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

Discretizing spherical integrals and its applications

Shaobo Lin 1, , Xingping Sun 2, and  Zongben Xu 1,

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.
Keywords: spherical convolution, Fredholm integral equation, Spherical basis function, approximation, sphere..
Mathematics Subject Classification: Primary: 41A25, 41A05; Secondary: 41A6.
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.  Google Scholar

[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.  Google Scholar

[3]

G. Brown and F. Dai, Approximation of smooth functions on compact two-point homogeneous spaces, J. Funct. Anal., 220 (2005), 401-423.  Google Scholar

[4]

F. Dai, Jackson-type inequality for doubling weights on the sphere, Constr. Approx., 24 (2006), 91-112.  Google Scholar

[5]

F. Dai, On generalized hyperinterpolation on the sphere, Proc. Amer. Math. Soc., 134 (2006), 2931-2941.  Google Scholar

[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.  Google Scholar

[7]

R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin, 1993.  Google Scholar

[8]

Z. Ditzian, Jackson-type inequality on the sphere, Acta Math. Hungar., 102 (2004), 1-35.  Google Scholar

[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.  Google Scholar

[10]

W. Freeden, T. Gervens and M. Schreiner, Constructive Approximation on The Sphere, Calderon Press, Oxford, 1998.  Google Scholar

[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. Google Scholar

[12]

W. Freeden, V. Michel and M. Stenger, Multiscale signal-to-noise thresholding,, Berichte der Arbeitsgruppe Technomathematik, ().   Google Scholar

[13]

W. Freeden and S. Perevrzev, Spherical Tikhonov regularization wavelets in satellite gravity gradiometry with random noise, J. Geod., 74 (2001), 730-736. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

S. Hubbert and T. M. Morton, A Duchon framework for the sphere, J. Approx. Theory, 129 (2004), 28-57.  Google Scholar

[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.  Google Scholar

[19]

K. Jetter, J. Stöckler and J. D. Ward, Error estimates for scattered data interpolation on spheres, Math. Comp., 68 (1999), 743-747.  Google Scholar

[20]

A. K. Kushpel and J. Levesley, Quasi-Interpolation on the 2-sphere using radial polynomials, J. Approx. Theory, 102 (2000), 141-154.  Google Scholar

[21]

P. Leopardi, Diameter bounds for equal area partitions of the unit sphere, Electron. Trans. Numer. Anal., 35 (2009), 1-16.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

H. N. Mhaskar, F. J. Narcowich and J. D. Ward, Spherical Marcinkiewicz-Zymund inequalities and positive quadrature, Math. Comp., 70 (2001), 1113-1130.  Google Scholar

[27]

H. Q. Minh, Reproducing kernel Hilbert spaces in learning theory, Ph. D. Thesis in Mathematics, Brown University, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, Vol. 17, Springer, Berlin, 1966.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

M. Neamtu and L. L. Schumaker, On the approximation order of splines on spherical triangulations, Adv. Comp. Math., 21 (2004), 3-20.  Google Scholar

[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.  Google Scholar

[35]

X. Sun and Z. Chen, Spherical basis functions and uniform distribution of points on spheres, J. Approx. Theory, 151 (2008), 186-207.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

G. Wagner, On a new method for constructing good point sets on spheres, Discrete Comput. Geom., 9 (1993), 111-129.  Google Scholar

[39]

G. Wahba, Convergence rates of certain approximate solutions to Fredholm integral equations of the first kind, J. Approx. Theory, 7 (1973), 167-185.  Google Scholar

[40]

K. Wang and L. Li, Harmonic Analysis and Approximation on The Unit Sphere, Science Press, Beijing, 2000. Google Scholar

show all references

References:
[1]

J. Bourgain and J. Lindenstrauss, Distribution of points on spheres and approximation by zonotopes, Israel J. Math., 64 (1988), 25-31.  Google Scholar

[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.  Google Scholar

[3]

G. Brown and F. Dai, Approximation of smooth functions on compact two-point homogeneous spaces, J. Funct. Anal., 220 (2005), 401-423.  Google Scholar

[4]

F. Dai, Jackson-type inequality for doubling weights on the sphere, Constr. Approx., 24 (2006), 91-112.  Google Scholar

[5]

F. Dai, On generalized hyperinterpolation on the sphere, Proc. Amer. Math. Soc., 134 (2006), 2931-2941.  Google Scholar

[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.  Google Scholar

[7]

R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin, 1993.  Google Scholar

[8]

Z. Ditzian, Jackson-type inequality on the sphere, Acta Math. Hungar., 102 (2004), 1-35.  Google Scholar

[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.  Google Scholar

[10]

W. Freeden, T. Gervens and M. Schreiner, Constructive Approximation on The Sphere, Calderon Press, Oxford, 1998.  Google Scholar

[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. Google Scholar

[12]

W. Freeden, V. Michel and M. Stenger, Multiscale signal-to-noise thresholding,, Berichte der Arbeitsgruppe Technomathematik, ().   Google Scholar

[13]

W. Freeden and S. Perevrzev, Spherical Tikhonov regularization wavelets in satellite gravity gradiometry with random noise, J. Geod., 74 (2001), 730-736. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

S. Hubbert and T. M. Morton, A Duchon framework for the sphere, J. Approx. Theory, 129 (2004), 28-57.  Google Scholar

[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.  Google Scholar

[19]

K. Jetter, J. Stöckler and J. D. Ward, Error estimates for scattered data interpolation on spheres, Math. Comp., 68 (1999), 743-747.  Google Scholar

[20]

A. K. Kushpel and J. Levesley, Quasi-Interpolation on the 2-sphere using radial polynomials, J. Approx. Theory, 102 (2000), 141-154.  Google Scholar

[21]

P. Leopardi, Diameter bounds for equal area partitions of the unit sphere, Electron. Trans. Numer. Anal., 35 (2009), 1-16.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

H. N. Mhaskar, F. J. Narcowich and J. D. Ward, Spherical Marcinkiewicz-Zymund inequalities and positive quadrature, Math. Comp., 70 (2001), 1113-1130.  Google Scholar

[27]

H. Q. Minh, Reproducing kernel Hilbert spaces in learning theory, Ph. D. Thesis in Mathematics, Brown University, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, Vol. 17, Springer, Berlin, 1966.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

M. Neamtu and L. L. Schumaker, On the approximation order of splines on spherical triangulations, Adv. Comp. Math., 21 (2004), 3-20.  Google Scholar

[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.  Google Scholar

[35]

X. Sun and Z. Chen, Spherical basis functions and uniform distribution of points on spheres, J. Approx. Theory, 151 (2008), 186-207.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

G. Wagner, On a new method for constructing good point sets on spheres, Discrete Comput. Geom., 9 (1993), 111-129.  Google Scholar

[39]

G. Wahba, Convergence rates of certain approximate solutions to Fredholm integral equations of the first kind, J. Approx. Theory, 7 (1973), 167-185.  Google Scholar

[40]

K. Wang and L. Li, Harmonic Analysis and Approximation on The Unit Sphere, Science Press, Beijing, 2000. Google Scholar

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