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

  • Thaís JordãoEmail author
  • Valdir A. Menegatto
Article
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Abstract

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.

Keywords

Kolmogorov widths decay rates eigenvalues Hölder condition sphere 

Mathematics Subject Classification

41A36 41A46 42A16 45C05 47B34 47G10 

Notes

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

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

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