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.     

Convergence Rates of Adaptive Methods,Besov Spaces,and Multilevel Approximation

This paper concerns characterizations of approximation classes associated with adaptive finite element methods with isotropic h-refinements. It is known from the seminal work of Binev, Dahmen, DeVore and Petrushev that such classes are related to Besov spaces. The range of parameters for which the inverse embedding results hold is rather limited, and recently, Gaspoz and Morin have shown, among other things, that this limitation disappears if we replace Besov spaces by suitable approximation spaces associated with finite element approximation from uniformly refined triangulations. We call the latter spaces multievel approximation spaces and argue that these spaces are placed naturally halfway between adaptive approximation classes and Besov spaces, in the sense that it is more natural to relate multilevel approximation spaces with either Besov spaces or adaptive approximation classes, than to go directly from adaptive approximation classes to Besov spaces. In particular, we prove embeddings of multilevel approximation spaces into adaptive approximation classes, complementing the inverse embedding theorems of Gaspoz and Morin. Furthermore, in the present paper, we initiate a theoretical study of adaptive approximation classes that are defined using a modified notion of error, the so-called total error, which is the energy error plus an oscillation term. Such approximation classes have recently been shown to arise naturally in the analysis of adaptive algorithms. We first develop a sufficiently general approximation theory framework to handle such modifications, and then apply the abstract theory to second-order elliptic problems discretized by Lagrange finite elements, resulting in characterizations of modified approximation classes in terms of memberships of the problem solution and data into certain approximation spaces, which are in turn related to Besov spaces. Finally, it should be noted that throughout the paper we paid equal attention to both conforming and non-conforming triangulations.     

Wavelet Characterizations for Anisotropic Besov Spaces

The goal of this paper is to provide wavelet characterizations for anisotropic Besov spaces. Depending on the anisotropy, appropriate biorthogonal tensor product bases are introduced and Jackson and Bernstein estimates are proved for two-parameter families of finite-dimensional spaces. These estimates lead to characterizations for anisotropic Besov spaces by anisotropy-dependent linear approximation spaces and lead further on to interpolation and embedding results. Finally, wavelet characterizations for anisotropic Besov spaces with respect to L_p-spaces with 0<p<∞ are derived.     

spline wavelets on triangulations

In this paper we investigate spline wavelets on general triangulations. In particular, we are interested in wavelets generated from piecewise quadratic polynomials. By using the Powell-Sabin elements, we set up a nested family of spaces of quadratic splines, which are suitable for multiresolution analysis of Besov spaces. Consequently, we construct wavelet bases on general triangulations and give explicit expressions for the wavelets on the three-direction mesh. A general theory is developed so as to verify the global stability of these wavelets in Besov spaces. The wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations.

     

Besov–Lipschitz and Mean Besov–Lipschitz Spaces of Holomorphic Functions on the Unit Ball

We give several characterizations of holomorphic mean Besov–Lipschitz spaces on the unit ball in ${\mathbb C^N} $ and appropriate Besov–Lipschitz spaces and prove the equivalences between them. Equivalent norms on the mean Besov–Lipschitz spaces involve different types of L ^p -moduli of continuity, while in characterizations of Hardy–Sobolev spaces we use not only the radial derivative but also the gradient. The characterization in terms of the best approximation by polynomials is also given.     

On Sobolev and Capacitary Inequalities for Contractive Besov Spaces over d-sets

We derive Sobolev inequalities for Besov spaces B ^p,p (F), 0<<1, 1p< on d-sets F in R ⁿ , dn, from a metric property of the Bessel capacity on R ⁿ . We first extend Kaimanovitch's result on the equivalence of Sobolev and capacitary inequalites for contractive p-norms in a general setting allowing unbounded Lévy kernels. A simple part of the Jonsson–Wallin trace theorem for Besov spaces and some basic properties of Bessel and Besov capacities on R ⁿ are then utilized in getting the desired inequalities. When p=2, the Besov space being considered is a non-local regular Dirichlet space and gives rise to a jump type symmetric Markov process M over the d-set. The upper bound of the transition function of M and metric properties of M -polar sets are then exhibited.     

Locally linearly independent bases for bivariate polynomial spline spaces

Locally linearly independent bases are constructed for the spaces S ^r _d () of polynomial splines of degree d3r+2 and smoothness r defined on triangulations, as well as for their superspline subspaces.     

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.     

Sobolev,Besov and Nikolskii fractional spaces: Imbeddings and comparisons for vector valued spaces on an interval

Summary We consider various fractional properties of regularity for vector valued functions defined on an interval I. In other words we study the functions in the Sobolev spaces W^s,p(I;E), in the Nikolskii spaces N^s,p(I;E), or in the Besov spaces B _{s, p} (I; E). Theses spaces are defined by integration and translation, and E is a Banach space. In particular we study the dependence on the parameters s, p and , that is imbeddings for different parameters. Moreover we compare each space to the others, and we give Lipschits continuity, existence of traces and q-integrability properties. These results rely only on integration techniques.     

RESTRICTED NONLINEAR APPROXIMATION AND SINGULAR SOLUTIONS OF BOUNDARY INTEGRAL EQUATIONS

This paper studies several problems, which are potentially relevant for the construction of adaptive numerical schemes. First, biorthogonal spline wavelets on [0,1] are chosen as a starting point for characterizations of functions in Besov spaces B _r,r ⁶ (0.1) with 0<σ<∞ and (1+σ)⁻¹相似文献   

The Fejér-Riesz inequality for the Besov spaces

A Fejér-Riesz inequality for the weighted Besov spaces B _p,q is given. Some characterizations of functions in B _p,q in terms of their Taylor coefficients are obtained.     

New properties of Besov and Triebel-Lizorkin spaces on RD-spaces

An RD-space ${\mathcal X}$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in ${\mathcal X}$ . In this paper, the authors first give several equivalent characterizations of RD-spaces and show that the definitions of spaces of test functions on ${\mathcal X}$ are independent of the choice of the regularity ${\epsilon\in (0,1)}$ ; as a result of this, the Besov and Triebel-Lizorkin spaces on ${\mathcal X}$ are also independent of the choice of the underlying distribution space. Then the authors characterize the norms of inhomogeneous Besov and Triebel-Lizorkin spaces by the norms of homogeneous Besov and Triebel-Lizorkin spaces together with the norm of local Hardy spaces in the sense of Goldberg. Also, the authors obtain the sharp locally integrability of elements in Besov and Triebel-Lizorkin spaces.     

Near Best Tree Approximation

Tree approximation is a form of nonlinear wavelet approximation that appears naturally in applications such as image compression and entropy encoding. The distinction between tree approximation and the more familiar n-term wavelet approximation is that the wavelets appearing in the approximant are required to align themselves in a certain connected tree structure. This makes their positions easy to encode. Previous work [4,6] has established upper bounds for the error of tree approximation for certain (Besov) classes of functions. This paper, in contrast, studies tree approximation of individual functions with the aim of characterizing those functions with a prescribed approximation error. We accomplish this in the case that the approximation error is measured in L ₂, or in the case p2, in the Besov spaces B _p ⁰(L _p ), which are close to (but not the same as) L _p . Our characterization of functions with a prescribed approximation order in these cases is given in terms of a certain maximal function applied to the wavelet coefficients.     

Intrinsic characterizations of Besov spaces on Lipschitz domains

The aim of this paper is to study the equivalence between quasi‐norms of Besov spaces on domains. We suppose that the domain Ω ? ?ⁿ is a bounded Lipschitz open subset in ?ⁿ. First, we define Besov spaces on Ω as the restrictions of the corresponding Besov spaces on ?ⁿ. Then, with the help of equivalent and intrinsic characterizations (the Peetre‐type characterization 3.10 and the characterization via local means 3.13) of these spaces, we get another equivalent and intrinsic quasi‐norm using, this time, generalized differences and moduli of smoothness. We extend the well‐known characterization of Besov spaces on ?ⁿ described in Theorem 2.4 to the case of Lipschitz domains.     

On Almost Interpolation

We obtain some characterizations of almost interpolation configurations of points with respect to finite-dimensional functional spaces. Particularly, a Schoenberg–Whitney type characterization which is valid for any multivariate spline space relative to an arbitrary partition of a domainA⊂^mis presented. As a closely related problem we investigate sectional structure of finite-dimensional spaces of real functions on a topological spaceA. It is shown that under some reasonable restrictions onAany space of this sort may be considered as piecewise almost Chebyshev.     

Generalized Besov spaces and Triebel-Lizorkin spaces

In this paper the classical Besov spaces Bsp.q and Triebel-Lizorkin spaces Fsp.q for s ∈R are generalized in an isotropy way with the smoothness weights {|2j|aln}∞j=0. These generalized Besov spaces and Triebel-Lizorkin spaces, denoted by Bap.q and Fap.q for a ∈Irk and k ∈N, respectively, keep many interesting properties, such as embedding theorems (with scales property for all smoothness weights), lifting properties for all parameters a, and duality for index 0 < p < ∞. By constructing an example, it is shown that there are infinitely many generalized Besov spaces and generalized Triebel-Lizorkin spaces lying between Bs,p.q and ∪tsBt,p.q,and between Fsp.q and ∪ts Ftp.q, respectively. Between Bs,p,q and ∪tsBt,p.qq,and between Fsp,qand ∪tsFtp.q,respectively.     

Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type

We introduce Besov type function spaces, based on the weak L ^p -spaces instead of the standard L ^p -spaces, and prove a local-in-time unique existence and a blow-up criterion of solutions in those spaces for the Euler equations of perfect incompressible fluid in . For the proof, we establish the Beale-Kato-Majda type logarithmic inequality and commutator type estimates in our weak spaces. Abbreviate title: Euler equations in Besov spaces of weak type     

Banach convolution algebras of functions II

LetG be a noncompact, locally compact group. By means of generalized dyadic decompositions ofG, translation invariant Banach spacesF(B, B, X) of (classes of) measurable functions onG are constructed, e. g. certain weighted amalgams ofL ^p -spaces. Basic properties of these spaces are derived and connections with spaces considered in the literature are indicated. As a main result, sufficient conditions are given which imply that a space of this type is a Banach algebra with respect to convolution.With 1 Figure     

Beyond Besov Spaces,Part 2: Oscillation Spaces

We study several extensions of Besov spaces; these extensions include the oscillation spaces O_p^s,swhich take into account correlations between the positions of large wavelet coefficients through the scales and, more generally, spaces defined through the distributions of suprema of wavelet coefficients on wavelet subtrees. These spaces are independent of the particular wavelet basis chosen. Examples of applications will be taken from image processing and multifractal analysis.