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Jackson and Bernstein Inequalities with Explicit Weight Dependence for Band-Limited Wavelet Approximation in Weighted Besov Spaces

  • Hung-Lung Lin
  • Kai-Cheng Wang
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Abstract

We prove direct and inverse estimates for band-limited biorthogonal wavelet systems in weighted Besov spaces with Muckenhoupt weights. On functions recovered from their wavelet expansion along the dyadic tree of the unit cube, the error of approximation by a prescribed number of terms decays at a rate set by the smoothness-to-dimension ratio, and this exponent cannot be increased. Under a multiresolution condition, Bernstein inequalities hold at a single scale and on every multiresolution subspace, and the linear approximation spaces are identified with weighted Lebesgue--Besov intersections. All constants are explicit; the weight enters only through displayed powers of the Muckenhoupt characteristic.

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Cite

BibTeXRIS
BibTeX
@misc{lin2026jackson,
  title = {{Jackson and Bernstein Inequalities with Explicit Weight Dependence for Band-Limited Wavelet Approximation in Weighted Besov Spaces}},
  author = {Hung-Lung Lin and Kai-Cheng Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.35520},
  eprint = {2609.35520},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.35520},
}
RIS
TY  - UNPB
TI  - Jackson and Bernstein Inequalities with Explicit Weight Dependence for Band-Limited Wavelet Approximation in Weighted Besov Spaces
AU  - Hung-Lung Lin
AU  - Kai-Cheng Wang
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.35520
UR  - https://arxiv.org/abs/2609.35520
AB  - We prove direct and inverse estimates for band-limited biorthogonal wavelet systems in weighted Besov spaces with Muckenhoupt weights. On functions recovered from their wavelet expansion along the dyadic tree of the unit cube, the error of approximation by a prescribed number of terms decays at a rate set by the smoothness-to-dimension ratio, and this exponent cannot be increased. Under a multiresolution condition, Bernstein inequalities hold at a single scale and on every multiresolution subspace, and the linear approximation spaces are identified with weighted Lebesgue--Besov intersections. All constants are explicit; the weight enters only through displayed powers of the Muckenhoupt characteristic.
ER  -