A polynomial approximation for arbitrary functions

We describe an expansion of Legendre polynomials, analogous to the Taylor expansion, for approximating arbitrary functions. We show that the polynomial coefficients in the Legendre expansion, and thus, the whole series, converge to zero much more rapidly compared to those in the Taylor expansion of the same order. Furthermore, using numerical analysis with a sixth-order polynomial expansion, we demonstrate that the Legendre polynomial approximation yields an error at least an order of magnitude smaller than that of the analogous Taylor series approximation. This strongly suggests that Legendre expansions, instead of Taylor expansions, should be used when global accuracy is important.     

关于渐近拟非扩张算子不动点迭代逼近的注记

研究Banach空间中渐近拟非扩张算子及渐近φ-半压缩算子不动点的迭代逼近问题，给出带误差Ishikawa型迭代序列收敛的充要条件，所得结果修正与改进了[1]的主要结果，且推广了[2]及其他一些文献的相关结果。     

Convergence of the Nested Multivariate Padé Approximants

The nested multivariate Padé approximants were recently introduced. In the case of two variablesxandy, they consist in applying the Padé approximation with respect to y to the coefficients of the Padé approximation with respect tox. The principal advantage of the method is that the computation only involves univariate Padé approximation. This allows us to obtain uniform convergence where the classical multivariate Padé approximants fail to converge.     

Well-posed problems in a layer with differential operators in boundary conditions

We obtain a criterion for the well-posedness of a boundary-value problem in the layer ⁿ x [0, T] for a linear differential evolution equation with constant complex coefficients in the class of functions of power growth under a two-point condition that contains two arbitrary differential operators with respect to space variables (one of the operators can be zero). Examples of well-posed and ill-posed problems of this form are given.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 8, pp. 1083–1090, August, 1992.     

Based on interval type-2 fuzzy-neural network direct adaptive sliding mode control for SISO nonlinear systems

In this paper, a novel direct adaptive interval type-2 fuzzy-neural tracking control equipped with sliding mode and Lyapunov synthesis approach is proposed to handle the training data corrupted by noise or rule uncertainties for nonlinear SISO nonlinear systems involving external disturbances. By employing adaptive fuzzy-neural control theory, the update laws will be derived for approximating the uncertain nonlinear dynamical system. In the meantime, the sliding mode control method and the Lyapunov stability criterion are incorporated into the adaptive fuzzy-neural control scheme such that the derived controller is robust with respect to unmodeled dynamics, external disturbance and approximation errors. In comparison with conventional methods, the advocated approach not only guarantees closed-loop stability but also the output tracking error of the overall system will converge to zero asymptotically without prior knowledge on the upper bound of the lumped uncertainty. Furthermore, chattering effect of the control input will be substantially reduced by the proposed technique. To illustrate the performance of the proposed method, finally simulation example will be given.     

Least squares coefficients for a quadrature formula for Laplace transform inversion

A linear iterative method of least squares approximation of functions by exponentials due to Miller [9] is adapted to derive a set of least squares coefficients for an approximate Laplace transform inversion formula eq. (1). An earlier assumption made by Zakian [2] - that the approximation to the Laplace transform inverse will improve provided the approximation to the Dirac delta function is improved - is shown to be not substantiated for a number of test functions.     

A novel scheme for the design of backstepping control for a class of nonlinear systems

A novel scheme is proposed for the design of backstepping control for a class of state-feedback nonlinear systems. In the design, the unknown nonlinear functions are approximated by the neural networks (NNs) identification models. The Lyapunov function of every subsystem consists of the tracking error and the estimation errors of NN weight parameters. The adaptive gains are dynamically determined in a structural way instead of keeping them constants, which can guarantee system stability and parameter estimation convergence. When the modeling errors are available, the indirect backstepping control is proposed, which can guarantee the functional approximation error will converge to a rather small neighborhood of the minimax functional approximation error. When the modeling errors are not available, the direct backstepping control is proposed, where only the tracking error is necessary. The simulation results show the effectiveness of the proposed schemes.     

An Extremal Problem Arising in the Dynamics of Two-Phase Materials That Directly Reveals Information about the Internal Geometry

In two-phase materials, each phase having a non-local response in time, it has been found that for some driving fields the response somehow untangles at specific times, and allows one to directly infer useful information about the geometry of the material, such as the volume fractions of the phases. Motivated by this, and to obtain an algorithm for designing appropriate driving fields, we find approximate, measure independent, linear relations between the values that Markov functions take at a given set of possibly complex points, not belonging to the interval [-1,1] where the measure is supported. The problem is reduced to simply one of polynomial approximation of a given function on the interval [-1,1] and, to simplify the analysis, Chebyshev approximation is used. This allows one to obtain explicit estimates of the error of the approximation, in terms of the number of points and the minimum distance of the points to the interval [-1,1]. Assuming this minimum distance is bounded below by a number greater than 1/2, the error converges exponentially to zero as the number of points is increased. Approximate linear relations are also obtained that incorporate a set of moments of the measure. In the context of the motivating problem, the analysis also yields bounds on the response at any particular time for any driving field, and allows one to estimate the response at a given frequency using an appropriately designed driving field that effectively is turned on only for a fixed interval of time. The approximation extends directly to Markov-type functions with a positive semidefinite operator valued measure, and this has applications to determining the shape of an inclusion in a body from boundary flux measurements at a specific time, when the time-dependent boundary potentials are suitably tailored. © 2022 Wiley Periodicals, Inc.     

Banach空间中渐近拟非扩展型映象不动点的迭代逼近

本文研究了一致凸Banach空间中渐近拟非扩展型映象和渐近非扩展型映象T的不动点的迭代逼近问题.利用范数不等式，在去掉[3]中一个较强条件的情况下，证明了T的具误差的Ishikawa迭代序列中收敛于T的某个不动点.所得定理推广和改进了已有的部分结果。     

Приближение функций многих переменных классов H p Ω полиномами по системе Хаара

In the paper, problems of approximation of functions indicated in the title are studied in the case when a majorant of their mixed modulus of continuity is given. Two forms of approximation (linear and nonlinear) are considered: the classical linear approximation by polynomials with respect to the Haar system when the indices lie in hyperbolic crosses; and the best M-term approximation by polynomials also with respect to the Haar system. For the corresponding quantities, exact estimates are obtained as to their order of magnitude.     

Restricted nonlinear approximation and singular solutions of boundary integral equations

This paper studies several problems, which are potentially relevant for the construction of adaptive numerical schemes. First, biorthogonal spline wavelets on [0,1 ] are chosen as a starting point for characterizations of functions in Besov spaces B , (0,1) with 0＜σ＜∞ and (1+σ)-1＜τ＜∞. Such function spaces are known to be related to nonlinear approximation. Then so called restricted nonlinear approximation procedures with respect to Sobolev space norms are considered. Besides characterization results Jackson type estimates for various tree-type and tresholding algorithms are investigated. Finally known approximation results for geometry induced singularity functions of boundary integeral equations are combined with the characterization results for restricted nonlinear approximation to show Besov space regularity results.     

Improved results on the robustness of stochastic approximation algorithms

This paper is a continuation of the research carried out in [1]–[2], where the robustness analysis for stochastic approximation algorithms is given for two cases: 1. The regression function and the Liapunov function are not zero at the sought-forx ⁰; 2. lim sup _n a131_{n i=1} ⁿ _i+1 is not zero, where { _i } are the measurement errors and {a _n} are the weighting coefficients in the algorithm. Allowing these deviations from zero to occur simultaneously but to remain small, this paper shows that the estimation error is still small even for a class of measurement errors more general than that considered in [2].This project is supported by the National Natural Science Foundation of China.     

Zum Franklin—Schneiderschen Satz

In this note we shall carry on further the simultaneous approximation of a, b and exp(bloga). In a recent paper BUNDSCHUH [2] proved a theorem which appears to be a sharpening of a theorem of SCHNEIDER [10]. But there is an error in the proof. We shall show, that under a supplementary condition, the theorem of BUNDSCHUH remains valid and as well, get an improvement to this theorem. We further give some results on linear forms in logarithms of two U-numbers with algebraic coefficients.     

High-Accuracy P-Stable Methods with Minimal Phase-Lag for $y "= f(t,y) ^*$

In this paper, we develop a one-parameter family of P-stable sixth-order and eighth-order two-step methods with minimal phase-lag errors for numerical integration of second order periodic initial value problems: $$ y''=f(t,y), \quad y(t_0)=y_0, \quad y'(t_0)=y'_0. $$ We determine the parameters so that the phase-lag (frequency distortion) of these methods are minimal. The resulting methods are P-stable methods with minimal phase-lag errors. The superiority of our present P-stable methods over the P-stable methods in [1-4] is given by comparative studying of the phase-lag errors and illustrated with numerical examples.     

Approximations to time-optimal boundary controls for weak generalized solutions of the wave equation

For the wave equation, time-optimal control problems with two-sided boundary controls of three basic types are considered in classes of weak generalized solutions. Stable algorithms for computing approximate optimal times and optimal boundary controls are developed by modifying algorithms proposed earlier for the case of strong generalized solutions. The approximate solutions are proved to converge as the parameters of the finite-dimensional approximation are asymptotically refined and the errors in the specified terminal functions are reduced.     

Analytic approximation and differentiability of joint chance constraints

ABSTRACT

An inner–outer approximation approach was recently developed to solve single chance constrained optimization (SCCOPT) problems. In this paper, we extend this approach to address joint chance constrained optimization (JCCOPT) problems. Using an inner–outer approximation, two smooth parametric optimization problems are defined whose feasible sets converge to the feasible set of JCCOPT from inside and outside, respectively. Any optimal solution of the inner approximation problem is a priori feasible to the JCCOPT. As the approximation parameter tends to zero, a subsequence of the solutions of the inner and outer problems, respectively, converge asymptotically to an optimal solution of the JCCOPT. As a main result, the continuous differentiability of the probability function of a joint chance constraint is obtained by examining the uniform convergence of the gradients of the parametric approximations.     

Biased rational chebyshev approximation

Chebyshev approximation on an interval [, ] by ordinary rational functions when positive deviations (errors) are magnified by a bias factor is considered. This problem is related to one-sided Chebyshev approximation for large bias factors. Best approximations are characterized by alternation. Non-degenerate best approximations can be determined by the Remez algorithm. A variant of the Fraser-Hart-Remez algorithm is implemented.     

Numerical solution of vector Sturm–Liouville problems with Dirichlet conditions and nonlinear dependence on the spectral parameter

A numerical-analytical iterative method is proposed for solving generalized self-adjoint regular vector Sturm–Liouville problems with Dirichlet boundary conditions. The method is based on eigenvalue (spectral) correction. The matrix coefficients of the equations are assumed to be nonlinear functions of the spectral parameter. For a relatively close initial approximation, the method is shown to have second-order convergence with respect to a small parameter. Test examples are considered, and the model problem of transverse vibrations of a hinged rod with a variable cross section is solved taking into account its rotational inertia.     

Max—min神经网络的一种有效学习算法

本文在Ａ．Ｂｌａｎｃｏ等人的算法的基础上，提出了ｍａｘ－ｍｉｎ神经网络的一种改进了的反馈学习算法，严格证明了该算法的迭代收敛性，理论分析及实例计算结果均表明，本文算法具有算法简单，收敛速度快，输出误差小等显著特点。     

Explicit generalized zolotarev polynomials with complex coefficients

We give explicitly a class of polynomials with complex coefficients of degreen which deviate least from zero on [−1, 1] with respect to the max-norm among all polynomials which have the same,m + 1, 2m ≤n, first leading coefficients. Form=1, we obtain the polynomials discovered by Freund and Ruschewyh. Furthermore, corresponding results are obtained with respect to weight functions of the type 1/√ρl, whereρl is a polynomial positive on [−1, 1].