Special uniform approximations of continuous vector-valued functions. Part II: special approximations in CX(T)CY(S)

In this paper, which is a continuation of Timofte (J. Approx. Theory 119 (2002) 291–299, we give special uniform approximations of functions from C_XY(T×S) and C_∞(T×S,XY) by elements of the tensor products C_X(T)C_Y(S), respectively C₀(T,X)C₀(S,Y), for topological spaces T,S and Γ-locally convex spaces X,Y (all four being Hausdorff).     

Bounded and Unitary Elements in Pro-C*-algebras

A pro-C*-algebra is a (projective) limit of C*-algebras in the category of topological *-algebras. From the perspective of non-commutative geometry, pro-C*-algebras can be seen as non-commutative k-spaces. An element of a pro-C*-algebra is bounded if there is a uniform bound for the norm of its images under any continuous *-homomorphism into a C*-algebra. The *-subalgebra consisting of the bounded elements turns out to be a C*-algebra. In this paper, we investigate pro-C*-algebras from a categorical point of view. We study the functor (−) _b that assigns to a pro-C*-algebra the C*-algebra of its bounded elements, which is the dual of the Stone-Čech-compactification. We show that (−) _b is a coreflector, and it preserves exact sequences. A generalization of the Gelfand duality for commutative unital pro-C*-algebras is also presented.     

About spherical functions onSL(2,ℝ)

In this paper, we will discuss some properties of the (n, m)-spherical functions on the Lie groupG = SL(2,ℝ), and obtain the decomposition off inC _c ⁴ (G) into these functions. Also we give the Fourier inversion formula for the (n, m)-spherical functions inC _c ³ (G).     

Exponential trichotomy and p-admissibility for evolution families on the real line

The aim of this paper is to obtain necessary and sufficient conditions for uniform exponential trichotomy of evolution families on the real line. We prove that if p ∈ (1,∞) and the pair (C_b(R,X),C_c(R,X)) is uniformly p-admissible for an evolution family ={U(t,s)}_t≥_s then is uniformly exponentially trichotomic. After that we analyze when the uniform p-admissibility of the pair (C_b(R, X), C_c(R, X)) becomes a necessary condition for uniform exponential trichotomy. As applications of these results we study the uniform exponential dichotomy of evolution families. We obtain that in certain conditions, the admissibility of the pair (C_b(R,X),L^p(R,X)) for an evolution family ={U(t,s)}_t≥_s is equivalent with its uniform exponential dichotomy.     

Finitely additive integration and local integration with respect to upper integrals

Summary We study the integration theory for general integral metrics when restricted to upper integrals q, finding improvements in the relation between the classes of the q-integrable and the q_l-integrable functions. We give new results and notions which lead to the desirable characterizations of q-integrable functions as q_l-integrable f with q(|f|) < ∞, and of q_l-integrable functions via the integrability of their upper truncations, under natural conditions which are fulfilled in most finitely additive integration theories.     

Sufficiency and Duality in Multiobjective Variational Problems with Generalized Type I Functions

Recently Hachimi and Aghezzaf introduced the notion of (F,α,ρ,d)-type I functions, a new class of functions that unifies several concepts of generalized type I functions. Here, we extend the concepts of (F,α,ρ,d)-type I and generalized (F,α,ρ, d)-type I functions to the continuous case and we use these concepts to establish various sufficient optimality conditions and mixed duality results for multiobjective variational problems. Our results apparently generalize a fairly large number of sufficient optimality conditions and duality results previously obtained for multiobjective variational problems.     

Spaces of p-integrable Functions with Respect to a Vector Measure

We study several properties of the Banach lattices L^p (m) and L^p_w (m) of p-integrable scalar functions and weakly p-integrable scalar functions with respect to a countably additive vector measure m. The relation between these two spaces plays a fundamental role in our analysis. This research has been partially supported by La Consejería de Educatión y Ciencia de la Junta de Andalucía.     

The approximation of logarithmic function by q-Bernstein polynomials in the case q > 1

Since in the case q > 1, q-Bernstein polynomials are not positive linear operators on C[0,1], the study of their approximation properties is essentially more difficult than that for 0<q<1. Despite the intensive research conducted in the area lately, the problem of describing the class of functions in C[0,1] uniformly approximated by their q-Bernstein polynomials (q > 1) remains open. It is known that the approximation occurs for functions admit ting an analytic continuation into a disc {z:|z| < R}, R > 1. For functions without such an assumption, no general results on approximation are available. In this paper, it is shown that the function f(x) = ln (x + a), a > 0, is uniformly approximated by its q-Bernstein polynomials (q > 1) on the interval [0,1] if and only if a ≥ 1.      

Spline Approximation Methods with Uniform Meshes in Algebras of Multiplication and Convolution Operators

The purpose of this paper is to obtain necessary and sufficient conditions for maximum defect spline approximation methods with uniform meshes to be stable. The methods are applied to operators belonging to the closed subalgebra of ℒ︁ (L² (ℝ)) generated by operators of multiplication by piecewise continuous functions on ℝ and convolution operators also with piecewise continuousgenerating functions. To that purpose, a C*-algebra of sequences is introduced, which contains the special sequences of approximating operators we are interested in. There is a direct relationship between the applicability of the approximation method to a given operator and invertibility of the corresponding sequence in this C*-algebra. Exploring this relationship, applicability criteria are derived by the use of C*-algebra and Banach algebra techniques (essentialization, localization andidentification of the local algebras by means of construction of locally equivalent representations). Finally, examples are presented, including explicit conditions for the applicability of spline Galerkin methods to Wiener-Hopf operators with piecewise continuous symbols.     

局部紧空间上模糊值模糊测度的正则性

研究局部紧Hausdorff空间上模糊值模糊测度。首先,给出模糊值模糊测度的自连续和一致自连续的定义,然后引进模糊值模糊测度正则性的概念,在自连续或一致自连续的条件下,证明了局部紧空间上模糊值模糊测度的一些定理。     

Gruppi Policiclici Dotati di un Automorfismo Uniforme di Ordinepq

An automorphismϕ of a groupG is said to be uniform il for everyg ∈G there exists anh ∈G such thatG=h ⁻¹ h ^ρ . It is a well-known fact that ifG is finite, an automorphism ofG is uniform if and only if it is fixed-point-free. In [7] Zappa proved that if a polycyclic groupG admits an uniform automorphism of prime orderp thenG is a finite (nilpotent)p′-group. In this paper we continue Zappa’s work considering uniform automorphism of orderpg (p andq distinct prime numbers). In particular we prove that there exists a constantμ (depending only onp andq) such that every torsion-free polycyclic groupG admitting an uniform automorphism of orderpq is nilpotent of class at mostμ. As a consequence we prove that if a polycyclic groupG admits an uniform automorphism of orderpq thenZ _μ (G) has finite index inG.

Al professore Guido Zappa per il suo 90⁰ compleanno     

The Mcshane Integral of Fuzzy-Valued Functions

In this paper, the Mcshane integrals of fuzzy-valued functions are defined, discussed, and characterized. It shows that, in a sense, the Mcshane integrals of interval-valued functions and fuzzy-valued functions are the Riemann-type definitions of the Aumann integral and (K) integral respectively.AMS Subject Classification (2000) 28EI0 (04A72)The authors are supported by the Special Fund of China for Ph.D. Instructors in Universities     

Uniform Convergence of Sequence of Holomorphic Functions in the Unite Disk

In this paper, suppose that ? _m (z)(m = 1,2,…) are of infinite order. Applying the notion of infinite type [4], we obtain the sufficient conditions for uniform convergence of sequence of holomorphic functions in the unite disk.     

模糊分析计算中的结构元方法

提出模糊结构元的概念，研究模糊结构元的性质，给出模糊数和模糊值函数的结构元表现定理。利用模糊数和模糊值函数的结构元表现形式，使得过去必须依赖扩张原理和表现定理来刻画的模糊数运算、模糊值函数的微积分运算等变得更加简单与直观。     

-Operator frames for a Banach space

In this paper, (p,Y)-Bessel operator sequences, operator frames and (p,Y)-Riesz bases for a Banach space X are introduced and discussed as generalizations of the usual concepts for a Hilbert space and of the g-frames. It is proved that the set of all (p,Y)-Bessel operator sequences for a Banach space X is a Banach space and isometrically isomorphic to the operator space B(X,ℓ^p(Y)). Some necessary and sufficient conditions for a sequence of operators to be a (p,Y)-Bessel operator sequence are given. Also, a characterization of an independent (p,Y)-operator frame for X is obtained. Lastly, it is shown that an independent (p,Y)-operator frame for X is just a (p,Y)-Riesz basis for X and has a unique dual (q,Y^*)-operator frame for X^*.     

Hermitian and Positive Integrated C-cosine Functions on Banach Spaces

Peculiar properties of hermitian and positive n-times integrated C-cosine functions on Banach spaces are investigated. Among them are: (1) Any nondegenerate positiven -times integrated C-cosine function is infinitely differentiable in operator norm; (2) An exponentially bounded, nondegenerateC -cosine function on L ^p () (1L ¹(), C₀ , in case C has dense range) is positive if and only if its generator is bounded, positive, and commutes with C.     

An application of <Emphasis Type="Italic">H</Emphasis>-differentiability to nonnegative and unrestricted generalized complementarity problems

This paper deals with nonnegative nonsmooth generalized complementarity problem, denoted by GCP(f,g). Starting with H-differentiable functions f and g, we describe H-differentials of some GCP functions and their merit functions. We show how, under appropriate conditions on H-differentials of f and g, minimizing a merit function corresponding to f and g leads to a solution of the generalized complementarity problem. Moreover, we generalize the concepts of monotonicity, P ₀-property and their variants for functions and use them to establish some conditions to get a solution for generalized complementarity problem. Our results are generalizations of such results for nonlinear complementarity problem when the underlying functions are C ¹, semismooth, and locally Lipschitzian.     

Limit properties of induced mappings

Given a class of mappings f between continua, near- stands for the class of uniform limits of sequences of mappings from . Let 2^f and C(f) mean the induced mappings between hyperspaces. Relations are studied between the conditions: f near- , 2^f near- and C(f) near- . A special attention is paid to the classes of open and of monotone mappings.     

Some properties of set-open topologies

The space of continuous real-valued functions with set-open topology is studied. The coincidence of the set-open topologies for various families λ is considered and conditions for a set-open topology to coincide with the uniform topology are investigated. The separation axioms in the spaces C ^λ(X) are studied. Criteria for the local compactness and σ-compactness of C ^λ(X) are given. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 34, General Topology, 2005.