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We prove a characterization for the Peetre type K-functional on M, a compact twopoint homogeneous space, in terms the rate of approximation of a family of multipliers operator defined to this purpose. This extends the well known results on the spherical setting. The characterization is employed to show that an abstract Hölder condition or finite order of differentiability condition imposed on kernels generating certain operators implies a sharp decay rates for their eigenvalues sequences. The latest is employed to obtain estimates for the Kolmogorov n-width of unit balls in Reproducing Kernel Hilbert Space (RKHS).

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