The construction and approximation of some neural networks operators

In this paper,the technique of approximate partition of unity is used to construct a class of neural networks operators with sigmoidal functions.Using the modulus of continuity of function as a metric,...     

一类解析算子函数空间上的复合算子

本文用算子函数论的方法，研究了解析算子函数的Banach空间X，X0上的复合算子．给出此复合算子为有界的条件，并刻划了此复合算子在X0上为紧的特征．     

Approximation by means of piecewise linear functions

In the present paper we use piecewise linear functions in order to obtain representations and estimates for the remainder in approximating continuous functions by positive linear operators. Applications of these results for Bernstein and Stancu’s operators are also presented. In addition, we give some partial results concerning the best constant problem for Bernstein operators with respect to the second order modulus of continuity.     

Interpolation by neural network operators activated by ramp functions

In this paper, a family of interpolation neural network operators are introduced. Here, ramp functions as well as sigmoidal functions generated by central B-splines are considered as activation functions. The interpolation properties of these operators are proved, together with a uniform approximation theorem with order, for continuous functions defined on bounded intervals. The relations with the theory of neural networks and with the theory of the generalized sampling operators are discussed.     

Spectral analysis of eigenparameter dependent boundary value transmission problems

In this paper, the singular second order differential operators are considered defined on the multi-interval. Some boundary and transmission conditions are imposed on the maximal domain functions with the spectral parameter. After constructing the differential operators associated with the boundary value transmission problems on the suitable Hilbert spaces, it is proved that these operators are the maximal dissipative operators. Finally constructing the model operators which are established with the help of the scattering functions, it is proved that all root vectors of the maximal dissipative operators are complete in the Hilbert spaces.     

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.     

On the analytic model of a class of semi-hyponormal operators

The analytic model of a class of semi-hyponormal operators is derived using three kernal functions. In addition, explicit forms of the kernal functions are given and the Pincus principal functions of the operators are calculated.     

Approximation properties of modified Gamma operators

In this paper we present a survey of rates of pointwise approximation of modified Gamma operators Gn for locally bounded functions and absolutely continuous functions by using some inequalities and results of probability theory with the method of Bojanic-Cheng. In the paper a kind of locally bounded functions is introduced with different growth conditions in the fields of both ends of interval (0,+∞), and it is found out that the operators have different properties compared to the Gamma operators discussed in [X.M. Zeng, Approximation properties of Gamma operators, J. Math. Anal. Appl. 311 (2005) 389-401]. And we obtain two main theorems. Theorem 1 gives an estimate for locally bounded functions which subsumes the approximation of functions of bounded variation as a special case. Theorem 2 gives an estimate for absolutely continuous functions which is best possible in the asymptotical sense.     

On the approximation of convex functions by Bernstein-type operators

As a consequence of Jensen's inequality, centered operators of probabilistic type (also called Bernstein-type operators) approximate convex functions from above. Starting from this fact, we consider several pairs of classical operators and determine, in each case, which one is better to approximate convex functions. In almost all the discussed examples, the conclusion follows from a simple argument concerning composition of operators. However, when comparing Szász-Mirakyan operators with Bernstein operators over the positive semi-axis, the result is derived from the convex ordering of the involved probability distributions. Analogous results for non-centered operators are also considered.     

Monotone Operators Representable by l.s.c. Convex Functions

A theorem due to Fitzpatrick provides a representation of arbitrary maximal monotone operators by convex functions. This paper explores representability of arbitrary (nonnecessarily maximal) monotone operators by convex functions. In the finite-dimensional case, we identify the class of monotone operators that admit a convex representation as the one consisting of intersections of maximal monotone operators and characterize the monotone operators that have a unique maximal monotone extension.Mathematics Subject Classifications (2000) 47H05, 46B99, 47H17.     

Hermitian Weighted Composition Operators and Bergman Extremal Functions

Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper, Hermitian weighted composition operators on weighted Hardy spaces of the unit disk are studied. In particular, necessary conditions are provided for a weighted composition operator to be Hermitian on such spaces. On weighted Hardy spaces for which the kernel functions are ${(1 - \overline{w}z)^{-\kappa}}$ for κ ≥ 1, including the standard weight Bergman spaces, the Hermitian weighted composition operators are explicitly identified and their spectra and spectral decompositions are described. Some of these Hermitian operators are part of a family of closely related normal weighted composition operators. In addition, as a consequence of the properties of weighted composition operators, we compute the extremal functions for the subspaces associated with the usual atomic inner functions for these weighted Bergman spaces and we also get explicit formulas for the projections of the kernel functions on these subspaces.     

Bézier variant of Baskakov-Beta-Stancu operators

In the year 1994, Gupta (Approx Theory Appl (N.S.) 10(3):74–78, 1994) introduced the integral modification of well known Baskakov operators with weights of Beta basis functions and obtained better approximation over the usual Baskakov Durrmeyer operators. The rate of convergence for Bézier variant of these operators for functions of bounded variations were discussed in Gupta (Int J Math Math Sci 32(8):471–479, 2002). The present paper is the extension of the previous work, here we consider the Bézier variant of Baskakov-Beta-Stancu operators. We estimate the rate of convergence of these operators for the bounded functions. In the end of the paper we suggest an open problem.     

Spectral analysis of singular ordinary differential operators with indefinite weights

In this paper we develop a perturbation approach to investigate spectral problems for singular ordinary differential operators with indefinite weight functions. We prove a general perturbation result on the local spectral properties of selfadjoint operators in Krein spaces which differ only by finitely many dimensions from the orthogonal sum of a fundamentally reducible operator and an operator with finitely many negative squares. This result is applied to singular indefinite Sturm-Liouville operators and higher order singular ordinary differential operators with indefinite weight functions.     

Weighted boundedness for rough Marcinkiewicz integrals with different weights

In this paper the authors give the weighted boundedness of the Marcinkiewicz integral operators for different weight functions, where the kernel functions of the operators discussed here are without smoothness not only on the sphere but also in the radial direction.     

Functions of commuting contractions under perturbation

The purpose of the paper is to obtain estimates for differences of functions of two pairs of commuting contractions on Hilbert space. In particular, Lipschitz type estimates, Hölder type estimates, Schatten–von Neumann estimates are obtained. The results generalize earlier known results for functions of self‐adjoint operators, normal operators, contractions and dissipative operators.     

Convolution factorability of bilinear maps and integral representations

In this paper we consider a special class of continuous bilinear operators acting in a product of Banach algebras of integrable functions with convolution product. In the literature, these bilinear operators are called ‘zero product preserving’, and they may be considered as a generalization of Lamperti operators. We prove a factorization theorem for this class, which establishes that each zero product preserving bilinear operator factors through a subalgebra of absolutely integrable functions. We obtain also compactness and summability properties for these operators under the assumption of some classical properties for the range spaces, as the Dunford–Pettis property or the Schur property and we give integral representations by some concavity properties of operators. Finally, we give some applications for integral transforms, and an integral representation for Hilbert–Schmidt operators.     

Abschätzung der Normen von Operatoren analytischer Funktionen

This article investigates the norms of certain interpolation operators of analytic functions on the unit disc. In particular, it is shown that the norms of interpolation operators being the identical operator for all n -degree polynomials have a lower bound of order ln n . This result is compared with a recent result regarding trigonometric interpolation of continuous functions on the unit circle. It is shown that opposed to the operators of analytic functions on the unit disc, the method of oversampling can be applied in order to uniformly bound the interpolation operators. Moreover, some practical implications with regard to communication engineering are discussed. It is concluded that in practice the results lead to non-linear interpolation operators.     

Hypercyclic Sequences of Differential and Antidifferential Operators

In this paper, we provide some extensions of earlier results about hypercyclicity of some operators on the Fréchet space of entire functions of several complex variables. Specifically, we generalize in several directions a theorem about hyper- cyclicity of certain infinite order linear differential operators with constant coefficients and study the corresponding property for certain kinds of “antidifferential” operators which are introduced in the paper. In addition, the existence of hypercyclic functions for certain sequences of differential operators with additional properties, for instance, boundedness or with some nonvanishing derivatives, is established.     

Local operators between spaces of ultradifferentiable functions and ultradistributions

In this article we characterize certain ultradifferential operators by the condition of being local. First, we examine the continuity properties of local linear operators on spaces of ultradifferentiable functions in the sense of Beurling and of Roumieu. Next, a structure theorem for vector-valued ultradistributions with support at the origin is proved. This result leads to a representation theorem for continuous local operators from spaces of ultradifferentiable functions into various spaces of ultradistributions. In combination with the continuity results we thus obtain in many cases the desired characterization.     

Approximation properties of Szász‐Mirakyan‐Kantorovich type operators

In this paper, we introduce and study new type Szász‐Mirakyan‐Kantorovich operators using a technique different from classical one. This allow to analyze the mentioned operators in terms of exponential test functions instead of the usual polynomial type functions. As a first result, we prove Korovkin type approximation theorems through exponential weighted convergence. The rate of convergence of the operators is obtained for exponential weights.