Generalized Convolution for the Discrete Jacobi Transformation

We study the translation and the convolution associated to the discrete Jacobi transformation on complex sequences of slow and rapid growth. Also we establish new topological properties for these spaces of sequences. This revised version was published online in June 2006 with corrections to the Cover Date.     

迁移算子离散本征值及其聚点的分布

讨论了各向异性、能量相关、非均匀有界凸体介质中迁移算子的谱,在省略某些不符合实际的 特殊条件的情况下,对这类算子离散本征值及其聚点的位置分布,同样获得了一个类似的结果.     

Probability Inequalities and Limit Theorems for Generalized L-Statistics

We obtain exponential upper bounds for tails of distributions of generalized L-statistics based on a sample from an exponential distribution. We prove the asymptotic normality of generalized L-statistics based on a sample from the uniform distribution on [0,1] and of L-statistics with decomposed kernels (without any restrictions on the sample distribution type).     

Limit Theorems for Non-Commutative Random Variables

We study the weak law of large numbers and the central limit theorem for non-commutative random variables. We first define the concepts of variance and expectation for probability measures on homogeneous spaces, and formulate the weak law of large numbers and the central limit theorem for probability measures on locally compact groups. Then, we consider the non-commutative case, where the homogeneous space is replaced by a C^*-algebra that is equipped with a locally compact group G of automorphisms. We define the concepts of variance and expectation in the non-commutative situation. Furthermore, we prove that the weak law of large numbers and the central limit theorem hold for non-commutative random variables on if they hold on the group G of automorphisms.     

Convolution-Dominated Operators on Discrete Groups

We study infinite matrices A indexed by a discrete group G that are dominated by a convolution operator in the sense that for x ∈ G and some . This class of “convolution-dominated” matrices forms a Banach-*-algebra contained in the algebra of bounded operators on ℓ ²(G). Our main result shows that the inverse of a convolution-dominated matrix is again convolution-dominated, provided that G is amenable and rigidly symmetric. For abelian groups this result goes back to Gohberg, Baskakov, and others, for non-abelian groups completely different techniques are required, such as generalized L ¹-algebras and the symmetry of group algebras. K. G. was supported by the Marie-Curie Excellence Grant MEXT-CT 2004-517154.     

Non-supercyclicity of Volterra Convolution and Related Operators

We show that if V ^α (α > 0) is the Riemann-Liouville fractional integration operator and T is an invertible operator on L ²(0, 1) which commutes with V , then TV ^α is not supercyclic on L ²(0, 1); in particular, many Volterra convolution operators are not supercyclic. The technique is based on an argument used by Gallardo-Gutiérrez and Montes-Rodríguez to show that V is not supercyclic.     

广义Φ-增生算子最速下降迭代序列的收敛定理

证明了任意实Banach空间广义Φ-增生算子的最速下降迭代序列的收敛定理,推广了ZeiqingLiu和周海云等人的近期结果.     

Discrete Approximation of Unbounded Operators and Approximation of their Spectra

Let E be a Banach space over and let the densely defined closed linear operator A: (A)E→E be discretely approximated by the sequence ((A_n, (A_n)))_n of operators A_n where each A_n is densely defined in the Banach space F_n. Let σ_a(A) be the approximate point spectrum of A and let σ(A_n) denote the -pseudospectrum of A_n. Generalizing our own result, we show that σ_a(A)lim inf σ(A_n)=_n ∩_kn σ(A_k) holds for every >0. We deduce that then for every compact set K lim_n dist(σ_a(A)∩K, σ_a(A_n))=0 provided there exists M>0 such that (λ−A_n)⁻¹M dist(λ, σ(A_n))⁻¹ holds for every n and every λ in the resolvent set ρ(A_n) of A_n. We finally treat the problem under which conditions σ_a(A) can be approximated from below. More precisely we investigate the problem: Under which assumptions does ∩_>0 ∩_{n kn} σ_, a(A_k)σ_a(A) hold where σ_, a(A) denotes the -approximate pseudospectrum?     

A Limit Theorem for Dirichlet L-Functions

We prove a limit theorem on the weak convergence of probability measures in the space of continuous functions for Dirichlet L-functions. The result generalizes a similar theorem for the Riemann zeta-function.     

Generalized KKM Type Theorems in FC-Spaces with Applications (I)

The class KKM(X,Y) (resp., s-KKM(X,Y,Z)) of set-valued mappings with KKM (resp., s-KKM) property is introduced in FC-spaces without any convexity structure. Some generalized KKM (resp., s-KKM) type theorems are proved in FC-spaces under much weak assumptions. As applications, some new section theorems and coincidence theorems are established in FC-spaces. These theorems generalize many known results in literature. The further applications of these results will be given in a follow-up paper.     

Existence theorem and algorithm for a general implicit variational inequality in Banach space

By using the generalized f-projection operator, the existence theorem of solutions for the general implicit variational inequality GIVI(T-ξ,K) is proved without assuming the monotonicity of operators in reflexive and smooth Banach space. An iterative algorithm for approximating solution of the general implicit variational inequality is suggested also, and the convergence for this iterative scheme is shown. These theorems extend the corresponding results of Wu and Huang [K.Q. Wu, N.J. Huang, Comput. Math. Appl. 54 (2007) 399–406], Wu and Huang [K.Q. Wu, N.J. Huang, Bull. Austral. Math. Soc. 73 (2006) 307–317], Zeng and Yao [L.C. Zeng, J.C. Yao, J. Optimiz. Theory Appl. 132 (2) (2007) 321–337] and Li [J. Li, J. Math. Anal. Appl. 306 (2005) 55–71].     

Fredholm Properties of Band-dominated Operators on Periodic Discrete Structures

Let (X, ~) be a combinatorial graph the vertex set X of which is a discrete metric space. We suppose that a discrete group G acts freely on (X, ~) and that the fundamental domain with respect to the action of G contains only a finite set of points. A graph with these properties is called periodic with respect to the group G. We examine the Fredholm property and the essential spectrum of band-dominated operators acting on the spaces l ^p (X) or c_0(X), where (X, ~) is a periodic graph. Our approach is based on the thorough use of band-dominated operators. It generalizes the necessary and sufficient results obtained in [39] in the special case and in [42] in case X = G is a general finitely generated discrete group. Submitted: May 21, 2007. Revised: September 25, 2007. Accepted: November 5, 2007.     

Limit Theorems for Sums of Heavy-tailed Variables with Random Dependent Weights

Let be independent and identically distributed random variables with heavy-tailed distributions. Consider a sequence of random weights , independent of and focus on the weighted sums , where μ involves a suitable centering. We establish sufficient conditions for these weighted sums to converge to non-trivial limit processes, as n→∞, when appropriately normalized. The convergence holds, for example, if is strictly stationary, dependent, and W ₁ has lighter tails than U ₁. In particular, the weights W _j s can be strongly dependent. The limit processes are scale mixtures of stable Lévy motions. We establish weak convergence in the Skorohod J ₁-topology. We also consider multivariate weights and show that they converge weakly in the strong Skorohod M ₁-topology. The M ₁-topology, while weaker than the J ₁-topology, is strong enough for the supremum and infimum functionals to be continuous. This research was partially supported by a fellowship of the Horace H. Rackham School of Graduate Studies at the University of Michigan and the NSF Grants BCS-0318209 and DMS-0505747 at Boston University.     

On Limit Theorems for Continued Fractions

It is shown that for sums of functionals of digits in continued fraction expansions the Kolmogorov-Feller weak laws of large numbers and the Khinchine-Lévy-Feller-Raikov characterization of the domain of attraction of the normal law hold.      

The Generalized Chern Character and Lefschetz Numbers in W*-Modules

We define N-theory as being an analogue of K-theory on the category of von Neumann algebras such that K ₀(A)N ₀(A) for any von Neumann algebra A. Moreover, it turns out to be possible to construct the extension of the Chern character to some homomorphism from N ₀(A) to an even Banach cyclic homology of A. Also, we define generalized Lefschetz numbers for an arbitrary unitary endomorphism U of an A-elliptic complex. We study them in the situation when U is an element of a representation of some compact Lie group.     

广义Bethe树指标可列马氏链场关于广义随机选择系统的一类局部极限定理

研究任意广义Bethe树指标马尔可夫链场二元泛函关于广义随机选择系统的一类局部极限定理.作为推论得到了广义随机选择系统中任意Cayley树上状态频率和状态序偶的一类极限定理.证明中采用了一种研究马尔可夫链场的较新颖的分析方法.     

Limit theorems for probability measures on simply connected nilpotent lie groups

LetX ₁,X ₂,... be i.i.d. random variables with values in a simply connected nilpotent Lie groupG. Assume the laws of to be weakly convergent to a probability measure onG, _n Aut(G), and (k _n)_n strictly increasing in. In this paper we want to characterize the possible limit laws. We obtain that every limit law is continuously embeddable and we prove a kind of functional limit theorem. Further, we study the connections between two different concepts of stability (resp. semistability) and limit laws. Finally, we describe the various domains of attraction of measures (resp. of convolution semigroups).     

Mixing Limit Theorems of Maximum of Absolute Sums and their Convergence Rates

Conditioned, in the sense of Rényi (Acta Math. Acad. Sci. Hungar. 9, 215–228 1958), limit theorem in the L_p-norm of the maximum of absolute sums of independent identically distributed random variables is established and its exact rate of convergence is given. The results are equivalent to establishing analogous results for the supremum of random functions of partial sums defined on C[0,1], i.e., the invariance principle. New methodologies are used to prove the results that are profoundly different from those used in Rényi (Acta Math. Acad. Sci. Hungar. 9, 215–228, 1958) and subsequent authors in proving the conditioned central limit theorem for partial sums.     

On K0-groups of operator algebras on Banach spaces

This paper merges some classifications of G-M-type Banach spaces simplifically, discusses the condition of K ₀(B(X)) = 0 for operator algebra B(X) on a Banach space X, and obtains a result to improve Laustsen's sufficient condition, gives an example to show that X ≈ X ² is not a sufficient condition of K ₀(B(X)) = 0.