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A discontinuous Galerkin leastsquares finite element method for solving Fisher's equation
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, Springeld, MO 65810, United States 
References:
[1] 
J. Bourgain and J. Lindenstrauss, Distribution of points on spheres and approximation by zonotopes,, Israel J. Math., 64 (1988), 25. 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. Google Scholar 
[3] 
G. Brown and F. Dai, Approximation of smooth functions on compact twopoint homogeneous spaces,, J. Funct. Anal., 220 (2005), 401. Google Scholar 
[4] 
F. Dai, Jacksontype inequality for doubling weights on the sphere,, Constr. Approx., 24 (2006), 91. Google Scholar 
[5] 
F. Dai, On generalized hyperinterpolation on the sphere,, Proc. Amer. Math. Soc., 134 (2006), 2931. 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. Google Scholar 
[7] 
R. A. DeVore and G. G. Lorentz, Constructive Approximation,, SpringerVerlag, (1993). Google Scholar 
[8] 
Z. Ditzian, Jacksontype inequality on the sphere,, Acta Math. Hungar., 102 (2004), 1. 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, (1998), 117. Google Scholar 
[10] 
W. Freeden, T. Gervens and M. Schreiner, Constructive Approximation on The Sphere,, Calderon Press, (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. Google Scholar 
[12] 
W. Freeden, V. Michel and M. Stenger, Multiscale signaltonoise 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. Google Scholar 
[14] 
W. Freeden, V. Michel and H. Nutz, Satellitetosatellite tracking and satellite gravity gradiometry (Advanced techniques for highresolution geopotential field determination),, J. Engin. Math., 43 (2002), 19. Google Scholar 
[15] 
Q. T. Le Gia, F. J. Narcowich, J. D. Ward and H. Wendland, Continuous and discrete leastsquares approximation by radial basis functions on spheres,, J. Approx. Theory, 143 (2006), 124. Google Scholar 
[16] 
S. M. Gomes, A. K. Kushpel and J. Levesley, Approximation in $L_2$ Sobolev spaces on the 2sphere by quasiinterpolation,, J. Four. Anal. Appl., 7 (2001), 283. Google Scholar 
[17] 
S. Hubbert and T. M. Morton, A Duchon framework for the sphere,, J. Approx. Theory, 129 (2004), 28. 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. Google Scholar 
[19] 
K. Jetter, J. Stöckler and J. D. Ward, Error estimates for scattered data interpolation on spheres,, Math. Comp., 68 (1999), 743. Google Scholar 
[20] 
A. K. Kushpel and J. Levesley, QuasiInterpolation on the 2sphere using radial polynomials,, J. Approx. Theory, 102 (2000), 141. Google Scholar 
[21] 
P. Leopardi, Diameter bounds for equal area partitions of the unit sphere,, Electron. Trans. Numer. Anal., 35 (2009), 1. 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. 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. Google Scholar 
[24] 
S. B. Lin, F. L. Cao, X. Y. Chang and Z. B. Xu, A general radial quasiinterpolation operator on the sphere,, J. Approx. Theory, 164 (2012), 1402. 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. Google Scholar 
[26] 
H. N. Mhaskar, F. J. Narcowich and J. D. Ward, Spherical MarcinkiewiczZymund inequalities and positive quadrature,, Math. Comp., 70 (2001), 1113. Google Scholar 
[27] 
H. Q. Minh, Reproducing kernel Hilbert spaces in learning theory,, Ph. D. Thesis in Mathematics, (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. Google Scholar 
[29] 
T. M. Morton and M. Neamtu, Error bounds for solving pseudodifferential equations on spheres by colloctation with zonal kernels,, J. Approx. Theory, 114 (2002), 242. Google Scholar 
[30] 
C. Müller, Spherical Harmonics,, Lecture Notes in Mathematics, (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. 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. Google Scholar 
[33] 
M. Neamtu and L. L. Schumaker, On the approximation order of splines on spherical triangulations,, Adv. Comp. Math., 21 (2004), 3. Google Scholar 
[34] 
E. A. Rakhmanov, E. B. Saff and Y. M. Zhou, Electrons on the sphere,, Computational methods and function theory 1994 (Penang), (1994), 293. Google Scholar 
[35] 
X. Sun and Z. Chen, Spherical basis functions and uniform distribution of points on spheres,, J. Approx. Theory, (2008), 186. Google Scholar 
[36] 
Y. T. Tsai and Z. C. Shih, Allfrequency precomputed radiance transfer using spherical radial basis functions and clustered tensor approximation,, ACM Trans. Graph., 25 (2006), 967. 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. Google Scholar 
[38] 
G. Wagner, On a new method for constructing good point sets on spheres,, Discrete Comput. Geom., 9 (1993), 111. 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. Google Scholar 
[40] 
K. Wang and L. Li, Harmonic Analysis and Approximation on The Unit Sphere,, Science Press, (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. 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. Google Scholar 
[3] 
G. Brown and F. Dai, Approximation of smooth functions on compact twopoint homogeneous spaces,, J. Funct. Anal., 220 (2005), 401. Google Scholar 
[4] 
F. Dai, Jacksontype inequality for doubling weights on the sphere,, Constr. Approx., 24 (2006), 91. Google Scholar 
[5] 
F. Dai, On generalized hyperinterpolation on the sphere,, Proc. Amer. Math. Soc., 134 (2006), 2931. 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. Google Scholar 
[7] 
R. A. DeVore and G. G. Lorentz, Constructive Approximation,, SpringerVerlag, (1993). Google Scholar 
[8] 
Z. Ditzian, Jacksontype inequality on the sphere,, Acta Math. Hungar., 102 (2004), 1. 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, (1998), 117. Google Scholar 
[10] 
W. Freeden, T. Gervens and M. Schreiner, Constructive Approximation on The Sphere,, Calderon Press, (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. Google Scholar 
[12] 
W. Freeden, V. Michel and M. Stenger, Multiscale signaltonoise 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. Google Scholar 
[14] 
W. Freeden, V. Michel and H. Nutz, Satellitetosatellite tracking and satellite gravity gradiometry (Advanced techniques for highresolution geopotential field determination),, J. Engin. Math., 43 (2002), 19. Google Scholar 
[15] 
Q. T. Le Gia, F. J. Narcowich, J. D. Ward and H. Wendland, Continuous and discrete leastsquares approximation by radial basis functions on spheres,, J. Approx. Theory, 143 (2006), 124. Google Scholar 
[16] 
S. M. Gomes, A. K. Kushpel and J. Levesley, Approximation in $L_2$ Sobolev spaces on the 2sphere by quasiinterpolation,, J. Four. Anal. Appl., 7 (2001), 283. Google Scholar 
[17] 
S. Hubbert and T. M. Morton, A Duchon framework for the sphere,, J. Approx. Theory, 129 (2004), 28. 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. Google Scholar 
[19] 
K. Jetter, J. Stöckler and J. D. Ward, Error estimates for scattered data interpolation on spheres,, Math. Comp., 68 (1999), 743. Google Scholar 
[20] 
A. K. Kushpel and J. Levesley, QuasiInterpolation on the 2sphere using radial polynomials,, J. Approx. Theory, 102 (2000), 141. Google Scholar 
[21] 
P. Leopardi, Diameter bounds for equal area partitions of the unit sphere,, Electron. Trans. Numer. Anal., 35 (2009), 1. 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. 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. Google Scholar 
[24] 
S. B. Lin, F. L. Cao, X. Y. Chang and Z. B. Xu, A general radial quasiinterpolation operator on the sphere,, J. Approx. Theory, 164 (2012), 1402. 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. Google Scholar 
[26] 
H. N. Mhaskar, F. J. Narcowich and J. D. Ward, Spherical MarcinkiewiczZymund inequalities and positive quadrature,, Math. Comp., 70 (2001), 1113. Google Scholar 
[27] 
H. Q. Minh, Reproducing kernel Hilbert spaces in learning theory,, Ph. D. Thesis in Mathematics, (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. Google Scholar 
[29] 
T. M. Morton and M. Neamtu, Error bounds for solving pseudodifferential equations on spheres by colloctation with zonal kernels,, J. Approx. Theory, 114 (2002), 242. Google Scholar 
[30] 
C. Müller, Spherical Harmonics,, Lecture Notes in Mathematics, (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. 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. Google Scholar 
[33] 
M. Neamtu and L. L. Schumaker, On the approximation order of splines on spherical triangulations,, Adv. Comp. Math., 21 (2004), 3. Google Scholar 
[34] 
E. A. Rakhmanov, E. B. Saff and Y. M. Zhou, Electrons on the sphere,, Computational methods and function theory 1994 (Penang), (1994), 293. Google Scholar 
[35] 
X. Sun and Z. Chen, Spherical basis functions and uniform distribution of points on spheres,, J. Approx. Theory, (2008), 186. Google Scholar 
[36] 
Y. T. Tsai and Z. C. Shih, Allfrequency precomputed radiance transfer using spherical radial basis functions and clustered tensor approximation,, ACM Trans. Graph., 25 (2006), 967. 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. Google Scholar 
[38] 
G. Wagner, On a new method for constructing good point sets on spheres,, Discrete Comput. Geom., 9 (1993), 111. 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. Google Scholar 
[40] 
K. Wang and L. Li, Harmonic Analysis and Approximation on The Unit Sphere,, Science Press, (2000). Google Scholar 
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