Properties of local orthonormal systems, Part II: Geometric characterization of Bernstein inequalities
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| Format: | Preprint |
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2023
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| _version_ | 1866913617272635392 |
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| author | Gulgowski, Jacek Kamont, Anna Passenbrunner, Markus |
| author_facet | Gulgowski, Jacek Kamont, Anna Passenbrunner, Markus |
| contents | Let $(Ω,\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 $S \subset L^\infty(Ω)$ be a finite-dimensional linear subspace, having an additional stability property on atoms $\mathscr A$. For these data, we consider the two dictionaries $\mathscr C = \{ f \cdot χ_A: f \in S, A \in \mathscr A\}$ and $Φ$, 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 $Φ$, respectively. It is known that in the classical Haar case, i.e. when $S = {\rm span} (χ_{[0,1]})$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_05647 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Properties of local orthonormal systems, Part II: Geometric characterization of Bernstein inequalities Gulgowski, Jacek Kamont, Anna Passenbrunner, Markus Functional Analysis 41A17, 41A46, 42C05, 42C40, 46E30 Let $(Ω,\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 $S \subset L^\infty(Ω)$ be a finite-dimensional linear subspace, having an additional stability property on atoms $\mathscr A$. For these data, we consider the two dictionaries $\mathscr C = \{ f \cdot χ_A: f \in S, A \in \mathscr A\}$ and $Φ$, 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 $Φ$, respectively. It is known that in the classical Haar case, i.e. when $S = {\rm span} (χ_{[0,1]})$ 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. |
| title | Properties of local orthonormal systems, Part II: Geometric characterization of Bernstein inequalities |
| topic | Functional Analysis 41A17, 41A46, 42C05, 42C40, 46E30 |
| url | https://arxiv.org/abs/2304.05647 |