On the theory of Banach space valued multifunctions. 2. Set valued martingales and set valued measures

This second part of the work on Banach space valued multifunctions begins with a detailed study of set valued martingales, which have their values in a Banach space. Several new convergence theorems are established for different modes of convergence. The profile of a multifunction in connection with set valued martingales is also studied. The notion of weak convergence of multifunctions is introduced and used to obtain additional convergence theorems for set valued martingales. In the last two sections of the paper set valued measures dealt with and an integral with respect to a set valued measure is introduced.     

有界闭凸值测度的集值积分

引入实值函数关于有界闭凸值测度的集值积分,并讨论了集值积分的收敛定理,证明了当集值测度为有界闭凸集值的有界变差集值测度时,关于弱紧凸集值测度的积分性质对有界闭凸集值测度仍然保持.推广了实值函数关于弱紧凸值测度的积分.     

p-Adic函数空间上Hardy型算子的多线性交换子估计

在p-adic域上研究分数次Hardy型算子与CMO(Qnp)函数生成的多线性交换子,建立了交换子在Lebesgue空间和Herz空间上的有界性.对Hardy算子的多线性交换子也得到了相应的结果.     

FLOATING QUANTIZATION EFFECTS ON MULTIRATE SAMPLED-DATA NONLINEAR CONTROL SYSTEMS

In this article,floating quantization effects on multirate sampled-data control systems are studied.It shows that the solutions of multirate digital feedback control systems with nonlinear plant and with floating quantization in the controller are uniformly ultimately bounded if the associated linear systems consisting of linearization of the plant and controller with no quantization are Schur stable.Moreover,it also shows that the difference between the response of multirate digital controllers without quantizers and the same plant with floating quantization in the controllers can be made as small as desired by selecting proper quantization level.     

Vector valued bilinear maximal operator and method of rotations

In this paper we obtain a bilinear analogue of Fefferman-Stein?s vector valued inequality for classical Hardy-Littlewood maximal function. Also, we prove the boundedness of bilinear Hardy-Littlewood maximal operator from Lp₁(Rn)×Lp₂(Rn)→L¹(Rn), where , by applying the method of rotations.     

基于单值中智集Choquet积分算子的群决策方法

单值中智集不仅能描述现实决策系统中不完整信息而且能描述不确定性和不一致信息,已有关于单值中智集的决策方法只能用来解决属性间相互独立的多属性决策问题.考虑到Choquet积分算子的特点,将Choquet积分算子应用到单值中智集中,用以解决属性间有关联关系的多属性群决策问题.首先应用单值中智集余弦相似度比较方法,提出了单值中智集Choquet积分算子,研究了其性质.然后建立了基于单值中智集Choquet积分算子的多属性群决策方法.最后通过实例分析说明了算法的可行性和有效性.     

A Regularization of Quantum Field Hamiltonians with the Aid of p–adic Numbers

Gaussian distributions on infinite-dimensional p-adic spaces are introduced and the corresponding L₂-spaces of p-adic-valued square integrable functions are constructed. Representations of the infinite-dimensional Weyl group are realized in p-adic L₂-spaces. There is a formal analogy with the usual Segal representation. But there is also a large topological difference: parameters of the p-adic infinite-dimensional Weyl group are defined only on some balls (these balls are additive subgroups). p-adic Hilbert space representations of quantum Hamiltonians for systems with an infinite number of degrees of freedom are constructed. Many Hamiltonians with potentials which are too singular to exist as functions over reals are realized as bounded symmetric operators in L₂-spaces with respect to a p-adic Gaussian distribution.     

Entropy-constrained functional quantization of Gaussian processes

The sharp asymptotics for the entropy-constrained -quantization errors of Gaussian measures on a Hilbert space and in particular, for Gaussian processes is derived. The condition imposed is regular variation of the eigenvalues of the covariance operator.

     

Non-Archimedean Analogues of Orthogonal and Symmetric Operators and p-adic Quantization

We study orthogonal and symmetric operators in non-Archimedean Hilbert spaces in the connection with p-adic quantization. This quantization describes measurements with finite precision. Symmetric (bounded) operators in the p-adic Hilbert spaces represent physical observables. We study spectral properties of one of the most important quantum operators, namely, the operator of the position (which is represented in the p-adic Hilbert L₂-space with respect to the p-adic Gaussian measure). Orthogonal isometric isomorphisms of p-adic Hilbert spaces preserve precisions of measurements. We study properties of orthogonal operators. It is proved that each orthogonal operator in the non-Archimedean Hilbert space is continuous. However, there exist discontinuous operators with the dense domain of definition which preserve the inner product. There also exist nonisometric orthogonal operators. We describe some classes of orthogonal isometric operators and we study some general questions of the theory of non-Archimedean Hilbert spaces (in particular, general connections between topology, norm and inner product).     

Embedding vector‐valued Besov spaces into spaces of γ ‐radonifying operators

It is shown that a Banach space E has type p if and only for some (all) d ≥ 1 the Besov space B^{(1/p – 1/2)d} _p,p (?^d ; E) embeds into the space γ (L²(?^d ), E) of γ ‐radonifying operators L²(?^d ) → E. A similar result characterizing cotype q is obtained. These results may be viewed as E ‐valued extensions of the classical Sobolev embedding theorems. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

BOUNDEDNESS OF WEYL QUANTIZATION

1.IntroductionQuantizationisafundamentalprocessinquantumphysics.Amongvariousquantizations,theoneformulatedbyH.Weylassociatestoacertainfunctiona(x,)(thesymbol)anoperatorSincequantizedphysicalquantitiesarerepresentedasoperatorsinthestatespaceH,whichisa...     

A class of vector‐valued dilation‐and‐modulation frames on the half real line

Vector‐valued frames were first introduced under the name of superframes by Balan in the context of signal multiplexing and by Han and Larson from the mathematical aspect. Since then, the wavelet and Gabor frames in have interested many mathematicians. The space models vector‐valued causal signal spaces because of the time variable being nonnegative. But it admits no nontrivial shift‐invariant system and thus no wavelet or Gabor frame since is not a group by addition (not as ). Observing that is a group by multiplication, we, in this paper, introduce a class of multiplication‐based dilation‐and‐modulation ( ) systems, and investigate the theory of frames in . Since is not closed under the Fourier transform, the Fourier transform does not fit . We introduce the notion of Θ_a transform in , and using Θ_a‐transform matrix method, we characterize frames, Riesz bases, and dual frames in and obtain an explicit expression of duals for an arbitrary given frame. An example theorem is also presented.     

Classification and geometric aspects of vector valued Fourier transforms

It is shown that for any locally compact abelian group ?? and 1 ≤ p ≤ 2, the Fourier type p norm with respect to ?? of a bounded linear operator T between Banach spaces, denoted by ‖T |???^??_p‖, satisfies ‖T |???^??_p‖ ≤ ‖T |???^??_p‖, where ?? is the direct product of ?₂, ?₃, ?₄, … It is also shown that if ?? is not of bounded order then Cⁿ_p ‖T |???^??_p‖ ≤ ‖T |???^??_p‖, where ?? is the circle group, n is a onnegative integer and C^p = . From these inequalities, for any locally compact abelian group ?? ‖T |???^??₂‖ ≤ ‖T |???^??₂‖, and moreover if ?? is not of bounded order then ‖T |???^??₂‖ = ‖T |???^??₂‖. The Hilbertian property and B‐convexity are discussed in the framework of Fourier type p norms. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lifting of polynomial symplectomorphisms and deformation quantization

We study the problem of lifting of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. In 1983, Anick proved the fundamental result on approximation of polynomial automorphisms. We obtain similar approximation theorems for symplectomorphisms and Weyl algebra authomorphisms. We then formulate the lifting problem. More precisely, we prove the possibility of lifting of a symplectomorphism to an automorphism of the power series completion of the Weyl algebra of the corresponding rank. The lifting problem has its origins in the context of deformation quantization of the a?ne space and is closely related to several major open problems in algebraic geometry and ring theory.

This paper is a continuation of the study [19 Kanel Belov, A., Razavinia, F., Zhang, W. (2017). Bergman’s centralizer theorem and quantization. Commun. Algebra 1–7.[Taylor &; Francis Online] , [Google Scholar]].     

Periodic decomposition of integer valued functions

We study those functions that can be written as a finite sum of periodic integer valued functions. On ℤ we give three different characterizations of these functions. For this we prove that the existence of a real valued periodic decomposition of a ℤ → ℤ function implies the existence of an integer valued periodic decomposition with the same periods. This result depends on the representation of the greatest common divisor of certain polynomials with integer coefficients as a linear combination of the given polynomials where the coefficients also belong to ℤ[x]. We give an example of an ℤ → {0, 1} function that has a bounded real valued periodic decomposition but does not have a bounded integer valued periodic decomposition with the same periods. It follows that the class of bounded ℤ → ℤ functions has the decomposition property as opposed to the class of bounded ℝ → ℤ functions. If the periods are pairwise commensurable or not prescribed, then we get more general results. Supported by OTKA grants T 43623 and K 61908.     

Existence of asymptotically almost periodic solutions for damped wave equations

In this paper, a class of nonlinear damped wave equations of the form αu^?(t)+u^″(t)=βAu(t)+γAu^′(t)+f(t,u(t)), t?0, satisfying αβ<γ with prescribed initial conditions are studied. Some sufficient conditions are established for the existence and uniqueness of an asymptotically almost periodic solution. These results have significance in the study of vibrations of flexible structures possessing internal material damping. Finally, an example is presented to illustrate the feasibility and effectiveness of the results.     

Weakly Compact Composition Operators on Locally Convex Spaces

Let E be a complete, barrelled locally convex space, let V = (v_n) be an increasing sequence of strictly positive, radial, continuous, bounded weights on the unit disc 𝔻 of the complex plane, and let φ be an analytic self map on 𝔻. The composition operators C_φ : f → f ○ φ on the weighted space of holomorphic functions HV (𝔻, E) which map bounded sets into relatively weakly compact subsets are characterized. Our approach requires a study of wedge operators between spaces of continuous linear maps between locally convex spaces which extends results of Saksman and Tylli [31, 32], and a representation of the space HV (𝔻, E) as a space of operators which complements work by Bierstedt , Bonet and Galbis [4] and by Bierstedt and Holtmanns [6].     

Calderón–Zygmund operators with operator‐valued kernel on homogeneous Besov spaces

We consider generalized Calderón–Zygmund operators whose kernel takes values in the space of all continuous linear operators between two Banach spaces. In the spirit of the T (1) theorem of David and Journé we prove boundedness results for such operators on vector‐valued Besov spaces. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions

We consider a large class of self-adjoint elliptic problems associated with the second derivative acting on a space of vector-valued functions. We present and survey several results that can be obtained by means of two different approaches to the study of the associated eigenvalues problems. The first, more general one allows to replace a secular equation (which is well known in some special cases) by an abstract rank condition. The second one, though available in general, seems to apply particularly well to a specific boundary condition, the sometimes dubbed anti-Kirchhoff condition in the literature, that arises in the theory of differential operators on graphs; it also permits to discuss interesting and more direct connections between the spectrum of the differential operator and some graph theoretical quantities, in particular some results on the symmetry of the spectrum in either case.