Entropy Numbers of Embeddings of Weighted Besov Spaces

We investigate the asymptotic behavior of the entropy numbers of the compact embedding $$ B^{s_1}_{p_1,q_1} \!\!(\mbox{\footnotesize\bf R}^d, \alpha) \hookrightarrow B^{s_2}_{p_2,q_2} \!\!({\xxR}). $$ Here $B^s_{p,q} \!({\mbox{\footnotesize\bf R}^d}, \alpha)$ denotes a weighted Besov space, where the weight is given by $w_\alpha (x) = (1+| x |^2)^{\alpha/2}$, and $B^{s_2}_{p_2,q_2} \!({\mbox{\footnotesize\bf R}^d})$ denotes the unweighted Besov space, respectively. We shall concentrate on the so-called limiting situation given by the following constellation of parameters: $s_2 < s_1$, $0 < p_1,p_2 \le \infty$, and $$ \alpha = s_1 - \frac{d}{p_1} - s_2 + \frac{d}{p_2} > d \, \max \Big(0, \frac{1}{p_2}-\frac{1}{p_1}\Big). $$ In almost all cases we give a sharp two-sided estimate.     

Compact Differences of Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions

Let be the weighted Banach space of analytic functions with a topology generated by weighted sup-norm. In the present article, we investigate the analytic mappings and which characterize the compactness of differences of two weighted composition operators on the space . As a consequence we characterize the compactness of differences of composition operators on weighted Bloch spaces.      

Bergman型空间的加权复合算子

Let g1,g2 be normal functions. For all 0 〈 p,q 〈 ∞, the necessary and sufficient conditions for weighted composition operators Tψ,φ ： A^Pg1 → A^Pg2 to be bounded or compact between Bergman type spaces on the unit ball of C^n are given.     

Function Spaces in Lipschitz Domains and Optimal Rates of Convergence for Sampling

Assume that we want to recover $f : \Omega \to {\bf C}$ in the $L_r$-quasi-norm ($0 < r \le \infty$) by a linear sampling method $$ S_n f = \sum_{j=1}^n f(x^j) h_j , $$ where $h_j \in L_r(\Omega )$ and $x^j \in \Omega$ and $\Omega \subset {\bf R}^d$ is an arbitrary bounded Lipschitz domain. We assume that $f$ is from the unit ball of a Besov space $B^s_{pq} (\Omega)$ or of a Triebel--Lizorkin space $F^s_{pq} (\Omega)$ with parameters such that the space is compactly embedded into $C(\overline{\Omega})$. We prove that the optimal rate of convergence of linear sampling methods is $$ n^{ -{s}/{d} + ({1}/{p}-{1}/{r})_+} , $$ nonlinear methods do not yield a better rate. To prove this we use a result from Wendland (2001) as well as results concerning the spaces $B^s_{pq} (\Omega) $ and $F^s_{pq}(\Omega)$. Actually, it is another aim of this paper to complement the existing literature about the function spaces $B^s_{pq} (\Omega)$ and $F^s_{pq} (\Omega)$ for bounded Lipschitz domains $\Omega \subset {\bf R}^d$. In this sense, the paper is also a continuation of a paper by Triebel (2002).     

Strongly Singular Integral Operators on Weighted Hardy Space

In this paper, we obtain that a strongly singular integral operator is bounded on space for 1 < p < ∞. We also obtain that a strongly singular integral operator is a bounded operator from to for some weight w and 0 < p ≤ 1. And by an atomic decomposition, we obtain that a strongly singular integral operator is a bounded operator on for some w and 0 < p ≤ 1. Supported by National 973 Program of China (Grant No. 19990751)     

The Essential Norm of Hankel Operators on the Weighted Bergman Spaces with Exponential Type Weights

Let denote the closed subspace of consisting of analytic functions in the unit disc . For certain class of subharmonic functions and , it is shown that the essential norm of Hankel operator is comparable to the distance norm from H_f to compact Hankel operators.     

Frames for Weighted Shift-invariant Spaces

In this paper we prove the equivalence of the frame property and the closedness for a weighted shift-invariant space $$ V^p_\mu(\Phi) = \left\{\sum \limits^{r}_{i=1} \sum \limits_{j \in \mathbb{Z}^d} c_{i}(j)\phi_{i}(\cdot-j)\left \vert {\{c_{i}(j)\}}_{j \in \mathbb{Z}^{d}} \in {\ell_{\mu}^{p}}\right.\right\}, \quad p \in [1, \infty], $$ which corresponds to ${{\Phi = \Phi^r = (\phi_1, \phi_2, . . . , \phi_r)^T \in (W^{1}_\omega)^r}}$ . We, also, construct a sequence Φ^2k+1 and the sequence of spaces ${{V^{p}_{\mu} (\Phi^{2k+1})}}$ , ${k \in {\mathbb N}}$ , on ${\mathbb R}$ , with the useful properties in sampling, approximations and stability.     

Zygmund 空间到 Bloch 空间上径向导数算子与积分型算子的乘积

设$H(\mathbb{B})$为单位球上全纯函数类,研究了单位球上 Zygmund 空间到 Bloch 空间上径向导数算子$\Re$与积分型算子$I_\varphi^g$乘积的有界性和紧性, 这里 $$ I_\varphi^g f(z)=\int_0^1 \Re f(\varphi(tz))g(tz)\frac{{\rm d}t}{t},\quad z\in\mathbb{B}, $$ 其中$g\in H(\mathbb{B}),\ g(0)=0$, $\varphi$ 是$\mathbb{B}$上全纯自映射.     

Tractability of Approximation for Weighted Korobov Spaces on Classical and Quantum Computers

We study the approximation problem (or problem of optimal recovery in the $L_2$-norm) for weighted Korobov spaces with smoothness parameter $\a$. The weights $\gamma_j$ of the Korobov spaces moderate the behavior of periodic functions with respect to successive variables. The nonnegative smoothness parameter $\a$ measures the decay of Fourier coefficients. For $\a=0$, the Korobov space is the $L_2$ space, whereas for positive $\a$, the Korobov space is a space of periodic functions with some smoothness and the approximation problem corresponds to a compact operator. The periodic functions are defined on $[0,1]^d$ and our main interest is when the dimension $d$ varies and may be large. We consider algorithms using two different classes of information. The first class $\lall$ consists of arbitrary linear functionals. The second class $\lstd$ consists of only function values and this class is more realistic in practical computations. We want to know when the approximation problem is tractable. Tractability means that there exists an algorithm whose error is at most $\e$ and whose information cost is bounded by a polynomial in the dimension $d$ and in $\e^{-1}$. Strong tractability means that the bound does not depend on $d$ and is polynomial in $\e^{-1}$. In this paper we consider the worst case, randomized, and quantum settings. In each setting, the concepts of error and cost are defined differently and, therefore, tractability and strong tractability depend on the setting and on the class of information. In the worst case setting, we apply known results to prove that strong tractability and tractability in the class $\lall$ are equivalent. This holds if and only if $\a>0$ and the sum-exponent $s_{\g}$ of weights is finite, where $s_{\g}= \inf\{s>0 : \xxsum_{j=1}^\infty\g_j^s\,<\,\infty\}$. In the worst case setting for the class $\lstd$ we must assume that $\a>1$ to guarantee that functionals from $\lstd$ are continuous. The notions of strong tractability and tractability are not equivalent. In particular, strong tractability holds if and only if $\a>1$ and $\xxsum_{j=1}^\infty\g_j<\infty$. In the randomized setting, it is known that randomization does not help over the worst case setting in the class $\lall$. For the class $\lstd$, we prove that strong tractability and tractability are equivalent and this holds under the same assumption as for the class $\lall$ in the worst case setting, that is, if and only if $\a>0$ and $s_{\g} < \infty$. In the quantum setting, we consider only upper bounds for the class $\lstd$ with $\a>1$. We prove that $s_{\g}<\infty$ implies strong tractability. Hence for $s_{\g}>1$, the randomized and quantum settings both break worst case intractability of approximation for the class $\lstd$. We indicate cost bounds on algorithms with error at most $\e$. Let $\cc(d)$ denote the cost of computing $L(f)$ for $L\in \lall$ or $L\in \lstd$, and let the cost of one arithmetic operation be taken as unity. The information cost bound in the worst case setting for the class $\lall$ is of order $\cc (d) \cdot \e^{-p}$ with $p$ being roughly equal to $2\max(s_\g,\a^{-1})$. Then for the class $\lstd$ in the randomized setting, we present an algorithm with error at most $\e$ and whose total cost is of order $\cc(d)\e^{-p-2} + d\e^{-2p-2}$, which for small $\e$ is roughly $$ d\e^{-2p-2}. $$ In the quantum setting, we present a quantum algorithm with error at most $\e$ that uses about only $d + \log \e^{-1}$ qubits and whose total cost is of order $$ (\cc(d) +d) \e^{-1-3p/2}. $$ The ratio of the costs of the algorithms in the quantum setting and the randomized setting is of order $$ \frac{d}{\cc(d)+d}\,\left(\frac1{\e}\right)^{1+p/2}. $$ Hence, we have a polynomial speedup of order $\e^{-(1+p/2)}$. We stress that $p$ can be arbitrarily large, and in this case the speedup is huge.     

Localization Operators and Exponential Weights for Modulation Spaces

We study localization operators within the framework of ultradistributions. More precisely, given a symbol a and two windows φ₁, φ₂, we investigate the multilinear mapping from to the localization operator Results are formulated in terms of modulation spaces with weights which may have exponential growth. We give sufficient and necessary conditions for a to be bounded or to belong to a Schatten class. As an application, we study symbols defined by ultra-distributions with compact support, that give trace class localization operators.     

Periodic Solutions of Third-order Differential Equations with Finite Delay in Vector-valued Functional Spaces

In this paper, we study the well-posedness of the third-order differential equation with finite delay(P_3): αu'"(t) + u"(t) = Au(t) + Bu'(t) + Fut +f(t)(t ∈ T := [0,2π]) with periodic boundary conditions u(0) = u(2π), u'(0) = u"(2π),u"(0)=u"(2π) in periodic Lebesgue-Bochner spaces Lp(T;X) and periodic Besov spaces B_(p,q)~s(T;X), where A and B are closed linear operators on a Banach space X satisfying D(A) ∩ D(B) ≠ {0}, α≠ 0 is a fixed constant and F is a bounded linear operator from Lp([-2π, 0]; X)(resp. Bp,qs([-2π, 0]; X)) into X, ut is given by ut(s) = u(t + s) when s ∈ [-2π,0]. Necessary and sufficient conditions for the Lp-well-posedness(resp. B_(p,q)~s-well-posedness)of(P_3) are given in the above two function spaces. We also give concrete examples that our abstract results may be applied.     

K-Theory for the Banach Algebra of Operators on James''s Quasi-reflexive Banach Spaces

We prove that the K-groups of the Banach algebra of bounded, linear operators on the pth James space , where 1 < p < , are given by and . Moreover, for each Banach space and each non-zero, closed ideal contained in the ideal of inessential operators, we show that and . This enables us to calculate the K-groups of for each Banach space which is a direct sum of finitely many James spaces and -spaces.     

Spectral Shorted Operators

If $$\mathcal{H}$$ is a Hilbert space, $$\mathcal{S}$$ is a closed subspace of $$\mathcal{H},$$ and A is a positive bounded linear operator on $$\mathcal{H},$$ the spectral shorted operator $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ is defined as the infimum of the sequence $$\sum (\mathcal{S},A^n )^{1/n} ,$$ where denotes $$\sum \left( {\mathcal{S},B} \right)$$ the shorted operator of B to $$\mathcal{S}.$$ We characterize the left spectral resolution of $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ and show several properties of this operator, particularly in the case that dim $${\mathcal{S} = 1.}$$ We use these results to generalize the concept of Kolmogorov complexity for the infinite dimensional case and for non invertible operators.     

The Monge Problem in Banach Spaces

In this paper, we generalize the Kantorovich functional to K?the-spaces for a cost or a profit function. We examine the convergence of probabilities with respect to this functional for some K?the-spaces. We study the Monge problem: Let be a K?the-space, P and Q two Borel probabilities defined on a Polish space M and a cost function . A K?the functional is defined by (P, Q) = inf where is the law of X. If c is a profit function, we note . (P, Q) = sup Under some conditions, we show the existence of a Monge function, φ, such that , or .      

沿超曲面的强奇异积分算子在调幅空间上的有界性

设■该文主要讨论了上述奇异积分算子在广义的调幅空间上的有界性,其中粗糙核Ω∈L~1(S~(n-2))h(y)为有界的径向函数,而γ(y)是满足一定条件的超曲面.     

Coreflections Versus Regular Epimorphisms in Categories of Topological Spaces

Let be an epireflective subcategory of the category Top of topological spaces which is not contained in the category of indiscrete spaces (e.g. Top, the category of Hausdorff spaces, the category of Tychonoff spaces) and be a coreflective subcategory of . In this paper we prove that the coreflector preserves regular epimorphisms if and only if or is contained in the category of discrete spaces.     

Embedding Derivatives and Integration Operators on Hardy Type Tent Spaces

Acta Mathematica Sinica, English Series - In this paper, we completely characterize the positive Borel measures μ on the unit ball $${\mathbb{B}_n}$$ such that the differential type operator...     

Exponential Dichotomy on the Real Line and Admissibility of Function Spaces

The purpose of this paper is to give characterizations for uniform exponential dichotomy of evolution families on the real line. We consider a general class of Banach function spaces denoted and we prove that if with and the pair is admissible for an evolution family then is uniformly exponentially dichotomic. By an example we show that the admissibility of the pair for an evolution family is not a sufficient condition for uniform exponential dichotomy. As applications, we deduce necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pairs and with      

Analytic Extension of Functions from Analytic Hilbert Spaces

Let M be an invariant subspace of Hv2. It is shown that for each f∈M⊥, f can be analytically extended across (?)Bd\σ(Sz1,…, Szd).