Sampling numbers and function spaces

We want to recover a continuous function using only its function values. Let us assume, that f is from the unit ball of some function space (for example a fractional Sobolev space or a Besov space) and the precision of the reconstruction is measured in the norm of another function space of this type. We describe the rate of convergence of the optimal sampling method (linear as well as nonlinear) in this setting.     

On the equivalence of brushlet and wavelet bases

We prove that the Meyer wavelet basis and a class of brushlet systems associated with exponential type partitions of the frequency axis form a family of equivalent (unconditional) bases for the Besov and Triebel-Lizorkin function spaces. This equivalence is then used to obtain new results on nonlinear approximation with brushlets in Triebel-Lizorkin spaces.     

Application of wavelet bases in linear and nonlinear approximation to functions from Besov spaces

Linear and nonlinear approximations to functions from Besov spaces B _{p, q} ^σ ([0, 1]), σ > 0, 1 ≤ p, q ≤ ∞ in a wavelet basis are considered. It is shown that an optimal linear approximation by a D-dimensional subspace of basis wavelet functions has an error of order D ^{-min(σ, σ + 1/2 ? 1/p)} for all 1 ≤ p ≤ ∞ and σ > max(1/p ? 1/2, 0). An original scheme is proposed for optimal nonlinear approximation. It is shown how a D-dimensional subspace of basis wavelet functions is to be chosen depending on the approximated function so that the error is on the order of D ^?σ for all 1 ≤ p ≤ ∞ and σ > max(1/p ? 1/2, 0). The nonlinear approximation scheme proposed does not require any a priori information on the approximated function.     

A note on convolution-type Calderón-Zygmund operators

For convolution-type Calderon-Zygmund operators, by the boundedness on Besov spaces and Hardy spaces, applying interpolation theory and duality, it is known that H5rmander condition can ensure the boundedness on Triebel-Lizorkin spaces Fp^0,q （1 〈 p,q 〈 ∞） and on a party of endpoint spaces FO,q （1 ≤ q ≤ 2）, hut this idea is invalid for endpoint Triebel-Lizorkin spaces F1^0,q （2 〈 q ≤ ∞）. In this article, the authors apply wavelets and interpolation theory to establish the boundedness on F1^0,q （2 〈 q ≤ ∞） under an integrable condition which approaches HSrmander condition infinitely.     

双线性Fourier乘子在Triebel-Lizorkin和Besov空间的有界性

本文利用Littlewood-Paley分解,Fourier变换和逆变换等方法,研究了双线性Fourier乘子在非齐次正光滑性Triebel-Lizorkin空间和Besov空间的有界性.     

Lipschitz曲线上Besov-Triebel-Lizorkin空间的嵌入定理

本文用离散的Calderón型再生公式。证明了Lipschitz曲线上Beasov空间与Triebel-Lizorkin空间的嵌入定理。     

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.     

Decomposition of Triebel-Lizorkin and Besov spaces in the context of Laguerre expansions

A pair of dual frames with almost exponentially localized elements (needlets) are constructed on based on Laguerre functions. It is shown that the Triebel-Lizorkin and Besov spaces induced by Laguerre expansions can be characterized in terms of respective sequence spaces that involve the needlet coefficients.     

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)     

级联算法在Besov和Triebel-Lizorkin空间上的有界性和收敛性(I)

本文通过引入级联算法的特征多项式和利用一维分解技术,完整地刻画了级联算法在Besov和Triebel-Lizorkin空间上的增量和收敛性。     

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.     

Besov 空间上的算子插值定理

本文研究了算子的插值问题.利用Riesz-Thorin定理的证明方法,并运用Daubechies小波得到了Besov空间上的线性算子的插值定理.     

Maximal regularity of second order delay equations in Banach spaces

We give necessary and sufficient conditions of Lp-maximal regularity(resp.B sp ,q-maximal regularity or F sp ,q-maximal regularity) for the second order delay equations:u″(t)=Au(t) + Gu't + F u t + f(t), t ∈ [0, 2π] with periodic boundary conditions u(0)=u(2π), u′(0)=u′(2π), where A is a closed operator in a Banach space X,F and G are delay operators on Lp([-2π, 0];X)(resp.Bsp ,q([2π, 0];X) or Fsp,q([-2π, 0;X])).     

BOUNDEDNESS OF CALDERN-ZYGMUND OPERATORS ON BESOV SPACES AND ITS APPLICATION

In this article, the author introduces a class of non-convolution Calder′on-Zygmund operators whose kernels are certain sums involving the products of Meyer wavelets and their convolutions. The boundedness on Besov spaces ■p0 ,q(1 ≤p,q ≤∞) is also obtained. Moreover, as an application, the author gives a brief proof of the known result that Hrmander condition can ensure the boundedness of convolution-type Calder′on-Zygmund operators on Besov spaces ■p0 ,q(1 ≤p,q ≤∞). However, the proof is quite different from the previous one.