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

TL;DR

This paper extends previous results showing that for $L_2$-approximation, linear algorithms based on function values are nearly as effective as all linear algorithms, now including separable Banach spaces and other function classes.

Contribution

It generalizes the equivalence of function-value-based algorithms to separable Banach spaces and broader function classes, beyond Hilbert spaces.

Findings

Function values suffice for $L_2$-approximation in broader settings.

Linear algorithms based on function values achieve near-optimal convergence rates.

Extension of previous Hilbert space results to Banach spaces and other function classes.

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.