Sequences of $m$-term deviations in Hilbert space

TL;DR

This paper investigates the behavior of m-term deviations in Hilbert spaces, establishing a dichotomy in their decay rates and constructing universal dictionaries in infinite dimensions.

Contribution

It proves a dichotomy for the decay of m-term deviations and constructs universal dictionaries in infinite-dimensional Hilbert spaces.

Findings

Deviations either decay exponentially or arbitrarily slowly.

Common dictionaries are not universal for all decreasing sequences.

Universal dictionaries exist only in infinite-dimensional Hilbert spaces.

Abstract

Let $D$ be a dictionary in a Hilbert space $H$, that is, a set of unit elements whose linear combinations are dense in $H$. We consider the least $m$-term deviation $\sigma_m(x)$ of an element $x\in H$: this is the distance of $x$ from the set of all $m$-term linear combinations of elements of $D$. We prove a dichotomy result: for any dictionary $D$, either the sequence $\{\sigma_m(x)\}_{m=0}^{\infty}$ decreases exponentially for every $x\in H$, or the rate of convergence $\sigma_m(x)\to 0$ can be arbitrarily slow. We seek universal dictionaries realizing all strictly decreasing null sequences as sequences of $m$-term deviations. All commonly used dictionaries turn out not to be universal. In particular, the least rational deviations in Hardy space $H^2$ do not form certain strictly monotone null sequences. There are no universal dictionaries in finite dimensional Hilbert spaces. We construct a universal dictionary in every infinite dimensional Hilbert space.