By utilizing symmetric functions,this paper presents explicit representations for Hermite interpolation and its numerical differentiation formula.And the corresponding error estimates are also provided.
The criteria of convergence, including a theorem of Grunwald- type and the rate of convergence in terms of the modulus omega phi (f,t) of Ditzian and Totik for truncated Hermite interpolation on ail arbitrary system o...The criteria of convergence, including a theorem of Grunwald- type and the rate of convergence in terms of the modulus omega phi (f,t) of Ditzian and Totik for truncated Hermite interpolation on ail arbitrary system of nodes are given.展开更多
The object of this paper is to establish the pointwise estimations of approximation of functions in C^1 and their derivatives by Hermite interpolation polynomials. The given orders have been proved to be exact in gen-...The object of this paper is to establish the pointwise estimations of approximation of functions in C^1 and their derivatives by Hermite interpolation polynomials. The given orders have been proved to be exact in gen- eral.展开更多
Multivariate Hermite interpolation is widely applied in many fields, such as finite element construction, inverse engineering, CAD etc.. For arbitrarily given Hermite interpolation conditions, the typical method is to...Multivariate Hermite interpolation is widely applied in many fields, such as finite element construction, inverse engineering, CAD etc.. For arbitrarily given Hermite interpolation conditions, the typical method is to compute the vanishing ideal I （the set of polynomials satisfying all the homogeneous interpolation conditions are zero） and then use a complete residue system modulo I as the interpolation basis. Thus the interpolation problem can be converted into solving a linear equation system. A generic algorithm was presented in [18], which is a generalization of BM algorithm [22] and the complexity is O（τ^3） where r represents the number of the interpolation conditions. In this paper we derive a method to obtain the residue system directly from the relative position of the points and the corresponding derivative conditions （presented by lower sets） and then use fast GEPP to solve the linear system with O（（τ ＋ 3）τ^2） operations, where τ is the displacement-rank of the coefficient matrix. In the best case τ = 1 and in the worst case τ = [τ/n], where n is the number of variables.展开更多
In this paper we study a class of multivariate Hermite interpolation problem on 2;nodes with dimension d ≥ 2 which can be seen as a generalization of two classical Hermite interpolation problems of d = 2. Two combina...In this paper we study a class of multivariate Hermite interpolation problem on 2;nodes with dimension d ≥ 2 which can be seen as a generalization of two classical Hermite interpolation problems of d = 2. Two combinatorial identities are firstly given and then the regularity of the proposed interpolation problem is proved.展开更多
A threshold scheme, which is introduced by Shamir in 1979, is very famous as a secret sharing scheme. We can consider that this scheme is based on Lagrange＇s interpolation formula. A secret sharing scheme has one key...A threshold scheme, which is introduced by Shamir in 1979, is very famous as a secret sharing scheme. We can consider that this scheme is based on Lagrange＇s interpolation formula. A secret sharing scheme has one key. On the other hand, a multi-secret sharing scheme has more than one key, that is, a multi-secret sharing scheme has p （〉_ 2） keys. Dealer distribute shares of keys among n participants. Gathering t （〈 n） participants, keys can be reconstructed. Yang et al. （2004） gave a scheme of a （t, n） multi-secret sharing based on Lagrange＇s interpolation. Zhao et al. （2007） gave a scheme of a （t, n） verifiable multi-secret sharing based on Lagrange＇s interpolation. Recently, Adachi and Okazaki give a scheme of a （t, n） multi-secret sharing based on Hermite interpolation, in the case ofp 〈 t. In this paper, we give a scheme ofa （t, n） verifiable multi-secret sharing based on Hermite interpolation.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
For the weighted approximation in Lp-norm,the authors determine the weakly asymptotic order for the p-average errors of the sequence of Hermite interpolation based on the Chebyshev nodes on the 1-fold integrated Wiene...For the weighted approximation in Lp-norm,the authors determine the weakly asymptotic order for the p-average errors of the sequence of Hermite interpolation based on the Chebyshev nodes on the 1-fold integrated Wiener space.By this result,it is known that in the sense of information-based complexity,if permissible information functionals are Hermite data,then the p-average errors of this sequence are weakly equivalent to those of the corresponding sequence of the minimal p-average radius of nonadaptive information.展开更多
In this paper sufficient conditions for mean convergence and rate of convergence of Hermite-Fejer type interpolation in the Lp norm on an arbitrary system of nodes are presented.
The present paper investigates the convergence of Hermite interpolation operators on the real line. The main result is： Given 0 〈 δo 〈 1/2, 0 〈 εo 〈 1. Let f ∈ C（-∞,∞） satisfy ｜y｜= O（e^（1/2-δo）xk^2,...The present paper investigates the convergence of Hermite interpolation operators on the real line. The main result is： Given 0 〈 δo 〈 1/2, 0 〈 εo 〈 1. Let f ∈ C（-∞,∞） satisfy ｜y｜= O（e^（1/2-δo）xk^2,） and ｜f（x）｜t= O（e^（1-εo ）x2^）. Then for any given point x ∈ R, we have limn→Hn,（f, x） = f（x）.展开更多
Hermite interpolation is a very important tool in approximation theory and nu- merical analysis, and provides a popular method for modeling in the area of computer aided geometric design. However, the classical Hermit...Hermite interpolation is a very important tool in approximation theory and nu- merical analysis, and provides a popular method for modeling in the area of computer aided geometric design. However, the classical Hermite interpolant is unique for a prescribed data set, and hence lacks freedom for the choice of an interpolating curve, which is a crucial requirement in design environment. Even though there is a rather well developed fractal theory for Hermite interpolation that offers a large flexibility in the choice of interpolants, it also has the short- coming that the functions that can be well approximated are highly restricted to the class of self-affine functions. The primary objective of this paper is to suggest a gl-cubic Hermite in- terpolation scheme using a fractal methodology, namely, the coalescence hidden variable fractal interpolation, which works equally well for the approximation of a self-affine and non-self-affine data generating functions. The uniform error bound for the proposed fractal interpolant is established to demonstrate that the convergence properties are similar to that of the classical Hermite interpolant. For the Hermite interpolation problem, if the derivative values are not actually prescribed at the knots, then we assign these values so that the interpolant gains global G2-continuity. Consequently, the procedure culminates with the construction of cubic spline coalescence hidden variable fractal interpolants. Thus, the present article also provides an al- ternative to the construction of cubic spline coalescence hidden variable fractal interpolation functions through moments proposed by Chand and Kapoor [Fractals, 15（1） （2007）, pp. 41-53].展开更多
In this paper the uniform convergence of Hermite-Fejer interpolation and Griinwald type theorem of higher order on an arbitrary system of nodes are presented.
The paper gives a new way of constructing Hermite Fejer and Hermite interpolatory polynomials with the nodes of the roots of first kind of Chebyshev polynomials and gives the approximation order of these two kinds of...The paper gives a new way of constructing Hermite Fejer and Hermite interpolatory polynomials with the nodes of the roots of first kind of Chebyshev polynomials and gives the approximation order of these two kinds of operators. The approximation orders are described with the best rate of approximation of f by polynomials of degree N=(q+1)n 1 in L p w spaces.展开更多
This paper introduces the definition of the Orthogonal Type Node Configuration and discusses the corresponding multivariate Lagrange, Hermite and Birkhoff interpolation problems in high dimensional space R s(s>2). ...This paper introduces the definition of the Orthogonal Type Node Configuration and discusses the corresponding multivariate Lagrange, Hermite and Birkhoff interpolation problems in high dimensional space R s(s>2). This node configuration can be considered to be a kind of extension of the Cross Type Node Configuration , in R 2 to high dimensional spaces. And the Mixed Type Node Configuration in R s(s>2) is also discussed in this paper in an example.展开更多
It is well-known that Hermite rational interpolation gives a better approximation than Hermite polynomial interpolation,especially for large sequences of interpolation points,but it is difficult to solve the problem o...It is well-known that Hermite rational interpolation gives a better approximation than Hermite polynomial interpolation,especially for large sequences of interpolation points,but it is difficult to solve the problem of convergence and control the occurrence of real poles.In this paper,we establish a family of linear Hermite barycentric rational interpolants r that has no real poles on any interval and in the case k=0,1,2,the function r^(k)(x)converges to f^(k)(x)at the rate of O(h^3d+3-k)as h→0 on any real interpolation interval,regardless of the distribution of the interpolation points.Also,the function r(x)is linear in data.展开更多
General interpolation formulae for barycentric interpolation and barycen- tric rational Hermite interpolation are established by introducing multiple parameters, which include many kinds of barycentric interpolation a...General interpolation formulae for barycentric interpolation and barycen- tric rational Hermite interpolation are established by introducing multiple parameters, which include many kinds of barycentric interpolation and barycentric rational Her- mite interpolation. We discussed the interpolation theorem, dual interpolation and special cases. Numerical example is given to show the effectiveness of the method.展开更多
Consider a kind of Hermit interpolation for scattered data of 3D by trivariate polynomial natural spline, such that the objective energy functional （with natural boundary conditions） is minimal. By the spline functi...Consider a kind of Hermit interpolation for scattered data of 3D by trivariate polynomial natural spline, such that the objective energy functional （with natural boundary conditions） is minimal. By the spline function methods in Hilbert space and variational theory of splines, the characters of the interpolation solution and how to construct it are studied. One can easily find that the interpolation solution is a trivariate polynomial natural spline. Its expression is simple and the coefficients can be decided by a linear system. Some numerical examples are presented to demonstrate our methods.展开更多
In this paper. a quantitative estimate for Hermite interpolant to function ψ(z)=(z^m-β~m)~l on the ze- ros of (z^n-α~n)~r is obtained Using this estimate. a rather wide exiension of the theorem of Walsh is proved a...In this paper. a quantitative estimate for Hermite interpolant to function ψ(z)=(z^m-β~m)~l on the ze- ros of (z^n-α~n)~r is obtained Using this estimate. a rather wide exiension of the theorem of Walsh is proved and five special cases of it are given.展开更多
In this paper,we present a class of novel Bernstein-like basis functions,which is an extension of classical Bernstein basis functions.The properties of this group of bases are analyzed and the Bézier-like curve w...In this paper,we present a class of novel Bernstein-like basis functions,which is an extension of classical Bernstein basis functions.The properties of this group of bases are analyzed and the Bézier-like curve with two shape parameters h1,h2is defined.The basis functions and Bézier-like curves have properties similar to cubic Bernstein basis functions and cubic Bézier curves,respectively.Furthermore,we construct Bézier-like curves with energy constraints and consider the C1and G1Hermite interpolation with minimal energy.Finally,some representative examples show the applicability and effectiveness of the proposed method.展开更多
This article presents a new method for G2 continuous interpolation of an arbitrary sequence of points on an implicit or parametric surfaee with prescribed tangent direction and curvature vector, respectively, at every...This article presents a new method for G2 continuous interpolation of an arbitrary sequence of points on an implicit or parametric surfaee with prescribed tangent direction and curvature vector, respectively, at every point. First, a G2 continuous curve is constructed in three-dimensional space. Then the curve is projected normally onto the given surface. The desired interpolation curve is just the projection curve, which can be obtained by numerieally solving the initialvalue problems for a system of first-order ordinary differential equations in the parametric domain for parametric case or in three-dimensional space for implicit ease. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification so that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface. Numerical experiments demonstrate that it is effective and potentially useful in numerical control （NC） machining, path planning for robotic fibre placement, patterns design on surface and other industrial and research fields.展开更多
基金Supported by the Education Department of Zhejiang Province (Y200806015)
文摘By utilizing symmetric functions,this paper presents explicit representations for Hermite interpolation and its numerical differentiation formula.And the corresponding error estimates are also provided.
基金Project 19671082 Supported by National Natural Science Foundation of China
文摘The criteria of convergence, including a theorem of Grunwald- type and the rate of convergence in terms of the modulus omega phi (f,t) of Ditzian and Totik for truncated Hermite interpolation on ail arbitrary system of nodes are given.
文摘The object of this paper is to establish the pointwise estimations of approximation of functions in C^1 and their derivatives by Hermite interpolation polynomials. The given orders have been proved to be exact in gen- eral.
基金Supported by the National Natural Science Foundation of China(11271156 and 11171133)the Technology Development Plan of Jilin Province(20130522104JH)
文摘Multivariate Hermite interpolation is widely applied in many fields, such as finite element construction, inverse engineering, CAD etc.. For arbitrarily given Hermite interpolation conditions, the typical method is to compute the vanishing ideal I （the set of polynomials satisfying all the homogeneous interpolation conditions are zero） and then use a complete residue system modulo I as the interpolation basis. Thus the interpolation problem can be converted into solving a linear equation system. A generic algorithm was presented in [18], which is a generalization of BM algorithm [22] and the complexity is O（τ^3） where r represents the number of the interpolation conditions. In this paper we derive a method to obtain the residue system directly from the relative position of the points and the corresponding derivative conditions （presented by lower sets） and then use fast GEPP to solve the linear system with O（（τ ＋ 3）τ^2） operations, where τ is the displacement-rank of the coefficient matrix. In the best case τ = 1 and in the worst case τ = [τ/n], where n is the number of variables.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1130105361432003)
文摘In this paper we study a class of multivariate Hermite interpolation problem on 2;nodes with dimension d ≥ 2 which can be seen as a generalization of two classical Hermite interpolation problems of d = 2. Two combinatorial identities are firstly given and then the regularity of the proposed interpolation problem is proved.
文摘A threshold scheme, which is introduced by Shamir in 1979, is very famous as a secret sharing scheme. We can consider that this scheme is based on Lagrange＇s interpolation formula. A secret sharing scheme has one key. On the other hand, a multi-secret sharing scheme has more than one key, that is, a multi-secret sharing scheme has p （〉_ 2） keys. Dealer distribute shares of keys among n participants. Gathering t （〈 n） participants, keys can be reconstructed. Yang et al. （2004） gave a scheme of a （t, n） multi-secret sharing based on Lagrange＇s interpolation. Zhao et al. （2007） gave a scheme of a （t, n） verifiable multi-secret sharing based on Lagrange＇s interpolation. Recently, Adachi and Okazaki give a scheme of a （t, n） multi-secret sharing based on Hermite interpolation, in the case ofp 〈 t. In this paper, we give a scheme ofa （t, n） verifiable multi-secret sharing based on Hermite interpolation.
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For the weighted approximation in Lp-norm,the authors determine the weakly asymptotic order for the p-average errors of the sequence of Hermite interpolation based on the Chebyshev nodes on the 1-fold integrated Wiener space.By this result,it is known that in the sense of information-based complexity,if permissible information functionals are Hermite data,then the p-average errors of this sequence are weakly equivalent to those of the corresponding sequence of the minimal p-average radius of nonadaptive information.
基金Project 19671082 supported by National Natural Science Foundation of China, I acknowledge endless help from Prof. Shi Ying-Guang during finishing this paper.
文摘In this paper sufficient conditions for mean convergence and rate of convergence of Hermite-Fejer type interpolation in the Lp norm on an arbitrary system of nodes are presented.
基金Open Funds of State Key Laboratory of Oil and Gas Reservoir and Exploitation,Southwest Petroleum University(No.PCN0613)the Natural Foundation of Zhejiang Provincial Education Department (No.Kyg091206029)
文摘The present paper investigates the convergence of Hermite interpolation operators on the real line. The main result is： Given 0 〈 δo 〈 1/2, 0 〈 εo 〈 1. Let f ∈ C（-∞,∞） satisfy ｜y｜= O（e^（1/2-δo）xk^2,） and ｜f（x）｜t= O（e^（1-εo ）x2^）. Then for any given point x ∈ R, we have limn→Hn,（f, x） = f（x）.
基金partially supported by the CSIR India(Grant No.09/084(0531)/2010-EMR-I)the SERC,DST India(Project No.SR/S4/MS:694/10)
文摘Hermite interpolation is a very important tool in approximation theory and nu- merical analysis, and provides a popular method for modeling in the area of computer aided geometric design. However, the classical Hermite interpolant is unique for a prescribed data set, and hence lacks freedom for the choice of an interpolating curve, which is a crucial requirement in design environment. Even though there is a rather well developed fractal theory for Hermite interpolation that offers a large flexibility in the choice of interpolants, it also has the short- coming that the functions that can be well approximated are highly restricted to the class of self-affine functions. The primary objective of this paper is to suggest a gl-cubic Hermite in- terpolation scheme using a fractal methodology, namely, the coalescence hidden variable fractal interpolation, which works equally well for the approximation of a self-affine and non-self-affine data generating functions. The uniform error bound for the proposed fractal interpolant is established to demonstrate that the convergence properties are similar to that of the classical Hermite interpolant. For the Hermite interpolation problem, if the derivative values are not actually prescribed at the knots, then we assign these values so that the interpolant gains global G2-continuity. Consequently, the procedure culminates with the construction of cubic spline coalescence hidden variable fractal interpolants. Thus, the present article also provides an al- ternative to the construction of cubic spline coalescence hidden variable fractal interpolation functions through moments proposed by Chand and Kapoor [Fractals, 15（1） （2007）, pp. 41-53].
基金Project 2921200 Supported by National Natural Science Foundation of China.
文摘In this paper the uniform convergence of Hermite-Fejer interpolation and Griinwald type theorem of higher order on an arbitrary system of nodes are presented.
文摘The paper gives a new way of constructing Hermite Fejer and Hermite interpolatory polynomials with the nodes of the roots of first kind of Chebyshev polynomials and gives the approximation order of these two kinds of operators. The approximation orders are described with the best rate of approximation of f by polynomials of degree N=(q+1)n 1 in L p w spaces.
文摘This paper introduces the definition of the Orthogonal Type Node Configuration and discusses the corresponding multivariate Lagrange, Hermite and Birkhoff interpolation problems in high dimensional space R s(s>2). This node configuration can be considered to be a kind of extension of the Cross Type Node Configuration , in R 2 to high dimensional spaces. And the Mixed Type Node Configuration in R s(s>2) is also discussed in this paper in an example.
基金Supported by the National Natural Science Foundation of China(Grant No.11601224)the Science Foundation of Ministry of Education of China(Grant No.18YJC790069)+1 种基金the Natural Science Foundation of Jiangsu Higher Education Institutions of China(Grant No.18KJD110007)the National Statistical Science Research Project of China(Grant No.2018LY28)。
文摘It is well-known that Hermite rational interpolation gives a better approximation than Hermite polynomial interpolation,especially for large sequences of interpolation points,but it is difficult to solve the problem of convergence and control the occurrence of real poles.In this paper,we establish a family of linear Hermite barycentric rational interpolants r that has no real poles on any interval and in the case k=0,1,2,the function r^(k)(x)converges to f^(k)(x)at the rate of O(h^3d+3-k)as h→0 on any real interpolation interval,regardless of the distribution of the interpolation points.Also,the function r(x)is linear in data.
基金supported by the grant of Key Scientific Research Foundation of Education Department of Anhui Province, No. KJ2014A210
文摘General interpolation formulae for barycentric interpolation and barycen- tric rational Hermite interpolation are established by introducing multiple parameters, which include many kinds of barycentric interpolation and barycentric rational Her- mite interpolation. We discussed the interpolation theorem, dual interpolation and special cases. Numerical example is given to show the effectiveness of the method.
基金Ph.D.Programs Foundation (200805581022) of Ministry of Education of China
文摘Consider a kind of Hermit interpolation for scattered data of 3D by trivariate polynomial natural spline, such that the objective energy functional （with natural boundary conditions） is minimal. By the spline function methods in Hilbert space and variational theory of splines, the characters of the interpolation solution and how to construct it are studied. One can easily find that the interpolation solution is a trivariate polynomial natural spline. Its expression is simple and the coefficients can be decided by a linear system. Some numerical examples are presented to demonstrate our methods.
基金The Project is supported by National Natural Science Foundation of China.
文摘In this paper. a quantitative estimate for Hermite interpolant to function ψ(z)=(z^m-β~m)~l on the ze- ros of (z^n-α~n)~r is obtained Using this estimate. a rather wide exiension of the theorem of Walsh is proved and five special cases of it are given.
基金Supported by the National Natural Science Foundation of China(Grant No.11801225)University Science Research Project of Jiangsu Province(Grant No.18KJB110005)。
文摘In this paper,we present a class of novel Bernstein-like basis functions,which is an extension of classical Bernstein basis functions.The properties of this group of bases are analyzed and the Bézier-like curve with two shape parameters h1,h2is defined.The basis functions and Bézier-like curves have properties similar to cubic Bernstein basis functions and cubic Bézier curves,respectively.Furthermore,we construct Bézier-like curves with energy constraints and consider the C1and G1Hermite interpolation with minimal energy.Finally,some representative examples show the applicability and effectiveness of the proposed method.
基金National Natural Science Foundation of China(60673026,50875130,50805075 and 50875126)
文摘This article presents a new method for G2 continuous interpolation of an arbitrary sequence of points on an implicit or parametric surfaee with prescribed tangent direction and curvature vector, respectively, at every point. First, a G2 continuous curve is constructed in three-dimensional space. Then the curve is projected normally onto the given surface. The desired interpolation curve is just the projection curve, which can be obtained by numerieally solving the initialvalue problems for a system of first-order ordinary differential equations in the parametric domain for parametric case or in three-dimensional space for implicit ease. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification so that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface. Numerical experiments demonstrate that it is effective and potentially useful in numerical control （NC） machining, path planning for robotic fibre placement, patterns design on surface and other industrial and research fields.
