Removal of the square-root oversampling factor for bounded-kernel RKHSs

Determine whether the factor \(\sqrt{n}\) in the general comparison \(g_{2n}(B_{\mathcal H_K})_\infty\lesssim \sqrt{n}\,c_n(B_{\mathcal H_K})_\infty\) can be removed when \(B_{\mathcal H_K}\) is the unit ball of a reproducing kernel Hilbert space with bounded kernel, without imposing the additional structural or decay assumptions used in the paper.

Background

For a bounded-kernel reproducing kernel Hilbert space $\mathcal H_K$, the Gelfand widths $c_n(B_{\mathcal H_K})_\infty$ benchmark recovery from arbitrary linear information, whereas the linear sampling widths $g_m^{\mathrm{lin}}(B_{\mathcal H_K})_\infty$ restrict recovery to point evaluations. A general comparison available for normed function spaces gives $g_{2n}^{\mathrm{lin}}\lesssim \sqrt{n}\,c_n$.

The paper proves several partial results for RKHS unit balls: decay-rate transfer under polynomial-type assumptions, a square-root bound without decay assumptions, and a direct comparison after logarithmic oversampling. The authors explicitly describe the logarithmic-oversampling result as only a partial answer to (Q2), so the complete removal of the $\sqrt n$ factor in the unrestricted bounded-kernel RKHS setting remains unresolved.