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

Abstract

Let $(\Omega,\mathscr F,\mathbb P) $ be a probability space and let $(\mathscr F_n)$ be a binary filtration, i.e. exactly one atom of $\mathscr F_{n-1}$ is divided into two atoms of $\mathscr F_n$ without any restriction on their respective measures. Additionally, denote the collection of atoms corresponding to this filtration by $\mathscr A$. Let $S \subset L^\infty(\Omega)$ be a finite-dimensional linear subspace, having an additional stability property on atoms $\mathscr A$. For these data, we consider the two dictionaries $\mathscr C = \{ f \cdot \chi_A: f \in S, A \in \mathscr A\}$ and $\Phi$, a local orthonormal system generated by $S$ and the filtration $(\mathscr F_n)$. We are interested in approximation spaces corresponding to the best $n$-term approximation in $L^p$ for $1<p<\infty$ by elements of $\mathscr C$ and $\Phi$, respectively. It is known that in the classical Haar case, i.e. when $S = {\rm span} (\chi_{[0,1]})$ and the binary filtration $(\mathscr F_n)$ is dyadic (that is, an atom $A \in \mathscr A$ 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 $\mathscr A$ and rings $\mathscr R = \{ A \setminus B: A, B \in \mathscr A, B \subset A \}\setminus \mathscr A$. We specialize this general result to some examples of interest, including general Haar systems and spaces $S$ consisting of (multivariate) polynomials.