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

Let $(\Omega ,ℱ,P)$ be a probability space and let ${\left({ℱ}_n\right)}_{n=0}^{\infty }$ be a binary filtration. i.e. exactly one atom of ${ℱ}_{n-1}$ is divided into two atoms of ${ℱ}_n$ without any restriction on their respective measures. Additionally, denote the collection of atoms corresponding to this filtration by $A$. Let $S\subset L^{\infty }\left(\Omega \right)$ be a finite-dimensional linear subspace, having an additional stability property on atoms $A$. For these data, we consider two dictionaries:

• •
  $C=\{f\cdot 1_A:f\in S,A\in A\}$,
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  $\Phi$ – a local orthonormal system generated by $S$ and the filtration ${\left({ℱ}_n\right)}_{n=0}^{\infty }$.

Let $L^p\left(S\right)={\overline{\mathrm{span}}}_{L^p\left(\Omega \right)}C={\overline{\mathrm{span}}}_{L^p\left(\Omega \right)}\Phi$, with $1<p<\infty$. We are interested in approximation spaces $A_q^{\alpha }(L^p\left(S\right),C)$ and $A_q^{\alpha }(L^p\left(S\right),\Phi )$, corresponding to the best $n$-term approximation in $L^p\left(S\right)$ by elements of $C$ and $\Phi$, respectively, where $\alpha >0$ and $0<q\le \infty$. It is known that in the classical Haar case, i.e. when $S=\mathrm{span}\left(1_{[0,1]}\right)$ and the binary filtration ${\left({ℱ}_n\right)}_{n=0}^{\infty }$ is dyadic (that is, an atom $A\in A$ is divided into two new atoms of equal measure), we have $A_q^{\alpha }(L^p\left(S\right),\Phi )=A_q^{\alpha }(L^p\left(S\right),C)$, cf. P. Petrushev (2003). This motivates us to ask the question whether this equality is true in the general setting described above. The answer to this question is governed by the validity of a specific Bernstein type inequality $BI(A,S,p,\tau )$, with parameters $1<p<\infty$, $0<\tau <p$.

The main result of this paper is a geometric characterization of this type of Bernstein inequality $BI(A,S,p,\tau )$, i.e. a characterization in terms of the behavior of functions from the space $S$ on atoms $A$ and rings $ℛ=\{A\setminus B:A,B\in A,B\subset A\}\setminus A$. We specialize this general result to some examples of interest, including general Haar systems and spaces $S$ consisting of (multivariate) polynomials.