Hyperbolic Wavelet Analysis of Classical Isotropic and Anisotropic Besov–Sobolev Spaces

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.

Rate-optimal sparse approximation of compact break-of-scale embeddings

“…Since wavelets are known to be a powerful tool in signal processing and numerical analysis [16,18], in this paper we shall focus on algorithms based on a system of hyperbolic wavelets. In contrast to classical isotropic wavelets, their tensor product structure is perfectly suited to resolve anisotropies which naturally arise in various applications, e.g., in physics, engineering, or medical image processing; see [34] and the references therein. On the other hand, these wavelets can be employed to characterize function spaces measuring dominating mixed smoothness [38] as well as spaces of isotropic regularity [34].…”

“…In contrast to classical isotropic wavelets, their tensor product structure is perfectly suited to resolve anisotropies which naturally arise in various applications, e.g., in physics, engineering, or medical image processing; see [34] and the references therein. On the other hand, these wavelets can be employed to characterize function spaces measuring dominating mixed smoothness [38] as well as spaces of isotropic regularity [34]. In order to ensure a fair comparison of the performance of linear and non-linear methods, we restrict the corresponding widths d m and σ m to a dictionary Ψ consisting of such hyperbolic wavelets; see Definition 2.5 below for details.…”

“…Let us recap some basics about hyperbolic wavelets as described in some more detail in [34,Section 4]. Let φ be a univariate scaling function and ψ the corresponding wavelet which fulfill the following conditions for some K ∈ N 0 :…”

“…For hyperbolic wavelets and Besov/Triebel-Lizorkin spaces S r p,q X(R d ) (with X ∈ {B, F }) of dominating mixed smoothness r, this has been done in [38]. Quite recently, it was found in [34] that exactly the same wavelets can be used to characterize also other anisotropic spaces X s p,q (R d ) of Besov-and Triebel-Lizorkin-type which in some cases coincide with the classical (isotropic!) spaces B s p,q (R d ) and F s p,q (R d ), respectively.…”

A higher order Faber spline basis for sampling discretization of functions

“…First we prove some auxiliary statements. We apply some known technique that was also used in [3], [25] and [40]. Let λ = (λ j,k ) j∈N −1 ,k∈Z be some sequence of real numbers that satisfy certain conditions (later we specify that λ ∈ b r p,θ or λ ∈ f r p,θ ).…”