The relationship between the order of approximation by neural network based on scattered threshold value nodes and the neurons involved in a single hidden layer is investigated. The results obtained show that the degr...The relationship between the order of approximation by neural network based on scattered threshold value nodes and the neurons involved in a single hidden layer is investigated. The results obtained show that the degree of approximation by the periodic neural network with one hidden layer and scattered threshold value nodes is increased with the increase of the number of neurons hid in hidden layer and the smoothness of excitation function.展开更多
In this paper we introduce a new kind of the mixed Hermite--Fejér interpolation with boundary condi- tions and obtain the mean approximation order.Our results include a new theorem of Varma and Prasad.Be- sides,w...In this paper we introduce a new kind of the mixed Hermite--Fejér interpolation with boundary condi- tions and obtain the mean approximation order.Our results include a new theorem of Varma and Prasad.Be- sides,we also get some other results about the mean approximation.展开更多
In this paper we introduce the two-parameter operators on Abelian group and establish their interpolation theorems of approximation, which are extensions of the interpolation theorems for nonlinear best approximation ...In this paper we introduce the two-parameter operators on Abelian group and establish their interpolation theorems of approximation, which are extensions of the interpolation theorems for nonlinear best approximation by R. Devore and are suitable for the approximation of oprators.展开更多
In this paper, by means of combining non-probabilistic convex modeling with perturbation theory, an improvement is made on the first order approximate solution in convex models of uncertainties. Convex modeling is ext...In this paper, by means of combining non-probabilistic convex modeling with perturbation theory, an improvement is made on the first order approximate solution in convex models of uncertainties. Convex modeling is extended to largely uncertain and non-convex sets of uncertainties and the combinational convex modeling is developed. The presented method not only extends applications of convex modeling, but also improves its accuracy in uncertain problems and computational efficiency. The numerical example illustrates the efficiency of the proposed method.展开更多
In the present paper, we deal with the complex Szasz-Durrmeyer operators and study Voronovskaja type results with quantitative estimates for these operators attached to analytic functions of exponential growth on comp...In the present paper, we deal with the complex Szasz-Durrmeyer operators and study Voronovskaja type results with quantitative estimates for these operators attached to analytic functions of exponential growth on compact disks. Also, the exact order of approximation is found.展开更多
When approximation order is an odd positive integer, a simple method is given to construct compactly supported orthogonal symmetric complex scaling function with dilation factor 3. Two corresponding orthogonal wavelet...When approximation order is an odd positive integer, a simple method is given to construct compactly supported orthogonal symmetric complex scaling function with dilation factor 3. Two corresponding orthogonal wavelets, one is symmetric and the other is antisymmetric about origin, are constructed explicitly. Additionally, when approximation order is an even integer 2, we also give a method to construct compactly supported orthogonal symmetric complex that illustrate the corresponding results. wavelets. In the end, there are several examples展开更多
In this paper, an interpolation polynomial operator F n(f; l,x) is constructed based on the zeros of a kind of Jacobi polynomials as the interpolation nodes. For any continuous function f(x)∈C b [-1,1] ...In this paper, an interpolation polynomial operator F n(f; l,x) is constructed based on the zeros of a kind of Jacobi polynomials as the interpolation nodes. For any continuous function f(x)∈C b [-1,1] (0≤b≤l) F n(f; l,x) converges to f(x) uniformly, where l is an odd number.展开更多
The results of accurate order of uniform approximation and simultaneous approximation in the early work "Jackson Type Theorems on Complex Curves" are improved from Fejer points to disturbed Fejer points in this arti...The results of accurate order of uniform approximation and simultaneous approximation in the early work "Jackson Type Theorems on Complex Curves" are improved from Fejer points to disturbed Fejer points in this article. Furthermore, the stability of convergence of Tn,∈(f,z) with disturbed sample values f(z^*) + Sk are also proved in this article.展开更多
Based on a node group <img src="Edit_effba4ca-e855-418a-8a72-d70cb1ec3470.png" width="240" height="46" alt="" />, the Newman type rational operator is constructed in the p...Based on a node group <img src="Edit_effba4ca-e855-418a-8a72-d70cb1ec3470.png" width="240" height="46" alt="" />, the Newman type rational operator is constructed in the paper. The convergence rate of approximation to a class of non-smooth functions is discussed, which is <img src="Edit_174e8f70-651b-4abb-a8f3-a16a576536dc.png" width="85" height="50" alt="" /> regarding to <em>X</em>. Moreover, if the operator is constructed based on further subdivision nodes, the convergence rate is <img src="Edit_557b3a01-7f56-41c0-bb67-deab88b9cc63.png" width="85" height="45" alt="" />. The result in this paper is superior to the approximation results based on equidistant nodes, Chebyshev nodes of the first kind and Chebyshev nodes of the second kind.展开更多
ωB-splines have many optimal properties and can reproduce plentiful commonly-used analytical curves.In this paper,we further propose a non-stationary subdivision method of hierarchically and efficiently generatingωB...ωB-splines have many optimal properties and can reproduce plentiful commonly-used analytical curves.In this paper,we further propose a non-stationary subdivision method of hierarchically and efficiently generatingωB-spline curves of arbitrary order ofωB-spline curves and prove its C^k?2-continuity by two kinds of methods.The first method directly prove that the sequence of control polygons of subdivision of order k converges to a C^k?2-continuousωB-spline curve of order k.The second one is based on the theories upon subdivision masks and asymptotic equivalence etc.,which is more convenient to be further extended to the case of surface subdivision.And the problem of approximation order of this non-stationary subdivision scheme is also discussed.Then a uniform ωB-spline curve has both perfect mathematical representation and efficient generation method,which will benefit the application ofωB-splines.展开更多
Let D be a smooth domain in the complex plane. In D consider the simultaneous ap- proximation to a function and its ith (0≤i≤q) derivatives by Hermite interpolation. The orders of uniform approximation and approxima...Let D be a smooth domain in the complex plane. In D consider the simultaneous ap- proximation to a function and its ith (0≤i≤q) derivatives by Hermite interpolation. The orders of uniform approximation and approximation in the mean, are obtained under some domain boundary conditions. Some known results are included as particular cases of the theorems of this paper.展开更多
There have been various studies on approximation ability of feedforward neural networks (FNNs). Most of the existing studies are, however, only concerned with density or upper bound estimation on how a multivariate fu...There have been various studies on approximation ability of feedforward neural networks (FNNs). Most of the existing studies are, however, only concerned with density or upper bound estimation on how a multivariate function can be approximated by an FNN, and consequently, the essential approximation ability of an FNN cannot be revealed. In this paper, by establishing both upper and lower bound estimations on approximation order, the essential approximation ability (namely, the essential approximation order) of a class of FNNs is clarified in terms of the modulus of smoothness of functions to be approximated. The involved FNNs can not only approximate any continuous or integrable functions defined on a compact set arbitrarily well, but also provide an explicit lower bound on the number of hidden units required. By making use of multivariate approximation tools, it is shown that when the functions to be approximated are Lipschitzian with order up to 2, the approximation speed of the FNNs is uniquely determined by modulus of smoothness of the functions.展开更多
The concept of paraunitary two-scale similarity transform (PTST) is introduced. We discuss the property of PTST, and prove that PTST preserves the orthogonal, approximation order and smoothness of the given orthogon...The concept of paraunitary two-scale similarity transform (PTST) is introduced. We discuss the property of PTST, and prove that PTST preserves the orthogonal, approximation order and smoothness of the given orthogonal multiscaling functions. What is more, by applying PTST, we present an algorithm of constructing high order balanced multiscaling functions by balancing the already existing orthogonal nonbalanced multiscaling functions. The corresponding transform matrix is given explicitly. In addition, we also investigate the symmetry of the balanced multiscaling functions. Finally, construction examples are given.展开更多
For the nearly exponential type of feedforward neural networks (neFNNs), it is revealed the essential order of their approximation. It is proven that for any continuous function defined on a compact set of Rd, there...For the nearly exponential type of feedforward neural networks (neFNNs), it is revealed the essential order of their approximation. It is proven that for any continuous function defined on a compact set of Rd, there exists a three-layer neFNNs with fixed number of hidden neurons that attain the essential order. When the function to be approximated belongs to the α-Lipschitz family (0 〈α≤ 2), the essential order of approxi- mation is shown to be O(n^-α) where n is any integer not less than the reciprocal of the predetermined approximation error. The upper bound and lower bound estimations on approximation precision of the neFNNs are provided. The obtained results not only characterize the intrinsic property of approximation of the neFNNs, but also uncover the implicit relationship between the precision (speed) and the number of hidden neurons of the neFNNs.展开更多
In this paper, we study the convergence order of a new polynominal operator H n(f;x,r) through Grnwald polynomial operator appoximating f(x)∈C j [-1,1] ,j≤r. The result of paper [1] is improved.
We present a concrete method of constructing multiresolution analysis on interval. The method generalizes the corresponding results of Cohen, Daubechies and Vial [Appl. Comput. Harmonic Anal., 1(1993), 54-81]. By th...We present a concrete method of constructing multiresolution analysis on interval. The method generalizes the corresponding results of Cohen, Daubechies and Vial [Appl. Comput. Harmonic Anal., 1(1993), 54-81]. By the use of the subdivision operator, the expressions of the constructed functions are more compact. Furthermore, the method reveals more clearly some properties of multiresolution analysis with certain approximation order.展开更多
Recently Brutman and Passow considered Newman-type rational interpolation to |x| induced by arbitrary sets of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods,one...Recently Brutman and Passow considered Newman-type rational interpolation to |x| induced by arbitrary sets of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods,one could establish the exact order of approximation for some special nodes.In the present note we consider the sets of interpolation nodes obtained by adjusting the Chebyshev roots of the second kind on the interval [0,1] and then extending this set to [-1,1] in a symmetric way.We show that in this case the exact order of approximation is O( 1 n 2 ).展开更多
The refinability and approximation order of finite element multi-scale vector are discussed in [1]. But the coefficients in the conditions of approximation order of finite element multi-scale vector are incorrect ther...The refinability and approximation order of finite element multi-scale vector are discussed in [1]. But the coefficients in the conditions of approximation order of finite element multi-scale vector are incorrect there. The main purpose of this note is to make a correction of the error in the main result of [1]. These Cuefficients are very important for the properties of wavelets, such as vanishing moments and regularity.展开更多
文摘The relationship between the order of approximation by neural network based on scattered threshold value nodes and the neurons involved in a single hidden layer is investigated. The results obtained show that the degree of approximation by the periodic neural network with one hidden layer and scattered threshold value nodes is increased with the increase of the number of neurons hid in hidden layer and the smoothness of excitation function.
文摘In this paper we introduce a new kind of the mixed Hermite--Fejér interpolation with boundary condi- tions and obtain the mean approximation order.Our results include a new theorem of Varma and Prasad.Be- sides,we also get some other results about the mean approximation.
文摘In this paper we introduce the two-parameter operators on Abelian group and establish their interpolation theorems of approximation, which are extensions of the interpolation theorems for nonlinear best approximation by R. Devore and are suitable for the approximation of oprators.
基金The project supported by the National Outstanding Youth Science Foundation of China the National Post Doctor Science Foundation of China
文摘In this paper, by means of combining non-probabilistic convex modeling with perturbation theory, an improvement is made on the first order approximate solution in convex models of uncertainties. Convex modeling is extended to largely uncertain and non-convex sets of uncertainties and the combinational convex modeling is developed. The presented method not only extends applications of convex modeling, but also improves its accuracy in uncertain problems and computational efficiency. The numerical example illustrates the efficiency of the proposed method.
文摘In the present paper, we deal with the complex Szasz-Durrmeyer operators and study Voronovskaja type results with quantitative estimates for these operators attached to analytic functions of exponential growth on compact disks. Also, the exact order of approximation is found.
基金supported by the National Natural Science Foundation of China (11071152, 11126343)the Natural Science Foundation of Guangdong Province(10151503101000025, S2011010004511)
文摘When approximation order is an odd positive integer, a simple method is given to construct compactly supported orthogonal symmetric complex scaling function with dilation factor 3. Two corresponding orthogonal wavelets, one is symmetric and the other is antisymmetric about origin, are constructed explicitly. Additionally, when approximation order is an even integer 2, we also give a method to construct compactly supported orthogonal symmetric complex that illustrate the corresponding results. wavelets. In the end, there are several examples
文摘In this paper, an interpolation polynomial operator F n(f; l,x) is constructed based on the zeros of a kind of Jacobi polynomials as the interpolation nodes. For any continuous function f(x)∈C b [-1,1] (0≤b≤l) F n(f; l,x) converges to f(x) uniformly, where l is an odd number.
基金Supported by NSF of Henan Province of China(20001110001)
文摘The results of accurate order of uniform approximation and simultaneous approximation in the early work "Jackson Type Theorems on Complex Curves" are improved from Fejer points to disturbed Fejer points in this article. Furthermore, the stability of convergence of Tn,∈(f,z) with disturbed sample values f(z^*) + Sk are also proved in this article.
文摘Based on a node group <img src="Edit_effba4ca-e855-418a-8a72-d70cb1ec3470.png" width="240" height="46" alt="" />, the Newman type rational operator is constructed in the paper. The convergence rate of approximation to a class of non-smooth functions is discussed, which is <img src="Edit_174e8f70-651b-4abb-a8f3-a16a576536dc.png" width="85" height="50" alt="" /> regarding to <em>X</em>. Moreover, if the operator is constructed based on further subdivision nodes, the convergence rate is <img src="Edit_557b3a01-7f56-41c0-bb67-deab88b9cc63.png" width="85" height="45" alt="" />. The result in this paper is superior to the approximation results based on equidistant nodes, Chebyshev nodes of the first kind and Chebyshev nodes of the second kind.
基金the National Natural Science Foundation of China(61772164,61761136010)the Natural Science Foundation of Zhejiang Province(LY17F020025).
文摘ωB-splines have many optimal properties and can reproduce plentiful commonly-used analytical curves.In this paper,we further propose a non-stationary subdivision method of hierarchically and efficiently generatingωB-spline curves of arbitrary order ofωB-spline curves and prove its C^k?2-continuity by two kinds of methods.The first method directly prove that the sequence of control polygons of subdivision of order k converges to a C^k?2-continuousωB-spline curve of order k.The second one is based on the theories upon subdivision masks and asymptotic equivalence etc.,which is more convenient to be further extended to the case of surface subdivision.And the problem of approximation order of this non-stationary subdivision scheme is also discussed.Then a uniform ωB-spline curve has both perfect mathematical representation and efficient generation method,which will benefit the application ofωB-splines.
文摘Let D be a smooth domain in the complex plane. In D consider the simultaneous ap- proximation to a function and its ith (0≤i≤q) derivatives by Hermite interpolation. The orders of uniform approximation and approximation in the mean, are obtained under some domain boundary conditions. Some known results are included as particular cases of the theorems of this paper.
文摘There have been various studies on approximation ability of feedforward neural networks (FNNs). Most of the existing studies are, however, only concerned with density or upper bound estimation on how a multivariate function can be approximated by an FNN, and consequently, the essential approximation ability of an FNN cannot be revealed. In this paper, by establishing both upper and lower bound estimations on approximation order, the essential approximation ability (namely, the essential approximation order) of a class of FNNs is clarified in terms of the modulus of smoothness of functions to be approximated. The involved FNNs can not only approximate any continuous or integrable functions defined on a compact set arbitrarily well, but also provide an explicit lower bound on the number of hidden units required. By making use of multivariate approximation tools, it is shown that when the functions to be approximated are Lipschitzian with order up to 2, the approximation speed of the FNNs is uniquely determined by modulus of smoothness of the functions.
基金supported by the National Natural Science Foundation of China(Grant Nos.90104004 and 10471002)the 973 Project of China(Grant Nos.G1999075105)+1 种基金the Natural Science Foundation of Guangdong Province(Grant Nos.032038,05008289)the Doctoral Foundation of Guangdong Province(Grant No.04300917).
文摘The concept of paraunitary two-scale similarity transform (PTST) is introduced. We discuss the property of PTST, and prove that PTST preserves the orthogonal, approximation order and smoothness of the given orthogonal multiscaling functions. What is more, by applying PTST, we present an algorithm of constructing high order balanced multiscaling functions by balancing the already existing orthogonal nonbalanced multiscaling functions. The corresponding transform matrix is given explicitly. In addition, we also investigate the symmetry of the balanced multiscaling functions. Finally, construction examples are given.
基金the National Natural Science Foundation of China (Grant Nos. 10371097 , 70531030).
文摘For the nearly exponential type of feedforward neural networks (neFNNs), it is revealed the essential order of their approximation. It is proven that for any continuous function defined on a compact set of Rd, there exists a three-layer neFNNs with fixed number of hidden neurons that attain the essential order. When the function to be approximated belongs to the α-Lipschitz family (0 〈α≤ 2), the essential order of approxi- mation is shown to be O(n^-α) where n is any integer not less than the reciprocal of the predetermined approximation error. The upper bound and lower bound estimations on approximation precision of the neFNNs are provided. The obtained results not only characterize the intrinsic property of approximation of the neFNNs, but also uncover the implicit relationship between the precision (speed) and the number of hidden neurons of the neFNNs.
文摘In this paper, we study the convergence order of a new polynominal operator H n(f;x,r) through Grnwald polynomial operator appoximating f(x)∈C j [-1,1] ,j≤r. The result of paper [1] is improved.
基金Research supported in part by NSF of China under Grant 10571010 and 10171007
文摘We present a concrete method of constructing multiresolution analysis on interval. The method generalizes the corresponding results of Cohen, Daubechies and Vial [Appl. Comput. Harmonic Anal., 1(1993), 54-81]. By the use of the subdivision operator, the expressions of the constructed functions are more compact. Furthermore, the method reveals more clearly some properties of multiresolution analysis with certain approximation order.
基金Supported by the National Natural Science Foundation of China (Grant No. 10601065)
文摘Recently Brutman and Passow considered Newman-type rational interpolation to |x| induced by arbitrary sets of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods,one could establish the exact order of approximation for some special nodes.In the present note we consider the sets of interpolation nodes obtained by adjusting the Chebyshev roots of the second kind on the interval [0,1] and then extending this set to [-1,1] in a symmetric way.We show that in this case the exact order of approximation is O( 1 n 2 ).
文摘The refinability and approximation order of finite element multi-scale vector are discussed in [1]. But the coefficients in the conditions of approximation order of finite element multi-scale vector are incorrect there. The main purpose of this note is to make a correction of the error in the main result of [1]. These Cuefficients are very important for the properties of wavelets, such as vanishing moments and regularity.