Data Assimilation and Sampling in Banach spaces

TL;DR

This paper develops a general theory for approximating functions in Banach spaces from measurements, focusing on approximation sets relevant to applications like PDEs and signal processing, and provides bounds and algorithms for near-optimal recovery.

Contribution

It introduces a comprehensive framework for recovering approximation sets in Banach spaces, connecting with classical concepts and extending results in sampling and data assimilation.

Findings

Provides tight bounds on optimal performance for approximation sets.

Develops algorithms for near-optimal function recovery.

Connects recovery problems with Banach space theory concepts.

Abstract

This paper studies the problem of approximating a function $f$ in a Banach space $X$ from measurements $l_j(f)$, $j=1,\dots,m$, where the $l_j$ are linear functionals from $X^*$. Most results study this problem for classical Banach spaces $X$ such as the $L_p$ spaces, $1\le p\le \infty$, and for $K$ the unit ball of a smoothness space in $X$. Our interest in this paper is in the model classes $K=K(\epsilon,V)$, with $\epsilon>0$ and $V$ a finite dimensional subspace of $X$, which consists of all $f\in X$ such that $dist(f,V)_X\le \epsilon$. These model classes, called {\it approximation sets}, arise naturally in application domains such as parametric partial differential equations, uncertainty quantification, and signal processing. A general theory for the recovery of approximation sets in a Banach space is given. This theory includes tight a priori bounds on optimal performance, and algorithms for finding near optimal approximations. We show how the recovery problem for approximation sets is connected with well-studied concepts in Banach space theory such as liftings and the angle between spaces. Examples are given that show how this theory can be used to recover several recent results on sampling and data assimilation.