Kolmogorov Widths on the Sphere via Eigenvalue Estimates for Hölderian Integral Operators

  • Thaís JordãoEmail author
  • Valdir A. Menegatto


Approximation processes in the reproducing kernel Hilbert space associated to a continuous kernel on the unit sphere \(S^m\) in the Euclidean space \({\mathbb {R}}^{m+1}\) are known to depend upon the Mercer’s expansion of the compact and self-adjoint \(L^2(S^m)\)-operator associated to the kernel. The estimation of the Kolmogorov nth width of the unit ball of the reproducing kernel Hilbert space in \(L^2(S^m)\) and the identification of the so-called optimal subspace usually suffice. These Kolmogorov widths can be computed through the eigenvalues of the integral operator associated to the kernel. This paper provides sharp upper bounds for the Kolmogorov widths in the case in which the kernel satisfies an abstract Hölder condition. In particular, we follow the opposite direction usually considered in the literature, that is, we estimate the widths from decay rates for the sequence of eigenvalues of the integral operator.


Kolmogorov widths decay rates eigenvalues Hölder condition sphere 

Mathematics Subject Classification

41A36 41A46 42A16 45C05 47B34 47G10 



  1. 1.
    Azevedo, D., Menegatto, V.A.: Sharp estimates for eigenvalues of integral operators generated by dot product kernels on the sphere. J. Approx. Theory 177, 57–68 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Berens, H., Butzer, P.L., Pawelke, S.: Limitierungsverfahren von Riehen mehrdimensionaler Kugelfunktionen und deren Saturationsverhalten. Publ. Res. Inst. Math. Sci. Ser. A 4, 201–268 (1968/1969)Google Scholar
  3. 3.
    Cucker, F., Zhou, D.-X.: Learning Theory: An Approximation Theory Viewpoint. With a foreword by Stephen Smale, Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  4. 4.
    Dai, F., Ditzian, Z.: Combinations of multivariate averages. J. Approx. Theory 131(2), 268–283 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    DeVore, R.A., Lorentz, G.G.: Constructive Approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303. Springer, Berlin (1993)Google Scholar
  6. 6.
    Ditzian, Z., Runovskii, K.: Averages on caps of \(S^{d-1}\). J. Math. Anal. Appl. 248(1), 260–274 (2000)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dunkl, C.F.: Operators and harmonic analysis on the sphere. Trans. Am. Math. Soc. 125, 250–263 (1966)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Folland, G.B.: Real Analysis. Modern Techniques and Their Applications, Pure and Applied Mathematics, 2nd edn. Wiley, New York (1999)zbMATHGoogle Scholar
  9. 9.
    Jordão, T., Menegatto, V.A.: Estimates for Fourier sums and eigenvalues of integral operators via multipliers on the sphere. Proc. Am. Math. Soc. 144(1), 269–283 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jordão, T., Menegatto, V.A.: Jackson Kernels: a tool for analysing the decay of eigenvalues sequences of integral operator on the sphere. Math. Inequal. Appl. 18(4), 1483–1500 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Lizorkin, P.I., Nikol’skii, S.M.: A theorem concerning approximation on the sphere. Anal. Math. 9(3), 207–221 (1983)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Luke, Y.L.: Inequalities for generalized hypergeometric functions. J. Approx. Theory 5(1), 41–65 (1972)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Minh, H.Q.: Some properties of Gaussian reproducing kernel Hilbert spaces and their implications for function approximation and learning theory. Constr. Approx. 32, 307–338 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Morimoto, M.: Analytic Functionals on the Sphere, Translations of Mathematical Monographs, vol. 178. American Mathematical Society, Providence (1998)CrossRefGoogle Scholar
  15. 15.
    Pinkus, A.: n-Widths in Approximation Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 7. Springer, Berlin (1985)Google Scholar
  16. 16.
    Platonov, S.S.: Approximations on compact symmetric spaces of rank 1 (Russian). Mat. Sb. 188(5), 113–130 (1997); translation in Sb. Math. 188(5), 753–769 (1997)Google Scholar
  17. 17.
    Schaback, R., Wendland, H.: Approximation by positive definite kernels. In: Buhmann, M., Mache, D. (eds.) Advanced Problems in Constructive Approximation, International Series in Numerical Mathematics, vol. 142, pp. 203–221. Birkhäuser, Basel (2002)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Departamento de MatemáticaICMC-USP - São CarlosSão CarlosBrazil

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