Properties of local orthonormal systems, Part II: Geometric characterization of Bernstein inequalities

Abstract

Let $(\Omega ,\mathcal{F},\mathbb{P})$ be a probability space and let $({\mathcal{F}}_n)$ be a binary filtration, i.e. exactly one atom of ${\mathcal{F}}_{n-1}$ is divided into two atoms of ${\mathcal{F}}_n$ without any restriction on their respective measures. Additionally, denote the collection of atoms corresponding to this filtration by $\mathcal{A}$. Let $S\subset L^{\infty }(\Omega )$ be a finite-dimensional linear subspace, having an additional stability property on atoms $\mathcal{A}$. For these data, we consider the two dictionaries $\mathcal{C}=\{f\cdot {\chi }_A:f\in S,A\in \mathcal{A}\}$ and $\Phi$, a local orthonormal system generated by $S$ and the filtration $({\mathcal{F}}_n)$. We are interested in approximation spaces corresponding to the best $n$-term approximation in $L^p$ for $1<p<\infty$ by elements of $\mathcal{C}$ and $\Phi$, respectively. It is known that in the classical Haar case, i.e. when $S=\mathrm{span}({\chi }_{[0,1]})$ and the binary filtration $({\mathcal{F}}_n)$ is dyadic (that is, an atom $A\in \mathcal{A}$ is divided into two new atoms of equal measure), those approximation spaces coincide, cf. [P. Petrushev, Multivariate $n$-term rational and piecewise polynomial approximation, J. Approx. Theory 121(1), 2003]. This motivates us to ask the question whether this is true in the general setting described above. The answer to this question is governed by the validity of a specific Bernstein type inequality. The main result of this paper is a geometric characterization of this type of Bernstein inequality, i.e. a characterization in terms of the behaviour of functions from the space $S$ on atoms $\mathcal{A}$ and rings $\mathcal{R}=\{A\setminus B:A,B\in \mathcal{A},B\subset A\}\setminus \mathcal{A}$. We specialize this general result to some examples of interest, including general Haar systems and spaces $S$ consisting of (multivariate) polynomials.