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Properties of local orthonormal systems, Part III: Variation spaces

Jacek Gulgowski, Anna Kamont, Markus Passenbrunner

Abstract

In [Y.~K.~Hu, K.~A.~Kopotun, X.~M.~Yu, Constr. Approx. 2000], the authors have obtained a characterization of best $n$-term piecewise polynomial approximation spaces as real interpolation spaces between $L^p$ and some spaces of bounded dyadic ring variation. We extend this characterization to the general setting of binary filtrations and finite-dimensional subspaces of $L^\infty$ as discussed in our earlier papers [J.~Gulgowski, A.~Kamont, M.~Passenbrunner, arXiv:2303.16470 and arXiv:2304.05647]. Furthermore, we study some analytical properties of thus obtained abstract spaces of bounded ring variation, as well as their connection to greedy approximation by corresponding local orthonormal systems.

Properties of local orthonormal systems, Part III: Variation spaces

Abstract

In [Y.~K.~Hu, K.~A.~Kopotun, X.~M.~Yu, Constr. Approx. 2000], the authors have obtained a characterization of best -term piecewise polynomial approximation spaces as real interpolation spaces between and some spaces of bounded dyadic ring variation. We extend this characterization to the general setting of binary filtrations and finite-dimensional subspaces of as discussed in our earlier papers [J.~Gulgowski, A.~Kamont, M.~Passenbrunner, arXiv:2303.16470 and arXiv:2304.05647]. Furthermore, we study some analytical properties of thus obtained abstract spaces of bounded ring variation, as well as their connection to greedy approximation by corresponding local orthonormal systems.
Paper Structure (22 sections, 30 theorems, 222 equations, 3 figures)

This paper contains 22 sections, 30 theorems, 222 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions and preliminaries
  3. Approximation spaces
  4. Greedy approximation
  5. Local orthonormal systems
  6. Variation spaces and their properties
  7. Definitions
  8. Interpolation results
  9. Comparison of variation spaces corresponding to binary and $\nu$-ary refinements
  10. Analysis of the variation space $V_{\sigma,p}$
  11. Variation spaces need not be separable
  12. Embeddings between Approximation and variation spaces
  13. The embedding of $\mathcal{A}_q^\alpha$ into $V_{\sigma,p}$
  14. The embedding of $V_{\sigma,p}$ into $\mathcal{A}_\infty^\alpha$
  15. The $K$-functional for the pair $(L^p(S), V_{\sigma,p})$ and greedy approximation.
  16. ...and 7 more sections

Key Result

Theorem 2.4

Fix $1<p<\infty$ and $0<\tau<p$ and let $\beta := 1/\tau - 1/p>0$. Then:

Figures (3)

  • Figure 1: The unit square $Q$, the points $(\xi_i)$, and the functions $g_i$ in the case $\nu = 5 = n+2$.
  • Figure 2: The first four iterations of the fractal $K$, with the same choice of $\nu$ and $(\xi_i)$ as in Figure \ref{['fig:counter2_alt']}.
  • Figure 3: The unit square $[0,1)^2$ and the respective positions of the squares $M_{j,m}$ in the case $n_1 = n_2 = 2$.

Theorems & Definitions (62)

  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 3.1
  • Theorem 3.2
  • Proposition 3.3
  • Lemma 3.4
  • Definition 3.5
  • Theorem 3.6
  • Proposition 3.7
  • ...and 52 more