Taiwanese Journal of Mathematics

The Covering Number for Some Mercer Kernel Hilbert Spaces on the Unit Sphere

Baohuai Sheng, Jianli Wang, and Zhixiang Chen

Full-text: Open access


In the present paper, we estimate the covering number for some reproducing kernel Hilbert spaces on the unit sphere. Both the upper bounds and the lower bounds are provided.

Article information

Taiwanese J. Math., Volume 15, Number 3 (2011), 1325-1340.

First available in Project Euclid: 18 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 41A46: Approximation by arbitrary nonlinear expressions; widths and entropy 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

reproducing kernel Hilbert spaces covering number spherical harmonics


Sheng, Baohuai; Wang, Jianli; Chen, Zhixiang. The Covering Number for Some Mercer Kernel Hilbert Spaces on the Unit Sphere. Taiwanese J. Math. 15 (2011), no. 3, 1325--1340. doi:10.11650/twjm/1500406302. https://projecteuclid.org/euclid.twjm/1500406302

Export citation


  • [3.] V. N. Vapnik, Statistical Learning Theory, New York, John Wiley and Sons, Inc., 1998.
  • [4.] F. Cucker and S. Smale, Best choices for regularization parameters in learning theory: on the bias-variance problem, Found. Comput. Math., 2 (2002), 413-428.
  • [5.] R. C. Williamson, A. J. Smola and B. Schölkopf, Generalization performance of regularization networks and support vector machine via entropy numbers of compact operators, IEEE Trans. Inform. Theory, 47(6) (2001), 2516-2532.
  • [6.] Y. Guo, P. L. Bartlett, J. Shawe-Taylor, and R. C. Williamson, Covering numbers for support vector machines, IEEE Trans. Inform. Theory, 48 (2002), 239-250.
  • [7.] D. X. Zhou, The covering number in learning theory, J. Complexity, 18 (2002), 739-767.
  • [8.] D. X. Zhou, Capacity of reproducing kernel spaces in learning theory, IEEE Trans. Inform. Theory, 49(7) (2003), 1743-1752.
  • [9.] H. W. Sun and D. X. Zhou, Reproducing kernel Hilbert spaces associated with analytic translation-invariant Mercer kernels, J. Fourier Anal. Appl., 14 (2008), 89-101.
  • [10.] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404.
  • [11.] C. Carmeli, E. De Vito and A. Toigo, Vector valued reproducing kernel Hilbert spaces of integrable functions and Mercer theorem, Analysis and Applications, 4(4) (2006), 377-408.
  • [12.] H. Sun, Mercer theorem for RKHS on noncompact sets, J. Complexity, 21 (2005), 337-349.
  • [13.] F. Cucker and D. X. Zhou, Learning Theory: An Approximation Theory Viewpoint, Cambridge University, New York, 2007.
  • [15.] H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics, Cambridge University Press, New York, 1996.
  • [17.] C. Muller, Spherical Harmonic, Springer-Verlag, Berlin, 1966.
  • [18.] H. Wendland, Scattered Aata Approximation, Cambridge University Press, Cambridge, 2005.
  • [19.] R. S. Womersly and I. H. Sloan, How good can polynomial interpolation on the sphere be? Advances in Computational Mathematics, 14(3) (2001), 195-226.
  • [20.] F. Dai, Multivariate polynomial inequalities with respect to doubling weights and $A_{\infty}$ weights, J. Functional Analysis, 235 (2006), 137-170.
  • [21.] B. H. Sheng and G. Z. Zhou, A way of constructing spherical approximation operators, Journal of Systems Science and Mathematical Sciences, 28(4) (2008), 456-467, (in Chinese).
  • [22.] B. H. Sheng, J. L. Wang and S. P. Zhou, A way of constructing zonal translation network operators with linear bounded operators, Taiwanese J. Mathematics, 12(1) (2008), 77-92.
  • [23.] B. H. Sheng, On the degree of approximation by spherical translations, Acta Mathematicae Applicatae Sinica, English Series, 22(4) (2006), 671-680.
  • \item[24.] H. N. Mhaskar, F. J. Narcowich and J. D. Ward, Approximation properties of zonal function networks using scattered data on the sphere, Advances in Computational Mathematics, 11 (1999), 121-127.
  • [25.] H. N. Mhaskar, F. J. Narcowich and D. Ward, Zonal function network frames on the sphere, Neural Networks, 16 (2003), 183-203.
  • [26.] B. H. Sheng, The Jackson theorem of approximation by zonal translation networks on the unit sphere, Advances in Mathematics $($Beijing$)$, 35(3) (2006), 325-335, (in Chinese).
  • [27.] H. N. Mhaskar, Weighted quadrature formulas and approximation by zonal function networks on the sphere, J. Complexity, 22 (2006), 348-370.
  • [28.] G. Brown and F. Dai, Approximation of smooth functions on compact two-point homogeneous spaces, J. Functional Analysis, 220 (2005), 401-423.
  • [29.] G. Brown, F. Dai and Y. S. Sun, Kolmogorov width of classes of smooth functions on the sphere $S^{d-1}$, Journal of Complexity, 18 (2002), 1001-1023.
  • [30.] G. Brown, F. Dai and Y. S. Sun, Kolmogorov width of classes of smooth functions on the sphere $S^{d-1}$, Advances in Mathematics $($Beijing$)$, 31(2) (2002), 181-184.
  • [31.] H. N. Mhaskar, F. J. Narcowich, N. Sivakumar and J. D. Ward, Approximation with interpolatory constraints, Proc. Amer. Math. Soc., 130(5) (2001), 1355-1361.
  • [32.] K. Jetter, J. Stöckler and J. D. Ward, Error estimates for scattered data interpolation on spheres, Mathematics of Computation, 68(226) (1999), 733-747.