A Short Note on the Comparison of Interpolation Widths, Entropy Numbers, and Kolmogorov Widths

2016/06/17 by Steinwart, Ingo · 1 citation
#41A15 #41A30 #41A45 #41A46 #46E35 #FOS: Mathematics #Functional Analysis (math.FA) #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1606.05500

Abstract

We compare the Kolmogorov and entropy numbers of compact operators mapping from a Hilbert space into a Banach space. We then apply these general findings to embeddings between reproducing kernel Hilbert spaces and L_∞(μ). Here we provide a sufficient condition for a gap of the order n^1/2 between the associated interpolation and Kolmogorov n-widths. Finally, we show that in the multi-dimensional Sobolev case, this gap actually occurs between the Kolmogorov and approximation widths.

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