We consider the questions connected with the approximation of a real continuous 1-periodic functions and give a new proof of the equivalence of the special Boman-Shapiro modulus of continuity with Peetre’s K-function...We consider the questions connected with the approximation of a real continuous 1-periodic functions and give a new proof of the equivalence of the special Boman-Shapiro modulus of continuity with Peetre’s K-functional. We also prove Jackson’s inequality for the approximation by trigonometric polynomials.展开更多
A partition-of-unity-based approach is proposed to derive an approximate model for a class of nonlinear systems. The precision of the approximate model is analyzed by using the modulus of continuity of continuous func...A partition-of-unity-based approach is proposed to derive an approximate model for a class of nonlinear systems. The precision of the approximate model is analyzed by using the modulus of continuity of continuous functions. The system stability of the approximate model is analyzed by using Lyapunov stability theory. A design algorithm for constructing tracking controllers with tracking performance related to tracking error is given based on the approximate model and the partition of unity method.展开更多
In this paper, we discuss some analytic properties of hyperbolic tangent function and estimate some approximation errors of neural network operators with the hyperbolic tan- gent activation function. Firstly, an equat...In this paper, we discuss some analytic properties of hyperbolic tangent function and estimate some approximation errors of neural network operators with the hyperbolic tan- gent activation function. Firstly, an equation of partitions of unity for the hyperbolic tangent function is given. Then, two kinds of quasi-interpolation type neural network operators are con- structed to approximate univariate and bivariate functions, respectively. Also, the errors of the approximation are estimated by means of the modulus of continuity of function. Moreover, for approximated functions with high order derivatives, the approximation errors of the constructed operators are estimated.展开更多
In order to study the approximation by reciprocals of polynomials with real coefficients, one always assumes that the approximated function has a fixed sign on the given interval. Sometimes, the approximated function ...In order to study the approximation by reciprocals of polynomials with real coefficients, one always assumes that the approximated function has a fixed sign on the given interval. Sometimes, the approximated function is permitted to have finite sign changes, such as l(l ≥ 1) times. Zhou Songping has studied the case l=1 and l≥2 in L^p spaces in order of priority. In this paper, we studied the case l ≥2 in Orlicz spaces by using the function extend, modified Jackson kernel, Hardy-Littlewood maximal function, Cauchy-Schwarz inequality, and obtained the Jackson type estimation.展开更多
We generalize several classical results on the integrability of trigonometric series and relations among the best approximation and the coefficients of trigonometric series. Theorem 3 and Theorem 4 are the first resul...We generalize several classical results on the integrability of trigonometric series and relations among the best approximation and the coefficients of trigonometric series. Theorem 3 and Theorem 4 are the first results on the relations among the weighted best approximation and the coefficients of trigonometric series.展开更多
We introduce a super-Lévy process and study maximal speed of all particles in the range and the support of the super-Lévy process. The state of historical super-Lévy process is a measure on the set of p...We introduce a super-Lévy process and study maximal speed of all particles in the range and the support of the super-Lévy process. The state of historical super-Lévy process is a measure on the set of paths. We study the maximal speed of all particles during a given time period, which turns out to be a function of the packing dimension of the time period. We calculate the Hausdorff dimension of the set of a-fast paths in the support and the range of the historical super-Lévy process.展开更多
In this paper we propose the q analogues of modified Baskakov-Sz^sz op- erators. We estimate the moments and established direct results in term of modulus of continuity. An estimate for the rate of convergence and wei...In this paper we propose the q analogues of modified Baskakov-Sz^sz op- erators. We estimate the moments and established direct results in term of modulus of continuity. An estimate for the rate of convergence and weighted approximation properties of the q operators are also obtained.展开更多
The purpose of this paper is to introduce ω2φ λ(f,t)α,β, and use it to prove the Steckin-Marchaud-type inequalities for BernsteinKantorovich Polynomials: where 0≤λ≤1, 0<α<2, 0≤β≤2, n∈N, and
The goal of this paper is to give a form of the operator involving the generating function of Apostol-Genocchi polynomials of orderα.Applying the Korovkin theorem,we arrive at the convergence of the operator with the...The goal of this paper is to give a form of the operator involving the generating function of Apostol-Genocchi polynomials of orderα.Applying the Korovkin theorem,we arrive at the convergence of the operator with the aid of moments and central moments.We determine the rate of convergence of the operator using several tools such as K-functional,modulus of continuity,second modulus of continuity.We also give a type of Voronovskaya theorem for estimating error.Moreover,we investigate some results about convergence properties of the operator in a weighted space.Finally,we give numerical examples to support our theorems by using the Maple.展开更多
We consider Jackson inequality in L^2 (B^d×T, Wκ,μ^B (x)), where the weight function Wκ,μ^B (X) is defined on the ball B^d and related to reflection group, and obtain the sharp Jackson inequalityEn-1,m-...We consider Jackson inequality in L^2 (B^d×T, Wκ,μ^B (x)), where the weight function Wκ,μ^B (X) is defined on the ball B^d and related to reflection group, and obtain the sharp Jackson inequalityEn-1,m-1(f)2≤κn,m(τ,r)ωr(f,t)2,τ≥2τn,λ,where Tn,λ is the first positive zero of the Gegenbauer cosine polynomial Cn^λ (cos θ)(n ∈ N).展开更多
In this paper,we study on the genuine modified Bernstein-Durrmeyer-Stancu operators Gn(f,x)and investigate some approximation properties of them.Furthermore,we present a Voronovskaja type theorem for these operators.W...In this paper,we study on the genuine modified Bernstein-Durrmeyer-Stancu operators Gn(f,x)and investigate some approximation properties of them.Furthermore,we present a Voronovskaja type theorem for these operators.We also give some graphs and numerical examples to illustrate the convergence properties of these operators for certain functions.展开更多
We study the moduli of continuity of a class of N-parameter Gaussian processes and get some results on'the packing dimension of the set of their fast points.
A new generalization of Stancu polynomials based on the q-integers and a nonnegative integer s is firstly introduced in this paper. Moreover, the shape-preserving and convergence properties of these polynomials are al...A new generalization of Stancu polynomials based on the q-integers and a nonnegative integer s is firstly introduced in this paper. Moreover, the shape-preserving and convergence properties of these polynomials are also investigated.展开更多
In this paper,the King’s type modification of(p,q)-Bleimann-Butzer and Hahn operators is defined.Some results based on Korovkin’s approximation theorem for these new operators are studied.With the help of modulus of...In this paper,the King’s type modification of(p,q)-Bleimann-Butzer and Hahn operators is defined.Some results based on Korovkin’s approximation theorem for these new operators are studied.With the help of modulus of continuity and the Lipschitz type maximal functions,the rate of convergence for these new operators are obtained.It is shown that the King’s type modification have better rate of convergence,flexibility than classical(p,q)-BBH operators on some subintervals.Further,for comparisons of the operators,we presented some graphical examples and the error estimation in the form of tables through MATLAB(R2015a)展开更多
The Lipschitz classes Lip(a,M) ,0 〈 a 〈 1 are defined for Orlicz space generated by the Young function M, and the degree of approximation by matrix transforms of f E Lip(or,M) is estimated by n-a.
The present paper deals with best onesided approximation rate in L p spaces n(f) L p of f∈C 2π. Although it is clear that the estimate n(f) L p≤C‖f‖ L p cannot be correct for all f∈L ...The present paper deals with best onesided approximation rate in L p spaces n(f) L p of f∈C 2π. Although it is clear that the estimate n(f) L p≤C‖f‖ L p cannot be correct for all f∈L p 2π in case p<∞, the question whether n(f) L p≤Cω(f,n -1) L p or n(f) L p≤CE n(f) L p holds for f∈C 2π remains totally untouched. Therefore it forms a basic problem to justify onesided approximation. The present paper will provide an answer to settle down the basis.展开更多
In the present paper, we consider Stancu type generalization of the summation integral type operators discussed in [15]. We apply hypergeometric series for obtaining moments of these operators. We also discuss about a...In the present paper, we consider Stancu type generalization of the summation integral type operators discussed in [15]. We apply hypergeometric series for obtaining moments of these operators. We also discuss about asymptotic formula and error estimation in terms of modules of continuity.展开更多
In this paper we propose the q analogues of modified Baskakov-Sz′asz operators.we estimate the moments and establish the direct results in term of modulus of continuity.An estimate for the rate of convergence and wei...In this paper we propose the q analogues of modified Baskakov-Sz′asz operators.we estimate the moments and establish the direct results in term of modulus of continuity.An estimate for the rate of convergence and weighted approximation properties of the q operators are also obtained.展开更多
In this paper,based on the basis composed of two sets of splines with distinct local supports,cubic spline quasi-interpolating operators are reviewed on nonuniform type-2 triangulation.The variation diminishing operat...In this paper,based on the basis composed of two sets of splines with distinct local supports,cubic spline quasi-interpolating operators are reviewed on nonuniform type-2 triangulation.The variation diminishing operator is defined by discrete linear functionals based on a fixed number of triangular mesh-points,which can reproduce any polynomial of nearly best degrees.And by means of the modulus of continuity,the estimation of the operator approximating a real sufficiently smooth function is reviewed as well.Moreover,the derivatives of the nearly optimal variation diminishing operator can approximate that of the real sufficiently smooth function uniformly over quasi-uniform type-2 triangulation.And then the convergence results are worked out.展开更多
文摘We consider the questions connected with the approximation of a real continuous 1-periodic functions and give a new proof of the equivalence of the special Boman-Shapiro modulus of continuity with Peetre’s K-functional. We also prove Jackson’s inequality for the approximation by trigonometric polynomials.
基金the National Natural Science Foundation of Guangdong Province (No.032035).
文摘A partition-of-unity-based approach is proposed to derive an approximate model for a class of nonlinear systems. The precision of the approximate model is analyzed by using the modulus of continuity of continuous functions. The system stability of the approximate model is analyzed by using Lyapunov stability theory. A design algorithm for constructing tracking controllers with tracking performance related to tracking error is given based on the approximate model and the partition of unity method.
基金Supported by the National Natural Science Foundation of China(61179041,61272023,and 11401388)
文摘In this paper, we discuss some analytic properties of hyperbolic tangent function and estimate some approximation errors of neural network operators with the hyperbolic tan- gent activation function. Firstly, an equation of partitions of unity for the hyperbolic tangent function is given. Then, two kinds of quasi-interpolation type neural network operators are con- structed to approximate univariate and bivariate functions, respectively. Also, the errors of the approximation are estimated by means of the modulus of continuity of function. Moreover, for approximated functions with high order derivatives, the approximation errors of the constructed operators are estimated.
基金supported by the National Natural Science Foundation of China (11161033)the Personnel Train Engineering Foundation of Inner Mongolia Normal University(RCPY-2-2012-K-036)
文摘In order to study the approximation by reciprocals of polynomials with real coefficients, one always assumes that the approximated function has a fixed sign on the given interval. Sometimes, the approximated function is permitted to have finite sign changes, such as l(l ≥ 1) times. Zhou Songping has studied the case l=1 and l≥2 in L^p spaces in order of priority. In this paper, we studied the case l ≥2 in Orlicz spaces by using the function extend, modified Jackson kernel, Hardy-Littlewood maximal function, Cauchy-Schwarz inequality, and obtained the Jackson type estimation.
文摘We generalize several classical results on the integrability of trigonometric series and relations among the best approximation and the coefficients of trigonometric series. Theorem 3 and Theorem 4 are the first results on the relations among the weighted best approximation and the coefficients of trigonometric series.
基金Project supported by the National Natural Science Foundation of China(No.10571159)the Ph.D.Programs Foundation of Ministry of Education of China(No.20060335032)and the Foundation of Hangzhou Dianzi University(No.KYS091506042)
文摘We introduce a super-Lévy process and study maximal speed of all particles in the range and the support of the super-Lévy process. The state of historical super-Lévy process is a measure on the set of paths. We study the maximal speed of all particles during a given time period, which turns out to be a function of the packing dimension of the time period. We calculate the Hausdorff dimension of the set of a-fast paths in the support and the range of the historical super-Lévy process.
文摘In this paper we propose the q analogues of modified Baskakov-Sz^sz op- erators. We estimate the moments and established direct results in term of modulus of continuity. An estimate for the rate of convergence and weighted approximation properties of the q operators are also obtained.
文摘The purpose of this paper is to introduce ω2φ λ(f,t)α,β, and use it to prove the Steckin-Marchaud-type inequalities for BernsteinKantorovich Polynomials: where 0≤λ≤1, 0<α<2, 0≤β≤2, n∈N, and
文摘The goal of this paper is to give a form of the operator involving the generating function of Apostol-Genocchi polynomials of orderα.Applying the Korovkin theorem,we arrive at the convergence of the operator with the aid of moments and central moments.We determine the rate of convergence of the operator using several tools such as K-functional,modulus of continuity,second modulus of continuity.We also give a type of Voronovskaya theorem for estimating error.Moreover,we investigate some results about convergence properties of the operator in a weighted space.Finally,we give numerical examples to support our theorems by using the Maple.
基金supported by National Natural Science Foundation of China(11071019)Beijing Natural Science Foundation(1132001)
文摘We consider Jackson inequality in L^2 (B^d×T, Wκ,μ^B (x)), where the weight function Wκ,μ^B (X) is defined on the ball B^d and related to reflection group, and obtain the sharp Jackson inequalityEn-1,m-1(f)2≤κn,m(τ,r)ωr(f,t)2,τ≥2τn,λ,where Tn,λ is the first positive zero of the Gegenbauer cosine polynomial Cn^λ (cos θ)(n ∈ N).
基金Supported by the National Natural Science Foundation of China(11601266)the Natural Science Foundation of Fujian Province of China(2020J01783)+1 种基金the Project for High-level Talent Innovation and Entrepreneurship of Quanzhou(2018C087R)the Program for New Century Excellent Talents in Fujian Province University and Fujian Provincial Scholarship for Overseas Study。
文摘In this paper,we study on the genuine modified Bernstein-Durrmeyer-Stancu operators Gn(f,x)and investigate some approximation properties of them.Furthermore,we present a Voronovskaja type theorem for these operators.We also give some graphs and numerical examples to illustrate the convergence properties of these operators for certain functions.
文摘We study the moduli of continuity of a class of N-parameter Gaussian processes and get some results on'the packing dimension of the set of their fast points.
文摘A new generalization of Stancu polynomials based on the q-integers and a nonnegative integer s is firstly introduced in this paper. Moreover, the shape-preserving and convergence properties of these polynomials are also investigated.
文摘In this paper,the King’s type modification of(p,q)-Bleimann-Butzer and Hahn operators is defined.Some results based on Korovkin’s approximation theorem for these new operators are studied.With the help of modulus of continuity and the Lipschitz type maximal functions,the rate of convergence for these new operators are obtained.It is shown that the King’s type modification have better rate of convergence,flexibility than classical(p,q)-BBH operators on some subintervals.Further,for comparisons of the operators,we presented some graphical examples and the error estimation in the form of tables through MATLAB(R2015a)
文摘The Lipschitz classes Lip(a,M) ,0 〈 a 〈 1 are defined for Orlicz space generated by the Young function M, and the degree of approximation by matrix transforms of f E Lip(or,M) is estimated by n-a.
基金Supported in part by National and Zhejiang Provincial Natural Science Foundations of China.Partof the work contained in this paper was done while the second named author was visiting Simon Fraser University,Canada,he thanks Dr.P.B.Borwein and the Depa
文摘The present paper deals with best onesided approximation rate in L p spaces n(f) L p of f∈C 2π. Although it is clear that the estimate n(f) L p≤C‖f‖ L p cannot be correct for all f∈L p 2π in case p<∞, the question whether n(f) L p≤Cω(f,n -1) L p or n(f) L p≤CE n(f) L p holds for f∈C 2π remains totally untouched. Therefore it forms a basic problem to justify onesided approximation. The present paper will provide an answer to settle down the basis.
文摘In the present paper, we consider Stancu type generalization of the summation integral type operators discussed in [15]. We apply hypergeometric series for obtaining moments of these operators. We also discuss about asymptotic formula and error estimation in terms of modules of continuity.
文摘In this paper we propose the q analogues of modified Baskakov-Sz′asz operators.we estimate the moments and establish the direct results in term of modulus of continuity.An estimate for the rate of convergence and weighted approximation properties of the q operators are also obtained.
基金The authors wish to express our great appreciation to Prof.Renhong Wang for his valuable suggestions.Also,the authors would like to thank Dr.Chongjun Li and Dr.Chungang Zhu for their helpThis work is supported by National Basic Research Program of China(973 Project No.2010CB832702)+3 种基金R and D Special Fund for Public Welfare Industry(Hydrodynamics,Grant No.201101014)National Science Funds for Distinguished Young Scholars(Grant No.11125208)Programme of Introducing Talents of Discipline to Universities(111 project,Grant No.B12032)This work is supported by the Fundamental Research Funds for the Central Universities,and Hohai University Postdoctoral Science Foundation 2016-412051.
文摘In this paper,based on the basis composed of two sets of splines with distinct local supports,cubic spline quasi-interpolating operators are reviewed on nonuniform type-2 triangulation.The variation diminishing operator is defined by discrete linear functionals based on a fixed number of triangular mesh-points,which can reproduce any polynomial of nearly best degrees.And by means of the modulus of continuity,the estimation of the operator approximating a real sufficiently smooth function is reviewed as well.Moreover,the derivatives of the nearly optimal variation diminishing operator can approximate that of the real sufficiently smooth function uniformly over quasi-uniform type-2 triangulation.And then the convergence results are worked out.