Jackson-type Inequalities with Explicit Weight Dependence for N-term Wavelet Approximation on Localized Weighted Besov Classes

Kai-Cheng Wang

Home/Archive/arXiv

arXiv · 2609.35520

Jackson-type Inequalities with Explicit Weight Dependence for N-term Wavelet Approximation on Localized Weighted Besov Classes

Kai-Cheng Wang

View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove a Jackson-type inequality for approximation by a prescribed number of terms of a band-limited biorthogonal wavelet system in homogeneous weighted Besov spaces with Muckenhoupt weights, on functions recovered from their wavelet expansion along the dyadic tree of the unit cube. The error decays at a rate set by the smoothness-to-dimension ratio. The explicit constant has a closed-form majorant, and the weight enters only through a displayed power of the Muckenhoupt characteristic. An extremal family shows that the exponent of this rate cannot be increased on the class, and an unweighted example shows that the localization is necessary.

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Kai-Cheng Wang. 2026-09-28. Jackson-type Inequalities with Explicit Weight Dependence for N-term Wavelet Approximation on Localized Weighted Besov Classes. https://arxiv.org/abs/2609.35520

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Exceptional sets for length under restricted families of projections onto lines in $\mathbb{R}^3$

It is shown that if $A \subseteq \mathbb{R}^3$ is a Borel set of Hausdorff dimension $\dim A>1$, and if $ρ_θ$ is orthogonal projection to the line spanned by $( \cos θ, \sin θ, 1 )$, then $ρ_θ(A)$ has positive length for all $θ$ outside a set of Hausdorff dimension at most $\frac{3-\dim A}{2}$.

math.CA↗

arXiv2026

Some one-dimensional elliptic problems with constraints

Given $m \in \mathbb{N} \setminus \{0\}$ and $ρ> 0$, we find solutions $(λ,u)$ to the problem \begin{equation*} \begin{cases} \bigl(-\frac{\mathrm{d}^2}{\mathrm{d} x^2}\bigr)^m u + λG'(u) = F'(u)\\ \int_{\mathbb{R}} K(u) \, \mathrm{d}x = ρ\end{cases} \end{equation*} in the following cases: $m=1$ or $2G(s) = K(s) = s^2$. In the former, we follow a bifurcation argument; in the latter, we use variational methods.

math.CA↗

arXiv2026

Besicovitch-Federer projection theorem for measures

In this paper we prove a Besicovitch-Federer type projection theorem for general finite Borel measures in $\mathbb{R}^n$. Assuming only that the conditional measures along typical affine $(n-m)$-planes are purely atomic, we characterize concentration on a purely $m$-unrectifiable set in terms of the singularity of the projected measure and the $μ$-almost everywhere injectivity of typical orthogonal projections. No regularity or absolute-continuity assumption on the projected measures is imposed, and the result is new even for measure of the form $μ=\mathcal H^m\llcorner E$. In this extended version, we further develop applications of the projection theorem to currents. First, we obtain a rectifiability criterion for Radon measures in terms of atomic disintegrations. We then use this criterion to prove that a finite-mass classical Federer-Fleming current satisfying the intrinsic integer-valued push-forward condition is rectifiable, and hence a real flat chain, without assuming any a priori flat-chain or metric-current structure. We also show that a Euclidean metric current is rectifiable whenever its slices are atomic along a set of projections of positive Grassmannian measure, with no assumption on the mass of its boundary. Finally, the projection theorem extends to separable locally compact metric spaces.

math.CA↗