Approximation tools and decay rates for eigenvalues of integral operators on a general setting

Abstract

We provide the characterization of the Peetre-type K-functional on a compact two-point homogeneous space in terms of the rate of approximation of a family of multipliers operators. This extends the well known results on the spherical setting. The characterization is employed to prove that an abstract Holder condition or finite order of differentiability assumption on generating positive kernels of integral operators implies a sharp decay rates for their eigenvalues sequences. Consequently, sharp upper bounds for the Kolmogorov n-width of unit balls in reproducing kernel Hilbert space are obtained.