On the estimate of the uniform deviation from the class of functions with bounded third derivative

The problem of the uniform approximation of a continuous function on a closed interval by a class of functions with a uniformly bounded third derivative is considered. It is shown that the value of best approximation of a function by this class cannot be estimated linearly in terms of its third-order modulus of continuity. At the same time, such estimates exist for classes with bounded first or second derivatives.     

Approximation by the class of functions with bounded second derivative

The paper studies the problem of uniform approximation of a continuous function on a closed interval by the class of functions with bounded second derivative. We prove an estimate of the value of best approximation of the function by this class via its second modulus of continuity. The obtained estimate is sharp for the class of continuous functions.     

An estimate for the least uniform deviation from the class of functions with bounded second derivative

We consider the problem of uniform approximation of a continuous function on an interval by means of the class of functions with bounded second derivative. We prove an estimate for the least uniform deviation of a function in terms of its local deviations on uniform and nonuniform three-point grids.     

Reflection of plane waves by rough surfaces in the sense of Born approximation

The topic of the present paper is the reflection of electromagnetic plane waves by rough surfaces, that is, by smooth and bounded perturbations of planar faces. Moreover, the contrast between the cover material and the substrate beneath the rough surface is supposed to be low. In this case, a modification of Stearns’ formula based on Born approximation and Fourier techniques is derived for a special class of surfaces. This class contains the graphs of functions where the interface function is a radially modulated almost periodic function. For the Born formula to converge, a sufficient and almost necessary condition is given. A further technical condition is defined, which guarantees the existence of the corresponding far field of the Born approximation. This far field contains plane waves, far‐field terms such as those for bounded scatterers, and, additionally, a new type of terms. The derived formulas can be used for the fast numerical computations of far fields and for the statistics of random rough surfaces. Copyright © 2013 John Wiley & Sons, Ltd.     

Approximation of Smooth Functions on the Semiaxis by Entire Functions of Bounded Half-Degree

In this paper, we study the pointwise approximation of functions defined on the real semiaxis and having an rth derivative bounded almost everywhere. The approximation is performed by means of entire functions of bounded half-degree, which were introduced by S. N. Bernstein. An asymptotically sharp estimate for pointwise approximation of this class of functions is obtained.     

A characterization of bounded convex approximation properties

The characterization of bounded approximation properties defined by arbitrary operator ideals due to Oja is extended to bounded convex approximation properties. As an application, it is shown that the unique extension property of a Banach space X enables to lift the metric convex approximation property from a Banach space X to its dual space X*.     

On rates of convergence for approximations to the generalized inverse

A rate of convergence result for a general class of approximation methods for the generalized inverse of a bounded linear operator with arbitrary range is presented. Applications are given to a number of iterative and noniterative approximation techniques.     

Pseudo-spectral Approximations for a Class of the Kdv-Burgers Type Equation

In this paper, the pseudo-spectral approximations for a class of the Kdv-Burgers type equation is presented. Convergence and stability of the approximation have been proved by Sobolev‘s inequalities and the bounded extensive method of the nonlinear function. Finally, the numerical examples are proposed.     

Approximation of functions subharmonic in a disk by the logarithm of the modulus of an analytic function

Yulmukhametov's result concerning the approximation of a function subharmonic in a bounded domain by the logarithm of the modulus of an analytic function is supplemented with an estimate of the exceptional set in the important case of a disk. We show that this approximation is unimprovable in a certain sense.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 8, pp. 1080–1083, August, 1994.     

Adaptive tuning of the fuzzy controller for robots

An adaptive tuning algorithm of the fuzzy controller is developed for a class of serial-link robot arms. The algorithm can on-line tune parameters of premise and consequence parts of fuzzy rules of the fuzzy basis function (FBF) controller. The main part of the fuzzy controller is a fuzzy basis function network to approximate unknown rigid serial-link robot dynamics. Under some mild assumptions, a stability analysis guarantees that both tracking errors and parameter estimate errors are bounded. Moreover, a robust technique is adopted to deal with uncertainties including approximation errors and external disturbances. Simulations of the proposed controller on the PUMA-560 robot arm demonstrate the effectiveness.     

COMPUTATIONAL COMPLEXITY IN WORST,STOCHASTIC AND AVERAGE CASE SETTING ON FUNCTIONAL APPROXIMATION PROBLEM OF MULTIVARIATE

The order of computational complexity of all bounded linear functional ap proximation problem is determined for the generalized Sobolev class W_p~(?)(Id), Nikolskii class H|∞~k(Id) in the worst (deterministic), stochastic and average case setting, from which it is concluded that the bounded linear functional approximation problem for the classes W_p~(?)(Id) and H_∞~k(Id) is intractable in worst case setting, but is tractable with respect to stochastic and average case setting.     

带有摄动的随机非线性系统的自适应神经网络控制

针对一类带有摄动的随机严格反馈非线性系统,引入积分型Lyapunov函数,利用神经网络的逼近能力,后推设计方法以及Young's不等式,构造出一类简单有效的自适应神经网络状态反馈控制器,在一定条件下,通过Lyapunov方法,证明了闭环系统的所有信号在二阶或四阶矩意义下半全局一致终结有界.仿真结果验证了所提控制方案的有效性.     

The approximation properties determined by operator ideals

We introduce the notion of the right approximation property with respect to an operator ideal A and solve the duality problem for the approximation property with respect to an operator ideal A, that is, a Banach space X has the approximation property with respect to A ^d whenever X* has the right approximation property with respect to an operator ideal A. The notions of the left bounded approximation property and the left weak bounded approximation property for a Banach operator ideal are introduced and new symmetric results are obtained. Finally, the notions of the p-compact sets and the p-approximation property are extended to arbitrary Banach operator ideals. Known results of the approximation property with respect to an operator ideal and the p-approximation property are generalized.     

On the rate of approximation of singular functions by step functions

We consider approximations of a monotone function on a closed interval by step functions having a bounded number of values: the dependence on the number of values of the rate of approximation in the norm of the spaces L _p is studied. A criterion for the singularity of the function in terms of the rate of approximation is obtained. For self-similar functions, we obtain sharp estimates of the rate of approximation in terms of the self-similarity parameters. Functions with arbitrarily fast and arbitrarily slow (down to the theoretic limit) rate of approximation are constructed.     

Discrete Approximations of Controlled Stochastic Systems with Memory: A Survey

This survey article considers discrete approximations of an optimal control problem in which the controlled state equation is described by a general class of stochastic functional differential equations with a bounded memory. Specifically, three different approximation methods, namely (i) semidiscretization scheme; (ii) Markov chain approximation; and (iii) finite difference approximation, are investigated. The convergence results as well as error estimates are established for each of the approximation methods.     

A subgradient algorithm for certain minimax and minisum problems

We present a subgradient algorithm for minimizing the maximum of a finite collection of functions. It is assumed that each function is the sum of a finite collection of basic convex functions and that the number of different subgradient sets associated with nondifferentiable points of each basic function is finite on any bounded set. Problems belonging to this class include the linear approximation problem and both the minimax and minisum problems of location theory. Convergence of the algorithm to an epsilon-optimal solution is proven and its effectiveness is demonstrated by solving a number of location problems and linear approximation problems.This research was partially supported by the Army Research Office, Triangle Park, NC, under contract number DAH-CO4-75-G-0150, and by NSF grants ENG 16-24294 and ENG 75-10225.     

齐型空间上的Lipschitz函数与Littlewood－Paley g－函数

在θ阶正规齐型空间上，如果算子列｛Ｓｋ｝ｋ∈Ｚ是恒等逼近，Ｄｋ＝Ｓｋ－Ｓｋ－１；本文给出一个用｛Ｄｋ｝ｋ∈Ｚ表达的ｆ∈Ｌｉｐα（Ｌｉｐｓｃｈｉｔｚ函数类，０＜α＜θ）的充分必要条件．作为其推论得到，对于ｆ∈ＬＩｐα，其Ｌｉｔｔｌｅｗｏｏｄ－Ｐａｌｅｙｇ函数ｇ（ｆ）（Ｘ）或者处处为无穷大，或者在Ｌｉｐα上有界．     

Approximation algorithms for graph approximation problems

Several versions of the graph approximation problem are under study. Approximation algorithms for these problems are proposed, and performance guarantees of the algorithms are obtained. In particular, it is shown that the problem of approximation by graphs with a bounded number of connected components belongs to the class APX.