Local means,wavelet bases and wavelet isomorphisms in Besov‐Morrey and Triebel‐Lizorkin‐Morrey spaces

We consider local means with bounded smoothness for Besov‐Morrey and Triebel‐Lizorkin‐Morrey spaces. Based on those we derive characterizations of these spaces in terms of Daubechies, Meyer, Bernstein (spline) and more general r‐regular (father) wavelets, finally in terms of (biorthogonal) wavelets which can serve as molecules and local means, respectively. Hereby both, local means and wavelet decompositions satisfy natural conditions concerning smoothness and cancellation (moment conditions). Moreover, the given representations by wavelets are unique and yield isomorphisms between the considered function spaces and appropriate sequence spaces of wavelet coefficients. These wavelet representations lead to wavelet bases if, and only if, the function spaces coincide with certain classical Besov‐Triebel‐Lizorkin spaces.     

Wavelets and local Triebel–Lizorkin spaces with the Lorentz index

In this paper, we apply wavelets to consider local norm function spaces with the Lorentz index. Triebel–Lizorkin–Lorentz spaces are based on the real interpolation of the Triebel–Lizorkin spaces. Triebel–Lizorkin–Morrey spaces are based on local norm of the Triebel–Lizorkin spaces. We give a unified depict of spaces that include these two kinds of spaces. Each index of the five index spaces represents a property of functions. We prove the wavelet characterization of the Triebel–Lizorkin–Lorentz–Morrey spaces and use such characterization to study some basic properties of these spaces.     

Atomic and Subatomic Decompositions in Anisotropic Function Spaces

This work deals with decompositions in anisotropic function spaces. Defining anisotropic atoms as smooth building blocks which are the counterpart of the atoms from the works of M. Frazier and B. Jawerth , it is shown that the study of anisotropic function spaces can be done with the help of some sequence spaces in a similar way as it is done in the isotropic case. It is also shown that the subatomic decomposition theorem for isotropic function spaces, recently proved by H. Triebel , can be extended to the anisotropic case if the mean smoothness parameter is sufficiently large.     

Anisotropic Lizorkin–Triebel spaces with mixed norms — traces on smooth boundaries

This article deals with trace operators on anisotropic Lizorkin–Triebel spaces with mixed norms over cylindrical domains with smooth boundary. As a preparation we include a rather self‐contained exposition of Lizorkin–Triebel spaces on manifolds and extend these results to mixed‐norm Lizorkin–Triebel spaces on cylinders in Euclidean space. In addition Rychkov's universal extension operator for a half space is shown to be bounded with respect to the mixed norms, and a support preserving right‐inverse of the trace is given explicitly and proved to be continuous in the scale of mixed‐norm Lizorkin–Triebel spaces. As an application, the heat equation is considered in these spaces, and the necessary compatibility conditions on the data are deduced.     

Pointwise multiplication by the characteristic function of the half-space on anisotropic vector-valued function spaces

We study the pointwise multiplier property of the characteristic function of the half-space on weighted mixed-norm anisotropic vector-valued function spaces of Bessel potential and Triebel–Lizorkin type.     

Pseudodifferential operators on mixed‐norm Besov and Triebel–Lizorkin spaces

We prove boundedness of pseudodifferential operators on anisotropic mixed‐norm Besov and Triebel–Lizorkin spaces. Our proof relies only on general maximal function estimates and provides a new perspective even in the case of spaces without mixed norms. Moreover, we cover the case of Fourier multipliers on the above mentioned spaces. As application we establish boundedness of pseudodifferential operators and Fourier multipliers on anisotropic mixed‐norm Sobolev spaces.     

Analysis of edge and corner points using parabolic dictionaries

Last decade saw the creation of a number of directional representation dictionaries that desire to address the weaknesses of the classical wavelet transform that arise due to its limited capacity for the analysis of edge-like features of two-dimensional signals. Salient features of these dictionaries are directional selectivity and anisotropic treatment of the axes, achieved through the parabolic scaling law. In this paper we will examine the adequacy of such dictionaries for the analysis of edge- and corner-like features of 2D regions through a comprehensive framework for directional parabolic dictionaries, called the continuous parabolic molecules. This work builds on a family of earlier studies and aims to give a broader perspective through the level of generality.     

Greedy bases in variable Lebesgue spaces

We compute the right and left democracy functions of admissible wavelet bases in variable Lebesgue spaces defined on \(\mathbb R^n\). As an application we give Lebesgue type inequalities for these wavelet bases. We also show that our techniques can be easily modified to prove analogous results for weighted variable Lebesgue spaces and variable exponent Triebel–Lizorkin spaces.     

Pointwise and directional regularity of nonharmonic Fourier series

We investigate how the regularity of nonharmonic Fourier series is related to the spacing of their frequencies. This is obtained by using a transform which simultaneously captures the advantages of the Gabor and wavelet transforms. Applications to the everywhere irregularity of solutions of some PDEs are given. We extend these results to the anisotropic setting in order to derive directional irregularity criteria.     

R-D optimized tree-structured compression algorithms with discrete directional wavelet transform

A new image coding method based on discrete directional wavelet transform (S-WT) and quad-tree decomposition is proposed here. The S-WT is a kind of transform proposed in [V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, P.L. Dragotti, Directionlets: anisotropic multidirectional representation with separable filtering, IEEE Trans. Image Process. 15(7) (2006)], which is based on lattice theory, and with the difference with the standard wavelet transform is that the former allows more transform directions. Because the directional property in a small region is more regular than in a big block generally, in order to sufficiently make use of the multidirectionality and directional vanishing moment (DVM) of S-WT, the input image is divided into many small regions by means of the popular quad-tree segmentation, and the splitting criterion is on the rate-distortion sense. After the optimal quad-tree is obtained, by means of the embedded property of SPECK, a resource bit allocation algorithm is fast implemented utilizing the model proposed in [M. Rajpoot, Model based optimal bit allocation, in: IEEE Data Compression Conference, 2004, Proceedings, DCC 2004.19]. Experiment results indicate that our algorithms perform better compared to some state-of-the-art image coders.     

Directional wavelets on n-dimensional spheres

Directional Poisson wavelets, being directional derivatives of Poisson kernel, are introduced on n-dimensional spheres. It is shown that, slightly modified and together with another wavelet family, they are an admissible wavelet pair according to the definition derived from the theory of approximate identities. We investigate some of the properties of directional Poisson wavelets, such as recursive formulae for their Fourier coefficients or explicit representations as functions of spherical variables (for some of the wavelets). We derive also an explicit formula for their Euclidean limits.     

Restricted non-linear approximation in sequence spaces and applications to wavelet bases and interpolation

Restricted non-linear approximation is a type of N-term approximation where a measure ν on the index set (rather than the counting measure) is used to control the number of terms in the approximation. We show that embeddings for restricted non-linear approximation spaces in terms of weighted Lorentz sequence spaces are equivalent to Jackson and Bernstein type inequalities, and also to the upper and lower Temlyakov property. As applications we obtain results for wavelet bases in Triebel–Lizorkin spaces by showing the Temlyakov property in this setting. Moreover, new interpolation results for Triebel–Lizorkin and Besov spaces are obtained.     

Decomposition for Morrey type Besov–Triebel spaces

In this paper, the author establishes the decomposition of Morrey type Besov–Triebel spaces in terms of atoms and molecules concentrated on dyadic cubes, which have the same smoothness and cancellation properties as those of the classical Besov–Triebel spaces. The results extend those of M. Frazier, B. Jawerth for Besov–Triebel spaces and those of A. L. Mazzucato for Besov–Morrey spaces (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Localisation of directional scale-discretised wavelets on the sphere

Scale-discretised wavelets yield a directional wavelet framework on the sphere where a signal can be probed not only in scale and position but also in orientation. Furthermore, a signal can be synthesised from its wavelet coefficients exactly, in theory and practice (to machine precision). Scale-discretised wavelets are closely related to spherical needlets (both were developed independently at about the same time) but relax the axisymmetric property of needlets so that directional signal content can be probed. Needlets have been shown to satisfy important quasi-exponential localisation and asymptotic uncorrelation properties. We show that these properties also hold for directional scale-discretised wavelets on the sphere and derive similar localisation and uncorrelation bounds in both the scalar and spin settings. Scale-discretised wavelets can thus be considered as directional needlets.     

N–Term Approximation in Anisotropic Function Spaces

In L₂(0, 1)²) infinitely many different biorthogonal wavelet bases may be introduced by taking tensor products of one–dimensional biorthogonal wavelet bases on the interval (0, 1). Most well–known are the standard tensor product bases and the hyperbolic bases. In [23, 24] further biorthogonal wavelet bases are introduced, which provide wavelet characterizations for functions in anisotropic Besov spaces. Here we address the following question: Which of those biorthogonal tensor product wavelet bases is the most appropriate one for approximating nonlinearly functions from anisotropic Besov spaces? It turns out, that the hyperbolic bases lead to nonlinear algorithms which converge as fast as the corresponding schemes with respect to specific anisotropy adapted bases.     

小波与函数空间

Triebe利用Littlewood Paley分解将大多数函数空间分类成两类三指标的函数空间：Besov空间和Triebel Lizorkin空间；但Littlewood Paley 分解很难直接分析Sobolev空间L^p的插值空间Lorentz空间，也很难分析Triebel Lizorkin空间F^{α,q}_1的预备对偶空间和对偶空间.运用小波，作者给出这些空间一个统一刻画：Triebel Lizorkin Lorentz 空间，Besov Lorentz空间和F^{α,q}_1的预备对偶空间和对偶空间；另外也研究这些空间的三个性质.     

Operator-valued Fourier Multipliers on Periodic Triebel Spaces

We establish operator-valued Fourier multiplier theorems on periodic Triebel spaces, where the required smoothness of the multipliers depends on the indices of the Triebel spaces. This is used to give a characterization of the maximal regularity in the sense of Triebel spaces for Cauchy problems with periodic boundary conditions.     

An Approach to Wavelet Isomorphisms of Function Spaces Via Atomic Representations

Both wavelet and atomic decomposition techniques are essential tools in the study of function spaces nowadays, but they both have their advantages and disadvantages. The celebrated bridge between both concepts was given by the compactly supported Daubechies wavelets which can be interpreted as atoms. In this paper we deal with the converse direction, that is, we present a fairly general approach how to construct compactly supported wavelets when an atomic decomposition is known already. The main idea is Taylor’s expansion combined with our new, so-called \(\varkappa \)-convergence assumption in the admitted sequence spaces. We finally exemplify our main result and collect some known and new settings where such a wavelet decomposition is obtained, e.g., in spaces of Besov or Triebel–Lizorkin type with a doubling weight.     

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON TRIEBEL SPACES

1IntroductionIn a series of recent publications operator-valued Fourier multipliers on vector-valued func-tion spaces were studied(see e.g.[1,2,3,5,6,7,14,16]).They are needed to establish existence anduniqueness as well as regularity of di?erential equat…     

Relations among Besov-type spaces,Triebel–Lizorkin-type spaces and generalized Carleson measure spaces

In this article, the authors construct some counterexamples to show that the generalized Carleson measure space and the Triebel–Lizorkin-type space are not equivalent for certain parameters, which was claimed to be true in Lin and Wang [C.-C. Lin and K.Wang, Equivalency between the generalized Carleson measure spaces and Triebel–Lizorkin-type spaces, Taiwanese J. Math. 15 (2011), pp. 919–926]. Moreover, the authors show that for some special parameters, the generalized Carleson measure space, the Triebel–Lizorkin-type space and the Besov-type space coincide with certain Triebel–Lizorkin space, which answers a question posed in Remark 6.11(i) of Yuan et al. [W. Yuan, W. Sickel and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics 2005, Springer-Verlag, Berlin, 2010]. In conclusion, the Triebel–Lizorkin-type space and the Besov-type space become the classical Besov spaces, when the fourth parameter is sufficiently large.