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.     

由Fourier系数确定的函数类的最佳m-项单边逼近渐近估计

结合最佳m项逼近和单边逼近的思想引进所谓最佳m项单边逼近的概念,给出由Fourier系数确定的光滑函数类通过三角函数系在Lp(1≤P≤∞)的最佳m-项单边逼近渐近估计以及m-项类贪婪单边逼近结果.     

A modified Bogoliubov method applied to a simple boson model

We use a non-gauge-invariant modification of the exact Hamiltonian to obtain a new Hamiltonian-like operator for a simple exactly solvable boson model. The eigenvalues of the new operator are close to those of the original Hamiltonian. We make a one-body approximation of the new two-body operator in the spirit of the Bogoliubov approximation. Because only the number operator appears, the c-number approximation is not required individually for the creation or annihilation operators in the ground state. For the simple model, the results using the new approximation are closer to the exact results than the usual Bogoliubov results over a wide range of parameters. The improvement increases dramatically as the model interaction strength increases.     

Legendre rational approximation on the whole line

The Legendre rational approximation is investigated. Some approximation results are established, which form the mathematical foundation of a new spectral method on the whole line. A model problem is considered. Numerical results show the efficiency of this new approach.     

Nonlinear Approximation with Dictionaries I. Direct Estimates

We study various approximation classes associated with m-term approximation by elements from a (possibly redundant) dictionary in a Banach space. The standard approximation class associated with the best m-term approximation is compared to new classes defined by considering m-term approximation with algorithmic constraints: thresholding and Chebychev approximation classes are studied, respectively. We consider embeddings of the Jackson type (direct estimates) of sparsity spaces into the mentioned approximation classes. General direct estimates are based on the geometry of the Banach space, and we prove that assuming a certain structure of the dictionary is sufficient and (almost) necessary to obtain stronger results. We give examples of classical dictionaries in L^p spaces and modulation spaces where our results recover some known Jackson type estimates, and discuss some new estimates they provide.     

A Charlier–Parseval approach to poisson approximation and its applications

We propose a new approach to Poisson approximation. The basic idea is very simple and based on properties of the Charlier polynomials and the Parseval identity. Such an approach quickly leads to new effective bounds for several Poisson approximation problems. We also give a selected survey on diverse Poisson approximation results.     

Direct and inverse results in Hölder norms

We present a general approach to obtain direct and inverse results for approximation in Hölder norms. This approach is used to obtain a collection of new results related with estimates of the best polynomial approximation and with the approximation by linear operators of non-periodic functions in Hölder norms.     

A new generalized Laguerre spectral approximation and its applications

A new family of generalized Laguerre polynomials is introduced. Various orthogonal projections are investigated. Some approximation results are established. As an example of their important applications, the mixed spherical harmonic-generalized Laguerre approximation is developed. A mixed spectral scheme is proposed for a three-dimensional model problem. Its convergence is proved. Numerical results demonstrate the high accuracy of this new spectral method.     

Some remarks on the Balog–Wooley decomposition theorem and quantities <Emphasis Type="Italic">D</Emphasis><Superscript>+</Superscript>, <Emphasis Type="Italic">D</Emphasis><Superscript>×</Superscript>

A finite element with new properties of approximation of higher derivatives is constructed, and a method for the construction of a finite element space in the planar case is proposed. The method is based on Yu.N. Subbotin’s earlier results as well as on the results obtained in this paper. The constructed piecewise polynomial function possesses the continuity property and new approximation properties.     

An approximate Newton method for non-smooth equations with finite max functions

A new version of finite difference approximation of the generalized Jacobian for a finite max function is constructed. Numerical results are reported for the generalized Newton methods using this approximation.     

有理平均逼近的新算法

有理平均逼近应比多项式逼近好,但至今研究很少.本文分析了此问题有奇性,并提出一种新算法:延拓 正则化,则以获得非常好的结果.数值实验也证实了此结论.     

Order of mean approximation by mixed quasi Hermite-Fejer interpolation

In this paper we introduce a new kind of the mixed Hermite-Fejer interpolation with boundary conditions and obtain the mean approximation order. Our results include a new theorem of Varma and Prasad. Besides, we also get some other results about the mean approximation.     

MONOTONE APPROXIMATION TO A SYSTEM WITHOUT MONOTONE NONLINEARITY

1. IntroductionDue to the development of various studies in electromagnetism, biology and someother fields, nonlinear systems hajve been paid extellsive attention both analytically andnumerically, e.g., see [1--12]. As we kll')w, a reasonable numerical method should notonly have the approximation error of higher order, but also preserve the main feature ofthe original problem. In this case? the numerical results might fit the physical processbetter. Since the usual approximations simulate the …     

Polynomial approximation in Sobolev spaces on the unit sphere and the unit ball

This work is a continuation of the recent study by the authors on approximation theory over the sphere and the ball. The main results define new Sobolev spaces on these domains and study polynomial approximations for functions in these spaces, including simultaneous approximation by polynomials and the relation between the best approximation of a function and its derivatives.     

Approximations by linear operators in spaces of fuzzy continuous functions

In this work, we further develop the Korovkin-type approximation theory by utilizing a fuzzy logic approach and principles of neoclassical analysis, which is a new branch of fuzzy mathematics and extends possibilities provided by the classical analysis. In the conventional setting, the Korovkin-type approximation theory is developed for continuous functions. Here we extend it to the space of fuzzy continuous functions, which contains a great diversity of functions that are not continuous. Furthermore, we give several applications, demonstrating that our new approximation results are stronger than the classical ones.     

进化泛函网络多维函数逼近理论及学习算法

提出了一种用于多维函数逼近的进化策略修正泛函网络基函数系数的新算法,并给出了其算法学习过程.利用进化策略的自适应性来确定基函数前的系数,改进了泛函网络的参数通过解方程组来得到这一传统方法.仿真结果表明,这种新的逼近算法简单可行,能够逼近给定的函数到预先给定的精度,具有较快的收敛速度和良好的逼近性能.     

A numerical approximation method for solving a three-dimensional space Galilei invariant fractional advection-diffusion equation

In this paper, a numerical approximation method for solving a three-dimensional space Galilei invariant fractional advection diffusion equation is presented. The convergence and stability of the numerical approximation method are discussed by a new technique of Fourier analysis. The solvability of the numerical approximation method also is analyzed. Finally, applying Richardson extrapolation technique, a high-accuracy algorithm is structured and the numerical example demonstrated the theoretical results.     

A general approach to approximation problems of the Bernstein type

The Bernstein approximation problem is treated in a general setting of Banach function spaces. An abstract theory is developed which includes as special cases the solutions of the Bernstein problem given by Mergelyan and Pollard. It also provides a unified setting for related results from polynomial and trigonometric approximation theory and yields as corollaries a number of new results.     

Restricted thresholding: recovering smoothness and preserving edges

Restricted non linear approximation is a generalization of the N‐term approximation in which a measure on the index set of the approximants controls the type, instead of the number, of elements in the approximation. Thresholding is a well‐known type of non linear approximation. We relate a generalized upper and lower Temlyakov property with the decreasing rate of the thresholding approximation. This relation is in the form of a characterization through some general discrete Lorentz spaces. Thus, not only we recover some results in the literature but find new ones. As an application of these results, we compress and reduce noise of some images with wavelets and shearlets and show, at least empirically, that the L²‐norm is not necessarily the best norm to measure the approximation error.     

优势关系下决策表的下近似约简方法研究

研究了优势关系下不协调决策表的下近似约简问题,引入新的下近似约简的定义,证明新的下近似约简与文献[7]定义的下近似约简等价。给出新的下近似约简的判定定理和辨识矩阵,与文献[7]的辨识矩阵相比,计算新的下近似约简的辨识矩阵的时间复杂度要低。因此,可以利用新的辨识矩阵来求决策表的下近似约简．