Function values are enough for L2-approximation: Part II

David Krieg,Mario Ullrich

doi:10.1016/j.jco.2021.101569

Abstract

In the first part we have shown that, for L2-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 linear widths 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.

Highlights

Let F be a set of complex-valued functions on a set D such that, for all x ∈ D, point evaluation δx : F → C, f → f (x) is continuous with respect to some metric dF on F
L2 which is the worst-case error of an optimal linear algorithm that uses at most n function values
We want to compare the power of function values with the power of arbitrary linear information for L2-approximation with linear algorithms

Summary

Introduction

Let F be a set of complex-valued functions on a set D such that, for all x ∈ D, point evaluation δx : F → C, f → f (x) is continuous with respect to some metric dF on F. L2 which is the worst-case error of an optimal linear algorithm that uses at most n function values. These numbers are sometimes called sampling widths of F. We. FUNCTION VALUES ARE ENOUGH – PART II want to compare the sampling widths with the linear widths of F , defined by (2). FUNCTION VALUES ARE ENOUGH – PART II want to compare the sampling widths with the linear widths of F , defined by (2) This is the worst-case error of an optimal linear algorithm that uses at most n linear functionals as information. We want to compare the power of function values ( called standard information) with the power of arbitrary linear information for L2-approximation with linear algorithms. We want to relate en and an for general function classes F

Objectives

Conclusion