Function values are enough for $L_2$-approximation: Part II

TL;DR

This work extends the relationship between sampling widths and linear widths from Hilbert spaces to general Banach spaces for $L_2$-approximation. It proves that, when the linear widths $(a_n(F,L_2))$ lie in $\ell_p$ for some $0<p<2$, the sampling widths satisfy $e_{cn}(F,L_2)\le c_p\sqrt{\log n}\,\big(\frac{1}{n}\sum_{k\ge n} a_k(F,L_2)^p\big)^{1/p}$, i.e., they share the same polynomial rate as the tail of the linear widths up to a $\sqrt{\log n}$ factor. The approach reduces to a countable dense subset, constructs a weighted least-squares estimator using random samples, and leverages Kadison-Singer to reduce the required number of samples while preserving the bound. While the results generalize to broad function classes and illustrate sharp bounds for Korobov-type spaces, the case $p=2$ remains open, signaling a fundamental limit in extending the theory to all square-summable widths. Overall, the paper clarifies when near-optimal sampling strategies can achieve the same convergence rates as optimal linear information beyond RKHS settings, highlighting the central role of $\ell_p$-tail decay in $a_n$.

Abstract

In the first part we have shown that, for $L_2$-approximation of functions from a separable Hilbert space in the worst-case setting, linear algorithms based on function values are almost as powerful as arbitrary linear algorithms if the approximation numbers are square-summable. That is, they achieve the same polynomial rate of convergence. In this sequel, we prove a similar result for separable Banach spaces and other classes of functions.

Theorems & Definitions (19)

• Theorem 1
• Example 1
• Theorem 2
• proof : Proof of Theorem \ref{['thm:main']}
• Lemma 3
• proof
• Lemma 4
• proof
• Remark 5
• Lemma 6