New bases for Triebel-Lizorkin and Besov spaces

We give a new method for construction of unconditional bases for general classes of Triebel-Lizorkin and Besov spaces. These include the , , potential, and Sobolev spaces. The main feature of our method is that the character of the basis functions can be prescribed in a very general way. In particular, if is any sufficiently smooth and rapidly decaying function, then our method constructs a basis whose elements are linear combinations of a fixed (small) number of shifts and dilates of the single function . Typical examples of such 's are the rational function and the Gaussian function This paper also shows how the new bases can be utilized in nonlinear approximation.

     

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.     

Characterizations of function spaces on the sphere using frames

In this paper we introduce a polynomial frame on the unit sphere of , for which every distribution has a wavelet-type decomposition. More importantly, we prove that many function spaces on the sphere , such as , and Besov spaces, can be characterized in terms of the coefficients in the wavelet decompositions, as in the usual Euclidean case . We also study a related nonlinear -term approximation problem on . In particular, we prove both a Jackson-type inequality and a Bernstein-type inequality associated to wavelet decompositions, which extend the corresponding results obtained by R. A. DeVore, B. Jawerth and V. Popov (``Compression of wavelet decompositions', Amer. J. Math. 114 (1992), no. 4, 737-785).

     

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.     

Sampling the miss-distance and transmission function

We use sampling techniques to reconstruct the characteristic function associated with the eigenvalues of two linked Sturm-Liouville operators by a transmission condition. The particular case when the transmission matrix is the identity yields the well-known miss-distance function which is used in the computation of eigenvalues by shooting methods.     

Poisson kernels and sparse wavelet expansions

We give a new characterization of a family of homogeneous Besov spaces by means of atomic decompositions involving poorly localized building blocks. Our main tool is an algorithm for expanding a wavelet into a series of dilated and translated Poisson kernels.

     

Continuous embeddings of Besov-Morrey function spaces

We study embeddings of spaces of Besov-Morrey type, M Bp1,q1s1,r1(Rd ) → M Bp2 ,q2s2 ,r2 (R d ), and obtain necessary and sufficient conditions for this. Moreover, we can also characterise the special weighted situation Bp1 ,r1s1 (R d , w) → M Bp2 ,q2s2 ,r2 (Rd ) for a Muckenhoupt A ∞ weight w, with wα(x) = |x|α , α -d1, as a typical example.     

Besov type function spaces defined on metric-measure spaces

The purpose of this article is to study the Besov type function spaces for maps which are defined on abstract metric-measure spaces. We extend some of the embedding theorems of the classical Besov spaces to the setting of abstract spaces.     

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.     

T1 theorem for Besov and Triebel-Lizorkin spaces

Using the discrete Calderon type reproducing formula and the PlancherelPolya characterization for the Besov and Triebel-Lizorkin spaces, the T1 theorem for the Besov and Triebel-Lizorkin spaces was proved.     

Tl theorem for Besov and Triebel-Lizorkin spaces

Using the discrete Calderon type reproducing formula and the Plancherel-Polya characterization for the Besov and Triebel-Lizorkin spaces, the T1 theorem for the Besov and Triebel-Lizorkin spaces was proved.     

On characterizations of isometries on function spaces

The paper gives characterization for an isometric isomorphism on little Bloch space, VMOA and holomorphic Besov space over the unit ball B_n in C~n.     

Wavelet para-bases and sampling numbers in function spaces on domains

This paper deals with wavelet frames (para-bases), local polynomial reproducing formulas, and sampling numbers in function spaces on arbitrary and on E-thick domains in Euclidean n-space. In an Appendix we collect some recent instruments for corresponding function spaces on Euclidean n-space.     

New characterizations of inhomogeneous Besov and Triebel-Lizorkin spaces over spaces of homogeneous type

In this paper we use the T1 theorem to prove a new characterization with minimum regularity and cancellation conditions for inhomogeneous Besov and Triebel-Lizorkin spaces over spaces of homogeneous type. These results are new even for R^n.     

Besov空间到Zygmund空间上的加权复合算子

研究了单位圆盘上的Besov空间B_(p,q)到Zygmund空间Z的加权复合算子u C_φ(u∈Z),利用函数空间上的算子理论相关知识,得到了u C_φ:B_(p,q)→Z的有界性和紧性的充分必要条件.     

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])).     

Bernstein numbers of embeddings of isotropic and dominating mixed Besov spaces

The purpose of the present paper is to investigate the decay of Bernstein numbers of the embedding from into the space . The asymptotic behaviour of Bernstein numbers of the identity will be also considered.     

Bilinear pseudodifferential operators with forbidden symbols on Lipschitz and Besov spaces

We show that bilinear pseudodifferential operators with symbols in the forbidden class are bounded on products of Lipschitz and Besov spaces.     

Nuclear embeddings in function spaces

Let be the usual Besov spaces in bounded Lipschitz domains Ω in (bounded intervals if ). The paper clarifies under which conditions the continuous embedding between two such spaces with is nuclear.     

Plancherel-Pôlya type inequality on spaces of homogeneous type and its applications

In this paper, using the discrete Calderon reproducing formula on spaces of homogeneous type obtained by the author, we obtain the Plancherel-Pôlya type inequalities on spaces of homogeneous type. These inequalities give new characterizations of the Besov spaces and the Triebel-Lizorkin spaces on spaces of homogeneous type introduced earlier by the author and E. T. Sawyer and also allow us to generalize these spaces to the case where . Moreover, using these inequalities, we can easily show that the Littlewood-Paley -function and -function are equivalent on spaces of homogeneous type, which gives a new characterization of the Hardy spaces on spaces of homogeneous type introduced by Macias and Segovia.