[Paper Review] Wavelets in function spaces on cellular domains

Benjamin Scharf|arXiv (Cornell University)|Feb 15, 2013

Mathematical Analysis and Transform Methods24 references3 citations

TL;DR

This paper establishes a theoretical framework for constructing u-Riesz wavelet bases in reinforced Triebel-Lizorkin function spaces on cellular domains, extending prior results from cubes to more general geometric structures. By introducing reinforced properties at boundaries and leveraging diffeomorphic decompositions, the work resolves challenges in wavelet representation for non-smooth domains, particularly in exceptional parameter regimes where standard wavelet bases fail.

ABSTRACT

Nowadays the theory and application of wavelet decompositions plays an important role not only for the study of function spaces (of Lebesgue, Hardy, Sobolev, Besov, Triebel-Lizorkin type) but also for its applications in signal and numerical analysis, partial differential equations and image processing. In this context it it a hard problem to construct wavelet bases for suitable function spaces on domains, e. g. the unit cube. A big step in this direction are the contributions of Hans Triebel from 2006 to 2008 where he constructed Riesz bases for classes of Besov- and Triebel-Lizorkin spaces on domains, starting with Daubechies wavelets. But there was a problem coming from the method: He had to exclude a big number of function spaces, in particular a large class of classical Sobolev spaces. The main goal of this thesis is a construction of Riesz bases of wavelet systems also for the exceptional cases using a modification of the function spaces - the so-called reinforced function spaces.

Motivation & Objective

• To extend the theory of u-Riesz wavelet bases from the unit cube to more general cellular domains, including $C^\infty$-domains and the unit ball.
• To address the lack of wavelet bases in exceptional parameter regimes for function spaces on domains, particularly when $s - \frac{k}{p} \notin \mathbb{N}_0$.
• To define and analyze reinforced function spaces $F_{p,q}^{s,\text{rinf}}(\Omega)$ on cellular domains with boundary conditions that ensure wavelet basis existence.
• To investigate the necessity and naturality of reinforced properties at internal boundaries arising from domain decomposition.
• To determine whether natural reinforce conditions can be introduced to extend wavelet bases to domains like the unit ball, where standard constructions fail.

Proposed method

• Adapts the theory of local means and atomic decompositions to function spaces on domains, particularly $F_{p,q}^{s,\text{rinf}}(\mathbb{R}^n \setminus \mathbb{R}^l)$, using Hardy inequalities at $l$-dimensional planes.
• Applies the concept of $u$-Riesz bases via wavelet-friendly extension operators and trace theorems on cubes and polyhedral domains.
• Uses diffeomorphic decomposition of cellular domains $\Omega$ into cubes or polyhedra, ensuring compatibility of wavelet bases across overlapping faces.
• Introduces reinforced function spaces $F_{p,q}^{s,\text{rinf}}(Q)$ on the unit cube $Q$ with boundary conditions $R_l^{r,p}$ to stabilize wavelet representations.
• Applies inductive construction techniques on the sphere $S^{n-1}$, decomposing it into northern/southern hemispheres and equator, to extend wavelet bases to the unit ball $B^n$.
• Analyzes the failure of natural reinforce conditions at internal boundaries (e.g., equator in $S^{n-1}$), showing that such conditions are unnatural and non-canonical.

Experimental results

Research questions

• RQ1 Can $u$-Riesz wavelet bases be constructed for reinforced Triebel-Lizorkin spaces $F_{p,q}^{s,\text{rinf}}(\Omega)$ on cellular domains $\Omega$?
• RQ2 Are the exceptional parameter values for wavelet basis existence (e.g., $s - \frac{k}{p} \notin \mathbb{N}_0$) removable via reinforced boundary conditions?
• RQ3 Is it possible to define natural reinforce properties $R_l^{r,p}$ at internal boundaries of decomposed domains, such as the equator in $S^{n-1}$?
• RQ4 How do the geometric properties of a domain, such as curvature or non-convexity, affect the existence of wavelet bases in function spaces?
• RQ5 Can the wavelet basis construction for $F_{p,q}^s(B^n)$ on the unit ball be extended to include exceptional cases using reinforced function space structures?

Key findings

• The paper constructs $u$-Riesz wavelet bases for reinforced Triebel-Lizorkin spaces $F_{p,q}^{s,\text{rinf}}(Q)$ on the unit cube $Q$, extending known results from non-reinforced spaces.
• For cellular domains $\Omega$ that are diffeomorphic to polyhedra, $u$-Riesz bases exist for $F_{p,q}^{s,\text{rinf}}(\Omega)$ in non-critical and non-exceptional cases.
• The construction fails in critical and exceptional cases ($s - \frac{k}{p} \notin \mathbb{N}_0$) without additional reinforce conditions.
• Reinforced properties $R_l^{r,p}$ at internal boundaries (e.g., the equator in $S^{n-1}$) are deemed unnatural and non-canonical, undermining their practical use.
• The wavelet basis construction for $F_{p,q}^s(B^n)$ on the unit ball cannot be extended to exceptional cases via natural reinforce conditions due to geometric and analytic inconsistencies.
• The decomposition of $S^{n-1}$ into hemispheres and equator allows inductive construction of wavelet bases, but only for non-exceptional parameters, highlighting a fundamental limitation in the method.