投影型插值算子的超收敛性质及其应用

本文首先将证明矩形剖分单元上的Ｌｏｂａｔｔｏ点,Ｇａｕｓｓ点和拟Ｌｏｂａｔｔｏ点分别是二维投影型插值算子函数,梯度和二阶导数的逼近佳点;然后考虑了二阶椭圆边值问题的有限元近似．通过建立投影型插值算子各种形式的超收敛基本估计,证明了投影型插值算子的各类...     

连续区间上积分值的MQ拟插值算子

在某些插值问题中,插值点处的函数值是未知的,而连续区间上的积分值是已知的.如何利用连续区间上积分值信息来解决函数重构是一个重要的问题.首先,文章利用连续区间上积分值的线性组合得到结点处函数值和一阶导数值的的四阶逼近.然后,构造了一类基于连续区间上积分值的MQ拟插值算子,它称之为积分值型MQ拟插值算子.最后,给出了该MQ拟插值算子的整体误差,它具有相应的四阶逼近阶.数值实验表明,该方法是有效可行的.     

Hermite插值算子在Orlicz空间内的逼近

插值算子逼近是逼近论中一个非常有趣的问题,尤其是以一些特殊的点为结点的插值算子的逼近问题很受人们的关注.研究了以第一类Chebyshev多项式零点为插值结点的Hermite插值算子在Orlicz范数下的逼近.     

二元三次样条空间S_3~(1,2)(△_(mn)~(2))的样条拟插值

本文首先利用由两组具有局部最小支集的样条所组成的基函数,构造非均匀2型三角剖分上二元三次样条空间S1,23(△(2)mn)的若干样条拟插值算子.这些变差缩减算子由样条函数B1ij支集上5个网格点或中心和样条函数B2ij支集上5个网格点处函数值定义.这些样条拟插值算子具有较好的逼近性,甚至算子Vmn(f)能保持近最优的三次多项式性.然后利用连续模,分析样条拟插值算子Vmn(f)一致逼近于充分光滑的实函数.最后推导误差估计.     

一种修正的拟Grünwald插值在Orlicz空间内的逼近度

修正了以第二类Chebyshev多项式的零点为插值结点组的拟Grünwald插值多项式,使之转化为积分形式,并利用不等式技巧和Hardy-Littlewood极大函数的方法,研究了此积分型拟Grünwald插值算子在带权Orlicz空间内的逼近问题,得出了意义相对广泛的逼近度估计的结果.     

一种修正的拟Grünwald插值在Orlicz空间内的逼近度

修正了以第二类Chebyshev多项式的零点为插值结点组的拟Grünwald插值多项式,使之转化为积分形式,并利用不等式技巧和Hardy-Littlewood极大函数的方法,研究了此积分型拟Grünwald插值算子在带权Orlicz空间内的逼近问题,得出了意义相对广泛的逼近度估计的结果.     

一种2型三角剖分上多元样条拟插值的精度提高策略

给定一个多元拟插值算子, 若其具有单位分解性质 (再生0次多项式), 我们提出一种利用其周围节点提高多项式再生性的方法. 所得算子不仅具有更高的逼近精度, 还不需要目标函数的任何导数信息. 然后利用此方法, 我们改进了2型三角剖分上的多元样条拟插值,使之具有更高的精度. 最后, 我们应用改进的拟插值算子数值求解时间发展偏微分方程. 数值实验验证了该方法的有效性.     

拟Hermite插值对解析函数类的逼近误差

在最大框架下研究基于第二类Tchebyshev节点组的拟Hermite插值算子和Hermite插值算子对一个解析函数类的逼近误差.对于一致范数,我们得到了相应量的精确值.对于L_p-范数(1≤p∞),我们得到了相应量的值或强渐近阶.     

局部线性插值算子的一个构造方法

利用具有紧支集函数平移变换的拟任值是逼近论中的构造算子的一个重要方法,但拟插值算子一般不具有插值性质,本提供了一个简单的方法,该法可以构造出许多具有拟插值优点,又具有插值性质的线性算子。     

拟Hermite-Fejr插值在一重积分Wiener空间下的平均误差

本文主要在Lp范数逼近意义下确定一类拟Hermite-Fejr插值多项式列在一重积分Wiener空间下平均误差的弱渐近阶.结果说明若概率空间不同,插值算子列在平均误差的意义下可能具有完全不同的逼近性质.在某些特殊情形下得到了其值或强渐近阶.     

Approximate approximations from scattered data

The aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions we propose a method to construct the approximate partition of unity and describe an application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators.     

The construction and approximation of feedforward neural network with hyp erb olic tangent function

In this paper, we discuss some analytic properties of hyperbolic tangent function and estimate some approximation errors of neural network operators with the hyperbolic tangent activation functionFirstly, an equation of partitions of unity for the hyperbolic tangent function is givenThen, two kinds of quasi-interpolation type neural network operators are constructed to approximate univariate and bivariate functions, respectivelyAlso, the errors of the approximation are estimated by means of the modulus of continuity of functionMoreover, for approximated functions with high order derivatives, the approximation errors of the constructed operators are estimated.     

Multiquadric quasi-interpolation method for fractional integral-differential equations

In this paper, Multiquadric quasi-interpolation method is used to approximate fractional integral equations and fractional differential equations. Firstly, we construct two operators for approximating the Hadamard integral-differential equation based on quasi interpolators, and verify their properties and order of convergence. Secondly, we obtain that the approximation order of the integral scheme is 3, and the approximation order of the differential scheme is $3-\mu$ for $\mu(0<\mu<1)$ order fractional Hadamard derivative. Finally, The results of numerical experiments show that the numerical results are in greement with the theoretical analysis.     

The new approximation operators with sigmoidal functions

The aim of this paper is to investigate approximation operators with logarithmic sigmoidal function of a class of two neural networks weights and a class of quasi-interpolation operators. Using these operators as approximation tools, the upper bounds of estimate errors are estimated for approximating continuous functions.     

A kind of multiquadric quasi-interpolation operator satisfying any degree polynomial reproduction property to scattered data

In this paper, by virtue of using the linear combinations of the shifts of f(x) to approximate the derivatives of f(x) and Waldron’s superposition idea (2009), we modify a multiquadric quasi-interpolation with the property of linear reproducing to scattered data on one-dimensional space, such that a kind of quasi-interpolation operator L_r+1f has the property of r+1(r∈Z,r≥0) degree polynomial reproducing and converges up to a rate of r+2. There is no demand for the derivatives of f in the proposed quasi-interpolation L_r+1f, so it does not increase the orders of smoothness of f. Finally, some numerical experiments are shown to compare the approximation capacity of our quasi-interpolation operators with that of Wu-Schaback’s quasi-interpolation scheme and Feng-Li’s quasi-interpolation scheme.     

A multiquadric quasi-interpolation with linear reproducing and preserving monotonicity

In this paper, we develop a multiquadric (MQ) quasi-interpolation which has the properties of linear reproducing and preserving monotonicity. Moreover, we give its approximation error by theoretic analysis and illustrate the effect by means of two examples. One of the examples is to approach the linear combination of two sine functions with different frequencies. Another is to approximate a function with discontinuity. From the results of the examples, we believe that the present MQ quasi-interpolation is feasible.     

Univariate hyperbolic tangent neural network approximation

Here we study the univariate quantitative approximation of real and complex valued continuous functions on a compact interval or all the real line by quasi-interpolation hyperbolic tangent neural network operators. This approximation is derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its high order derivative. Our operators are defined by using a density function induced by the hyperbolic tangent function. The approximations are pointwise and with respect to the uniform norm. The related feed-forward neural network is with one hidden layer.     

Error estimates of quasi-interpolation and its derivatives

Quasi-interpolation of radial basis functions on finite grids is a very useful strategy in approximation theory and its applications. A notable strongpoint of the strategy is to obtain directly the approximants without the need to solve any linear system of equations. For radial basis functions with Gaussian kernel, there have been more studies on the interpolation and quasi-interpolation on infinite grids. This paper investigates the approximation by quasi-interpolation operators with Gaussian kernel on the compact interval. The approximation errors for two classes of function with compact support sets are estimated. Furthermore, the approximation errors of derivatives of the approximants to the corresponding derivatives of the approximated functions are estimated. Finally, the numerical experiments are presented to confirm the accuracy of the approximations.     

APPROXIMATE IMPLICITIZATION BASED ON RBF NETWORKS AND MQ QUASI-INTERPOLATION

In this paper, we propose a new approach to solve the approximate implicitization problem based on RBF networks and MQ quasi-interpolation. This approach possesses the advantages of shape preserving, better smoothness, good approximation behavior and relatively less data etc. Several numerical examples are provided to demonstrate the effectiveness and flexibility of the proposed method.