Full Recovery from Point Values: an Optimal Algorithm for Chebyshev Approximability Prior

TL;DR

This paper introduces a practical algorithm for optimal recovery of functions from pointwise samples within Chebyshev spaces, ensuring minimal worst-case error in function reconstruction.

Contribution

It provides the first explicit construction of a linear optimal recovery algorithm for functions in Chebyshev spaces of dimension at least three.

Findings

Algorithm guarantees full recovery from point values.

Applicable to Chebyshev spaces containing constants.

Enhances practical implementation of optimal recovery methods.

Abstract

Given pointwise samples of an unknown function belonging to a certain model set, one seeks in Optimal Recovery to recover this function in a way that minimizes the worst-case error of the recovery procedure. While it is often known that such an optimal recovery procedure can be chosen to be linear, e.g. when the model set is based on approximability by a subspace of continuous functions, a construction of the procedure is rarely available. This note uncovers a practical algorithm to construct a linear optimal recovery map when the approximation space is a Chevyshev space of dimension at least three and containing the constant functions.