Adaptive Decomposition by Weighted Inner Functions: A Generalization of Fourier Series

In recent study adaptive decomposition of functions into basic functions of analytic instantaneous frequencies has been sought. Fourier series is a particular case of such decomposition. Adaptivity addresses certain optimal property of the decomposition. The present paper presents a fast decomposition of functions in the $\mathcal {L}^{2}(\partial {\mathbb{D}})$ spaces into a series of inner and weighted inner functions of increasing frequencies.     

A Generalization of Hankel Operators

We introduce a class of operators, called λ-Hankel operators, as those that satisfy the operator equation S*X−XS=λX, where S is the unilateral forward shift and λ is a complex number. We investigate some of the properties of λ-Hankel operators and show that much of their behaviour is similar to that of the classical Hankel operators (0-Hankel operators). In particular, we show that positivity of λ-Hankel operators is equivalent to a generalized Hamburger moment problem. We show that certain linear spaces of noninvertible operators have the property that every compact subset of the complex plane containing zero is the spectrum of an operator in the space. This theorem generalizes a known result for Hankel operators and applies to λ-Hankel operators for certain λ. We also study some other operator equations involving S.     

Minimizing Multivariable Functions by Minimization-Preserving Operators

Given a real function f of class defined on the unit cube Iⁿ=[0,1]ⁿ , n ≥ 2, our purpose consists in finding an algorithm to approximate to by a dimensional reduction. The method deals with α-dense curves γ_α in the domain Iⁿ with arbitrary small density α and a minimization-preserving operator T (briefly M.P.O.) applied to the univariable function By reiterating the action of this M.P.O. we obtain an algorithm to determine a global minimizer t₀^* of f_α. The value f_α(t₀^*), taken as an approximation to f*, only depends on the density α of the curve chosen to densify the domain of the objective function.     

三角函数的推广

利用函数方程定义仿正切函数类,再用仿正切函数定义仿正、余弦函数类,并对这些函数类进行研究.所定义的函数类中的函数具有通常三角函数的性质及运算规律,故而可将其作为三角函数的某种推广.通过实例证实.了通常的正切函数其实是仿正切函数的一种解析延拓.     

On Ptak's Generalization of Hankel Operators

In 1997 Ptak defined generalized Hankel operators as follows: Given two contractions and , an operator is said to be a generalized Hankel operator if and X satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations of T ₁ and T ₂. This approach, call it (P), contrasts with a previous one developed by Ptak and Vrbova in 1988, call it (PV), based on the existence of a previously defined generalized Toeplitz operator. There seemed to be a strong but somewhat hidden connection between the theories (P) and (PV) and we clarify that connection by proving that (P) is more general than (PV), even strictly more general for some T ₁ and T ₂, and by studying when they coincide. Then we characterize the existence of Hankel operators, Hankel symbols and analytic Hankel symbols, solving in this way some open problems proposed by Ptak.     

预解算子族的一个推广

本文引入了正则预解族的概念,给出了其若干性质及两个特征,所得结果推广了预解算子族的相应结果．     

Decomposition of the Space of Regular Operators

Let E be a directed vector space with the Riesz decomposition property and F be a complete v?ctor lattice. It is proved that if G is a solid subspace of E then the subspace of all regular operators annihilating on G is a component in the space of all regular operators from E into F.¹²     

Approximations of Set-Valued Functions by Metric Linear Operators

In this paper we introduce new approximation operators for univariate set-valued functions with general compact images in Rⁿ. We adapt linear approximation methods for real-valued functions by replacing linear combinations of numbers with new metric linear combinations of finite sequences of compact sets, thus obtaining "metric analogues" of these operators for set-valued functions. The new metric linear combination extends the binary metric average of Artstein to several sets and admits any real coefficients. Approximation estimates for the metric analogue operators are derived. As examples we study metric Bernstein operators, metric Schoenberg operators, and metric polynomial interpolants.     

Approximation of Unbounded Functions by Linear Positive Operators

A unified class of linear positive operators has been defined. Using these operators some approximation estimates have been obtained for unbounded functions. For particular linear positive operators these results sharpen and improve the earlier estimates due to Fuhua Cheng (J. Approx. Theory, 1984) and Xiehua Sun (J. Approx. Theory, 1988).     

The Approximation of The Continuous Functions by Means of Some Linear Positive Operators

In this paper we give positive answers to some problems posed by H.Gonska, respectively A.Lupa?. We show here that there exist positive linear operators H _n : C[0, 1] → π_n satisfying      

On the West Decomposition of Riesz Operators

Let T be a Riesz operator on a Banach space and let {_n} be anenumeration of the nonzero eigenvalues of T, each repeated accordingto algebraic multiplicity. Then a sufficient condition thatT admit a West decomposition T = C + Q, where C is compact andQ is quasinilpotent, is that n[_n] < . Furthermore, C andQ can be chosen to have a quasinilpotent commutator.     

The Properties of Functions and Approximation by Summation Rational Operators on the Real Axis

In the space $C(\mathbb{R})$ , we construct special summation operators of Jackson type with rational kernels and under a suitable choice of poles we obtain order estimates for deviations of these operators from functions belonging to certain classes.     

On Bounded Variation Functions by General MKZD Operators

In this paper, we study the rate of convergence for functions of bounded variation for the recently introduced Bzier variant of the Meyer-Knig-Zeller-Durrmeyer operators.     

微分方程递归算子的一种推广

微分方程的多数重要的递归算子都是积微分算子。在试图获得整个对称族时人们常常遇到困难。有时甚至由于缺乏精确性而导致伪对称。本文给出递归算子一种推广,其在某种程度上消除了这些问题。文中还给出了若干重要例子说明这种推广及其重要性。     

Approximation of Continuous Functions by de La Vallee-Poussin Operators

We study the asymptotic (as σ → ∞) behavior of upper bounds of the deviations of functions belonging to the classes and from the so-called de la Vallee-Poussin operators. We obtain asymptotic equalities that, in some important cases, give a solution of the Kolmogorov-Nikol’skii problem for the de la Vallee-Poussin operators on the classes and .__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 230–238, February, 2005.     

由斜坡函数激发的神经网络算子逼近

首先,引入一种由斜坡函数激发的神经网络算子,建立了其对连续函数逼近的正、逆定理,给出了其本质逼近阶.其次,引入这种神经网络算子的线性组合以提高逼近阶,并且研究了这种组合的同时逼近问题.最后,利用Steklov函数构造了一种新的神经网络算子,建立了其在L~p[a,b]空间逼近的正、逆定理.