Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2305.01641

A structural characterization of Compactly Supported OEP-based balanced dual multiframelets

Ran Lu
View original source ↗Read source document ↗Export metadata ↓

Abstract

Compared to scalar framelets, multiframelets have certain advantages, such as relatively smaller supports on generators, high vanishing moments, etc. The balancing property of multiframelets is very desired, as it reflects how efficient vector-valued data can be processed under the corresponding discrete multiframelet transform. Most of the literature studying balanced multiframelets is from the point of view of the function setting, but very few approaches are from the aspect of multiframelet filter banks. In this paper, we study structural characterizations of balanced dual multiframelets from the point of view of the Oblique Extension Principle (OEP). The OEP naturally connects framelets with filter banks, which makes it a very handy tool for analyzing the properties of framelets. With the OEP, we shall characterize compactly supported balanced dual multiframemets through the concept of balanced moment correction filters, which is the key notion that will be introduced in our investigation. The results of this paper demonstrate what essential structures a balanced dual multiframelet has in the most general setting, and bring us a more complete picture to understand balanced multiframelets and their underlying discrete multiframelet transforms.

Explore related subjects

math.FAcs.CVmath.CA

Keep this discovery

BibTeXRIS

Ran Lu. 2023-04-03. A structural characterization of Compactly Supported OEP-based balanced dual multiframelets. https://arxiv.org/abs/2305.01641

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

KEEP EXPLORING

Related papers

arXiv2026

Typical dynamical properties of operators on $\ell_p$

We investigate the typical dynamical properties of hypercyclic operators in $\mathcal{L}_M(X)$, the set of all bounded linear operators on $X$ whose norms are at most $M$, when $X=\ell_p$, $1< p<\infty$. We show that, with respect to SOT$^*$, a typical operator $T\in \mathcal{L}_M(X)$ is weakly mixing, is weakly disjoint from a given hypercyclic operator $S$, is not topologically ergodic, and satisfies $(T,T^2,\dotsc,T^k)$ is disjoint hypercyclic for any $k\geq 2$. We also study the typical dynamical properties for the concrete family $\mathcal{M}=\{I+B_w\in \mathcal{L}(X)\colon w\in c_0(\mathbb{Z})\}$, endowed with the norm topology, where $B_w$ is a bilateral weighted backward shift.

math.FA↗
arXiv2026

A bi-Lipschitz characterization of strong minimum-attainment for Lipschitz maps

We completely characterize the denseness of strongly minimum-attaining Lipschitz functions, a minimum analogue for strongly norm-attaining Lipschitz functions, in terms of bi-Lipschitz embeddings. More precisely, our main result shows that the set of strongly minimum-attaining Lipschitz functions defined on a complete metric space $M$ fails the denseness if and only if $M$ is bi-Lipschitz equivalent to a subset of $\mathbb{R}$ with positive Lebesgue measure, or equivalently, if $M$ admits a bi-Lipschitz embedding into $\mathbb{R}$ and $M$ has positive 1-dimensional Hausdorff measure. As a consequence, we provide an isometric characterization of the pure 1-unrectifiability of $M$ in terms of strongly minimum-attaining Lipschitz maps defined on bi-Lipschitz copies of closed subsets of $M$. Several counterexamples showing that the main result cannot be naturally extended to the vector-valued setting are also presented.

math.FA↗
arXiv2026

On weak dominance of t-conorms over t-norms

The weak dominance of aggregation operators, particularly between triangular norms (t-norms) and triangular conorms (t-conorms), has attracted considerable attention in aggregation operator theory. While several characterizations have been obtained for Archimedean and continuous cases, a general criterion for continuous t-conorms over continuous t-norms remains to be fully clarified. In this paper, we provide a complete characterization of a continuous t-conorm weakly dominating a continuous t-norm. We first reduce the problem for ordinal sum operators to that for their single Archimedean components, and then express the weak dominance condition entirely in terms of the additive generators of these components. Our approach covers both strict and nilpotent cases uniformly, and recovers the known results for Archimedean operators as a special case.

math.FA↗