A Short Note on the Comparison of Interpolation Widths, Entropy Numbers, and Kolmogorov Widths
Provides theoretical insights for approximation theory, but is incremental as it extends known comparisons to specific function spaces.
This paper compares Kolmogorov and entropy numbers for compact operators between Hilbert and Banach spaces, applies results to RKHS-to-L∞ embeddings, and shows a gap of order n^{1/2} between interpolation and Kolmogorov widths, with a similar gap in multi-dimensional Sobolev cases.
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 $L_\infty(μ)$. 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.