh, Some Properties of Gaussian reproducing kernel Hilbert spaces and their implications for function approximation and learning theory, Constr. Approx., 32 (2010), 307-338. [29] T. M. Morton and M. Neamtu, Error bounds for solving pseudo-differential equations on spheres by colloctation with zonal kernels, J. Approx. Theory, 114 (2002), 242-268. [30] C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, Vol. 17, Springer, Berlin, 1966. [31] F. J. Narcowich and J. D. Ward, Scattered data interpolation on spheres: Error estimates and locally supported basis functions, SIAM J. Math. Anal., 33 (2002), 1393-1410. [32] F. J. Narcowich, X. P. Sun, J. D. Ward and H. Wendland, Direct and inverse sobolev error estimates forscattered data interpolation via spherical basis functions, Found. Comp. Math., 7 (2007), 369-370. [33] M. Neamtu and L. L. Schumaker, On the approximation order of splines on spherical triangulations, Adv. Comp. Math., 21 (2004), 3-20. [34] E. A. Rakhmanov, E. B. Saff and Y. M. Zhou, Electrons on the sphere, Computational methods and function theory 1994 (Penang), 293-309, Ser. Approx. Decompos. 5, World Sci. Publ., River Edge, NJ, 1995. [35] X. Sun and Z. Chen, Spherical basis functions and uniform distribution of points on spheres, J. Approx. Theory, 151 (2008), 186-207. [36] Y. T. Tsai and Z. C. Shih, All-frequency precomputed radiance transfer using spherical radial basis functions and clustered tensor approximation, ACM Trans. Graph., 25 (2006), 967-976. [37] Y. T. Tsai, C. C. Chang, Q. Z. Jiang and S. C. Weng, Importance sampling of products from illumination and BRDF using spherical radial basis functions, Visual Comp., 24 (2008), 817-826. [38] G. Wagner, On a new method for constructing good point sets on spheres, Discrete Comput. Geom., 9 (1993), 111-129. [39] G. Wahba, Convergence rates of certain approximate solutions to Fredholm integral equations of the first kind, J. Approx. Theory, 7 (1973), 167-185. [40] K. Wang and L. Li, Harmonic Analysis and Approximation on The Unit Sphere, Science Press, Beijing, 2000.
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