A higher order Faber spline basis for sampling discretization of functions

Derevianko, Nadiia; Ullrich, Tino

doi:10.1016/j.jat.2020.105449

Cited by 8 publications

(25 citation statements)

References 29 publications

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“…Choosing ε > 0 small enough, we then obtain from (11) that condition (15) in Remark 4.8 is satisfied. Hence, for a finite partial summation of ( 12) we have…”

Section: Proof Of Theorem 46mentioning

confidence: 99%

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

Schäfer

Ullrich

Vedel

2021

J Fourier Anal Appl

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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.

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“…Choosing ε > 0 small enough, we then obtain from (11) that condition (15) in Remark 4.8 is satisfied. Hence, for a finite partial summation of ( 12) we have…”

Section: Proof Of Theorem 46mentioning

confidence: 99%

“…Besides its elegance and simplicity, notably its connection to the Faber system [18] and other spline functions, such as e.g. the Chui-Wang wavelet [11], makes it interesting from a numerical perspective (see also [15]). Fast implementations are possible based on the fast Haar transform [29,38] and variations thereof (see e.g.…”

Section: Introductionmentioning

confidence: 99%

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

Schäfer

Ullrich

Vedel

2021

J Fourier Anal Appl

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“…Discover truncated quarklet systems. At first we make some estimates with the goal to reach a situation where we can work with truncated quarklet systems, see (13). Since w < 1 and w < q we find…”

Section: This In Combination Withmentioning

confidence: 99%

“…For a definition of those wavelets we refer to Theorem 1 in [4]. In [13] the biorthogonal Chui-Wang wavelets are used to prove equivalent quasi-norms for F s r,q (R), see Theorem 5.1 and also Theorem 6.2 for higher order Chui-Wang wavelets.…”

Section: Lower Estimatesmentioning

confidence: 99%

Quarklet Characterizations for Triebel-Lizorkin Spaces

Hovemann¹,

Dahlke²

2021

Preprint

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In this paper we prove that under some conditions on the parameters the onedimensional Triebel-Lizorkin spaces F s r,q (R) can be described in terms of quarklets. So for functions from Triebel-Lizorkin spaces we obtain a quarkonial decomposition as well as a new equivalent quasi-norm. For that purpose we use quarklets that are constructed out of biorthogonal compactly supported Cohen-Daubechies-Feauveau spline wavelets, where the primal generator is a cardinal B-spline. Moreover we introduce some sequence spaces apposite to our quarklet system and study their properties. Finally we also obtain a quarklet characterization for the Triebel-Lizorkin-Morrey spaces E s u,r,q (R).

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“…In particular in imaging science it plays an important role in practical applications. Recently, it has attracted renewed attention with a series of publications [44,45,21,22,23,24,14].…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic wavelet analysis of classical isotropic and anisotropic Besov-Sobolev spaces

Schäfer

Ullrich

Vedel

2019

Preprint

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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.

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A higher order Faber spline basis for sampling discretization of functions

Cited by 8 publications

References 29 publications

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

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

Quarklet Characterizations for Triebel-Lizorkin Spaces

Hyperbolic wavelet analysis of classical isotropic and anisotropic Besov-Sobolev spaces

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