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

Let (Ω,ℱ,P) be a probability space and let (ℱ_n)_n=0^∞ 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⊂L^∞(Ω) be a finite-dimensional linear subspace, having an additional stability property on atoms A. For these data, we consider two dictionaries: • C={f⋅1_A:f∈S,A∈A}, • Φ – a local orthonormal system generated by S and the filtration (ℱ_n)_n=0^∞. Let L^p(S)=span¯_L^p(Ω)C=span¯_L^p(Ω)Φ, with 1<p<∞. We are interested in approximation spaces A_q^α(L^p(S),C) and A_q^α(L^p(S),Φ), corresponding to the best n-term approximation in L^p(S) by elements of C and Φ, respectively, where α>0 and 0<q≤∞. It is known that in the classical Haar case, i.e. when S=span(1_[0,1]) and the binary filtration (ℱ_n)_n=0^∞ is dyadic (that is, an atom A∈A is divided into two new atoms of equal measure), we have A_q^α(L^p(S),Φ)=A_q^α(L^p(S),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,τ), with parameters 1<p<∞, 0<τ<p. The main result of this paper is a geometric characterization of this type of Bernstein inequality BI(A,S,p,τ), i.e. a characterization in terms of the behavior of functions from the space S on atoms A and rings ℛ={A∖B:A,B∈A,B⊂A}∖A. We specialize this general result to some examples of interest, including general Haar systems and spaces S consisting of (multivariate) polynomials.