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.
Highlights
With the development of wavelet analysis from the beginning of the 1980s until the present time we nowadays have several powerful tools at hand to perform signal analysis with the aim to extract important information out of a signal
In this paper we further develop this idea of describing anisotropy with the help of the hyperbolic wavelet transform
For this reason we introduce a new family of anisotropic function spaces which are defined via a hyperbolic Littlewood–Paley analysis and
Summary
With the development of wavelet analysis from the beginning of the 1980s until the present time we nowadays have several powerful tools at hand to perform signal analysis with the aim to extract important information out of a signal. The common underlying idea is the fact that a few wavelet coefficients contain a rather complete information of the signal to be analyzed Due to their construction principle (dyadic dilations and integer translates of a few basic “mother” functions) classical wavelets are not well-suited for the analysis of, say, anisotropic signals. In this paper we further develop this idea of describing anisotropy with the help of the hyperbolic wavelet transform For this reason we introduce a new family of anisotropic function spaces which are defined via a hyperbolic Littlewood–Paley analysis and. As an important consequence of this equality (2), we can further prove that it is possible to characterize (e.g. detect and classify) classical anisotropies described by the via the wavelet coefficients of a universal hyperbolic wavelet basis.
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