[Paper Review] Hyperbolic wavelet analysis of classical isotropic and anisotropic Besov-Sobolev spaces

Martin Schäfer, Tino Ullrich+1 more|arXiv (Cornell University)|Dec 17, 2019

Advanced Harmonic Analysis Research44 references4 citations

TL;DR

This paper introduces anisotropic hyperbolic Besov and Triebel-Lizorkin spaces via a hyperbolic Littlewood-Paley analysis with anisotropy encoded in smoothness weights, enabling characterization of anisotropic smoothness using a single universal hyperbolic wavelet basis. The key contribution is that in the Sobolev range, these new spaces coincide with classical anisotropic Besov-Triebel spaces, allowing universal detection of anisotropy without prior knowledge of the anisotropy vector or basis adaptation.

ABSTRACT

In this paper we introduce new function spaces which we call anisotropic hyperbolic Besov and Triebel-Lizorkin spaces. Their definition is based on a hyperbolic Littlewood-Paley analysis involving an anisotropy vector only occurring in the smoothness weights. Such spaces provide a general and natural setting in order to understand what kind of anisotropic smoothness can be described using hyperbolic wavelets (in the literature also sometimes called tensor-product wavelets), a wavelet class which hitherto has been mainly used to characterize spaces of dominating mixed smoothness. A centerpiece of our present work are characterizations of these new spaces based on the hyperbolic wavelet transform. Hereby we treat both, the standard approach using wavelet systems equipped with sufficient smoothness, decay, and vanishing moments, but also the very simple and basic hyperbolic Haar system. The second major question we pursue is the relationship between the novel hyperbolic spaces and the classical anisotropic Besov-Lizorkin-Triebel scales. As our results show, in general, both approaches to resolve an anisotropy do not coincide. However, in the Sobolev range this is the case, providing a link to apply the newly obtained hyperbolic wavelet characterizations to the classical setting. In particular, this allows for detecting classical anisotropies via the coefficients of a universal hyperbolic wavelet basis, without the need of adaption of the basis or a-priori knowledge on the anisotropy.

Motivation & Objective

• To develop a general framework for anisotropic smoothness using hyperbolic wavelet analysis, independent of prior knowledge of anisotropy direction.
• To define new function spaces—hyperbolic Besov and Triebel-Lizorkin spaces—based on a hyperbolic Littlewood-Paley decomposition with anisotropy in smoothness weights.
• To establish exact characterizations of these new spaces using both smooth hyperbolic wavelets and the hyperbolic Haar system.
• To clarify the relationship between the new hyperbolic spaces and classical anisotropic Besov-Triebel-Lizorkin spaces, particularly in the Sobolev range.
• To demonstrate that a universal hyperbolic wavelet basis can detect classical anisotropies without basis adaptation or a-priori anisotropy information.

Proposed method

• Defining new function spaces via a hyperbolic Littlewood-Paley decomposition: $\Delta_{\bar{j}}(f) := \mathcal{F}^{-1}[\theta_{j_1} \otimes \cdots \otimes \theta_{j_d} \mathcal{F}f]$, where $\bar{j} \in \mathbb{N}_0^d$.
• Introducing anisotropy through a vector $\bar{\alpha} = (\alpha_1, \ldots, \alpha_d) > 0$ with $\sum \alpha_i = d$, which enters only in the smoothness weights of the new spaces $\widetilde{A}^{s,\bar{\alpha}}_{p,q}(\mathbb{R}^d)$.
• Using the hyperbolic wavelet transform to characterize the new spaces, including both smooth wavelets with sufficient decay, smoothness, and vanishing moments, and the simpler hyperbolic Haar system.
• Establishing norm equivalences between the new hyperbolic spaces and sequence norms of hyperbolic wavelet coefficients, proving exact characterizations.
• Applying the wavelet isomorphism theorem (Theorem 4.2) to relate function norms to coefficient sequence norms in $\ell^p$ and $\ell^q$ spaces.
• Analyzing the behavior of random wavelet series with i.i.d. signs to derive lower bounds on $L_p$-norms, showing $\mathbb{E}_{\bar{\varepsilon}}(\|f_{N,\bar{\varepsilon}}\|_p^p) \asymp N^{p/2}$ for $p \geq 2$, and $\|f_N\|_p \asymp \sqrt{N}$.

Experimental results

Research questions

• RQ1 Can a single universal hyperbolic wavelet basis characterize anisotropic smoothness without prior knowledge of the anisotropy direction?
• RQ2 How do the newly defined hyperbolic Besov-Triebel-Lizorkin spaces relate to classical anisotropic function spaces?
• RQ3 In which smoothness regimes do the hyperbolic wavelet-based spaces coincide with classical anisotropic spaces?
• RQ4 What is the behavior of wavelet coefficient sequences in $L_p$-norms for random wavelet series with i.i.d. signs?
• RQ5 Can the hyperbolic wavelet transform detect anisotropy in signals with mixed smoothness, even when the anisotropy is not known in advance?

Key findings

• The new hyperbolic Besov and Triebel-Lizorkin spaces $\widetilde{A}^{s,\bar{\alpha}}_{p,q}(\mathbb{R}^d)$ are exactly characterized by the hyperbolic wavelet transform, both for smooth wavelets and the hyperbolic Haar system.
• For $p \geq 2$, the expected $L_p$-norm of a random wavelet series with i.i.d. signs satisfies $\mathbb{E}_{\bar{\varepsilon}}(\|f_{N,\bar{\varepsilon}}\|_p^p) \asymp N^{p/2}$, implying $\|f_N\|_p \asymp \sqrt{N}$.
• In the Sobolev range ($F^{s,\bar{\alpha}}_{p,2}$), the new hyperbolic spaces coincide with classical anisotropic Triebel-Lizorkin spaces, establishing a bridge between the two frameworks.
• The hyperbolic wavelet transform allows universal detection of anisotropy: a single basis can resolve anisotropic smoothness without prior knowledge of the anisotropy vector or basis adaptation.
• For $p < 1$, the $F^0_{p,q}$-norm of a localized frequency projection satisfies $\|f_N\|_{F^0_{p,q}} \gtrsim 1$, while for $p \geq 1$, it grows as $N^{1/p}$, showing distinct behavior depending on $p$.
• The wavelet coefficient sequence norms are equivalent to the function norms in the new hyperbolic spaces, confirming the isomorphism between function spaces and sequence spaces via the hyperbolic wavelet transform.