Properties of local orthonormal systems, Part III: Variation spaces

TL;DR

This paper extends the characterization of approximation spaces as interpolation spaces to more general settings involving binary filtrations and finite-dimensional subspaces, and explores their analytical properties and relation to greedy approximation.

Contribution

It generalizes previous results on approximation spaces and bounded variation to broader contexts and analyzes their properties and connections to orthonormal systems.

Findings

Characterization of approximation spaces as interpolation spaces in new settings

Analysis of analytical properties of bounded ring variation spaces

Connection established between these spaces and greedy approximation methods

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.