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.     

Atomic and molecular decompositions of anisotropic Besov spaces

In this work we develop the theory of weighted anisotropic Besov spaces associated with general expansive matrix dilations and doubling measures with the use of discrete wavelet transforms. This study extends the isotropic Littlewood- Paley methods of dyadic -transforms of Frazier and Jawerth [19, 21] to non-isotropic settings.Several results of isotropic theory of Besov spaces are recovered for weighted anisotropic Besov spaces. We show that these spaces are characterized by the magnitude of the -transforms in appropriate sequence spaces. We also prove boundedness of an anisotropic analogue of the class of almost diagonal operators and we obtain atomic and molecular decompositions of weighted anisotropic Besov spaces, thus extending isotropic results of Frazier and Jawerth [21].The author was partially supported by the NSF grant DMS-0441817.     

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.     

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.     

Embeddings of anisotropic Besov spaces into Sobolev spaces

We study the embeddings of (homogeneous and inhomogeneous) anisotropic Besov spaces associated to an expansive matrix A into Sobolev spaces, with a focus on the influence of A on the embedding behavior. For a large range of parameters, we derive sharp characterizations of embeddings.     

Anisotropic smoothness spaces via level sets

It has been understood for sometime that the classical smoothness spaces, such as the Sobolev and Besov classes, are not satisfactory for certain problems in image processing and nonlinear PDEs. Their deficiency lies in their isotropy. Functions in these smoothness spaces must be simultaneously smooth in all directions. The anisotropic generalizations of these spaces also have the deficiency that they are biased in coordinate directions. While they allow different smoothness in certain directions, these directions must be aligned to the coordinate axes. In the application areas mentioned above, it would be desirable to measure smoothness in new ways that would allow one to have more local control over the smoothness directions. We introduce one possible approach to this problem based on defining smoothness via level sets. We present this approach in the case of functions defined on ?^d. Our smoothness spaces depend on two smoothness indices (s₁, s₂). The first reflects the smoothness of the level sets of the function, while the second index reflects how smoothly the level sets themselves are changing. As a motivation, we start with d = 2 and investigate Besov smooth domains. © 2007 Wiley Periodicals, Inc.     

Estimates of n ‐widths of Besov classes on two‐point homogeneous manifolds

Estimates of Kolmogorov and linear n ‐widths of Besov classes on compact globally symmetric spaces of rank 1 (i.e., on S^d, P^d (?), P^d (?), P^d (?), P¹⁶(Cay)) are established. It is shown that these estimates have sharp orders in different important cases. A new characterisation of Besov spaces is also given (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hankel operators on the Paley-Wiener space in ℝ d

The basic theory of Besov spaces inI ^d of Paley-Wiener type is developed. This kind of Besov spaces turns out to be quite a success to characterize the Schatten-von Neumann ideal criteria for Hankel operators acting on Paley-Wiener spaces inI ^d.     

Decomposition of Spaces of Distributions Induced by Hermite Expansions

Decomposition systems with rapidly decaying elements (needlets) based on Hermite functions are introduced and explored. It is proved that the Triebel-Lizorkin and Besov spaces on ℝ ^d induced by Hermite expansions can be characterized in terms of the needlet coefficients. It is also shown that the Hermite-Triebel-Lizorkin and Besov spaces are, in general, different from the respective classical spaces. The first author has been supported by NSF Grant DMS-0709046 and the second author by NSF Grant DMS-0604056.     

Decay Estimates for Solutions of Linear Elasticity for Anisotropic Media

We analyse the time decay of solutions to the Cauchy problem for the linear hyperbolic system of elasticity for anisotropic media. As an example, we will consider media with hexagonal symmetry. First we derive decay estimates for special initial data using the method of stationary phase in several variables and degenerate phase function based on the Malgrange preparation theorem. Asymptotic expansions are given to prove the sharpness of the weaker time decay found for zinc and beryl than in the isotropic case. A method using Besov spaces leads to ℒ︁^p–ℒ︁^q-estimates.     

Some Boundary — Contact Problems of the Elasticity Theory with Mixed Boundary Conditions Outside the Contact Surface

n — Dimensional (n ≥ 2) boundary-contact problems of statics of the elasticity theory for homogeneous anisotropic media are investigated when the contact of two bounded domains occurs from the outside on some part of boundaries with mixed boundary conditions. Theorems on the existence and uniqueness of solutions of boundary-contact problems in Besov and Bessel potential spaces are obtained. The smoothness of solutions is studied in closed domains occupied by elastic media.     

The Dirichlet,Neumann and Mixed Boundary Value Problems of the Theory of Elasticity in n-Dimensional Domains with Boundaries Containing Closed Cuspidal Edges

The Dirichlet, Neumann and mixed problems of statics of the theory of elasticity for anisotropic homogeneous media are studied in n-dimensional (n ≧ 2) domains with boundaries containing closed cuspidal edges. Theorems of the existence and uniqueness of solutions of these problems in the Besov and Bessel potential spaces are obtained. Smoothness of solutions in a closed domain occupied by elastic medium is investigated.     

Piecewise Linear Bases and Besov Spaces on Fractal Sets

For a class of closed sets F R ⁿ admitting a regular sequence of triangulations or generalized triangulations, the analogues on F of the Faber—Schauder and Franklin bases are discussed. The characterizations of the Besov spaces on F in the terms of coefficients of functions with respect to these bases are proved. As a consequence, analogous characterizations of the Besov spaces on some fractal domains (including the Sierpinski gasket and the von Koch curve) by coefficients of functions with respect to the wavelet bases constructed in [26] are obtained.     

Besov regularity for second order elliptic boundary value problems with variable coefficients

This paper is concerned with some theoretical foundations for adaptive numerical methods for elliptic boundary value problems. The approximation order that can be achieved by such an adaptive method is determined by certain Besov regularity of the weak solution. We study Besov regularity for second order elliptic problems in bounded domains in ℝ ^d . The investigations are based on intermediate Schauder estimates and on some potential theoretic framework. Moreover, we use characterizations of Besov spaces by wavelet expansions. This work has been supported by the Deutsche Forschungsgemeinschaft (Da 360/1-1)     

Imbedding theorems of Besov spaces in Banach lattices

The paper examines imbeddings of Besov spaces B _{E, θ} ^ω in ideal spaces (Banach lattices) given that ω ∈ S_k). In particular, the symmetric hull of the space B _{E, θ} ^ω is described (E is a symmetric space), an inequality of different metrics is obtained, and imbeddings in Orlicz and Lorentz spaces and in some weighted spaces are studied. Most of the results are easily extended to the anisotropic case. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 159, pp. 69–82, 1987.     

Maximal B-regular boundary value problems with parameters

This study focuses on non-local boundary value problems (BVP) for elliptic differential-operator equations (DOE) defined in Banach-valued Besov (B) spaces. Here equations and boundary conditions contain certain parameters. This study found some conditions that guarantee the maximal regularity and fredholmness in Banach-valued B-spaces uniformly with respect to these parameters. These results are applied to non-local boundary value problems for a regular elliptic partial differential equation with parameters on a cylindrical domain to obtain algebraic conditions that guarantee the same properties.     

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     

A discrete characterization of Besov spaces

The characterization of isotropic Besov spaces B _p,q ^a (I^d) for a>d/p in terms of progressive differences of a function on dyadic points is obtained. Moreover, for with a_i>1/p, an analogous characterization of anisotropic Besov spaces is presented. This work was supported by KBN grant 2 P301 019 06.     

Tractable embeddings of Besov spaces into small Lebesgue spaces

This paper deals with dimension‐controllable (tractable) embeddings of Besov spaces on n‐dimensional torus into small Lebesgue spaces. Our techniques rely on the approximation structure of Besov spaces, extrapolation properties of small Lebesgue spaces and interpolation.     

Best N Term Approximation Spaces for Tensor Product Wavelet Bases

We consider best N term approximation using anisotropic tensor product wavelet bases ("sparse grids"). We introduce a tensor product structure ⊗_q on certain quasi-Banach spaces. We prove that the approximation spaces A^α_q(L₂) and A^α_q(H¹) equal tensor products of Besov spaces B^α_q(L_q), e.g., A^α_q(L₂([0,1]^d)) = B^α_q(L_q([0,1])) ⊗_q · ⊗_q B^α_{q · ·}(L_q([0,1])). Solutions to elliptic partial differential equations on polygonal/polyhedral domains belong to these new scales of Besov spaces.