Wavelet characterization of exponentially weighted Besov space with dominating mixed smoothness and its application to function approximation

TL;DR

This paper introduces a new class of exponentially weighted Besov spaces with mixed smoothness, characterizes them using wavelets, and applies these results to develop approximation formulas like sparse grids.

Contribution

It defines exponentially weighted Besov spaces with mixed smoothness and provides their wavelet characterization, extending the understanding of these function spaces.

Findings

Wavelet characterization of $VB_{p,q}^{ ext{delta,w}}(\

Development of approximation formulas such as sparse grids for these spaces.

Abstract

Although numerous studies have focused on normal Besov spaces, limited studies have been conducted on exponentially weighted Besov spaces. Therefore, we define exponentially weighted Besov space $VB_{p,q}^{\delta,w}(\mathbb{R}^d)$ whose smoothness includes normal Besov spaces, Besov spaces with dominating mixed smoothness, and their interpolation. Furthermore, we obtain wavelet characterization of $VB_{p,q}^{\delta,w}(\mathbb{R}^d)$. Next, approximation formulas such as sparse grids are derived using the determined formula. The results of this study are expected to provide considerable insight into the application of exponentially weighted Besov spaces with mixed smoothness.