PreprintCC0arXiv
A self-dual Dunkl wavelet frame and Besov--Triebel--Lizorkin spaces
Abstract
We construct a self-dual Parseval wavelet frame associated with the Dunkl transform and use it to characterize the corresponding Besov and Triebel-Lizorkin spaces for arbitrary smoothness and the full range of admissible indices. We also establish differential atomic decompositions with atoms supported on dilated Euclidean dyadic cubes, satisfying ordinary derivative estimates and polynomial moment indices. A key difficulty is the interaction between Euclidean dyadic cubes and the orbit geometry associated with the Dunkl operators. As applications, we obtain fractional chain rules and a Jackson inequality for nonlinear wavelet approximation in Lp(dw).
Related
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- Triebel - Lizorkin spaces of dominating mixed smoothness are algebras with respect to pointwise multiplication
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- Mixed-norm Herz-type Triebel-Lizorkin spaces
BibTeX
@misc{bui2026selfdual,
title = {{A self-dual Dunkl wavelet frame and Besov--Triebel--Lizorkin spaces}},
author = {The Anh Bui},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27391},
eprint = {2610.27391},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27391},
note = {CC0},
}
RIS
TY - UNPB
TI - A self-dual Dunkl wavelet frame and Besov--Triebel--Lizorkin spaces
AU - The Anh Bui
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27391
UR - https://arxiv.org/abs/2610.27391
AB - We construct a self-dual Parseval wavelet frame associated with the Dunkl transform and use it to characterize the corresponding Besov and Triebel-Lizorkin spaces for arbitrary smoothness and the full range of admissible indices. We also establish differential atomic decompositions with atoms supported on dilated Euclidean dyadic cubes, satisfying ordinary derivative estimates and polynomial moment indices. A key difficulty is the interaction between Euclidean dyadic cubes and the orbit geometry associated with the Dunkl operators. As applications, we obtain fractional chain rules and a Jackson inequality for nonlinear wavelet approximation in L^p(dw).
ER -