S.M.Lozinskii proved the exact convergence rate at the zero of Lagrange interpolation polynomials to |x| based on equidistant nodes in [-1,1]. In 2000, M. Rever generalized S.M.Lozinskii's result to |x|α(0 <≤...S.M.Lozinskii proved the exact convergence rate at the zero of Lagrange interpolation polynomials to |x| based on equidistant nodes in [-1,1]. In 2000, M. Rever generalized S.M.Lozinskii's result to |x|α(0 <≤ α≤ 1). In this paper we will present the exact rate of convergence at the point zero for the interpolants of |x|α1(1 < α < 2)..展开更多
It is a classical result of Bernstein that the sequence of Lagrange interpolation polynomials to \x\ at e-qually spaced nodes in [-1.1] diverges everywhere. except at zero and the end-points. In this paper we show tha...It is a classical result of Bernstein that the sequence of Lagrange interpolation polynomials to \x\ at e-qually spaced nodes in [-1.1] diverges everywhere. except at zero and the end-points. In this paper we show that the sequence of Lagrange interpolation polynomials corresponding to the functions which possess better smoothness on equidistant nodes in [-1.1] still diverges every -where in the interval except at zero and the end-points.展开更多
It is a classical result of Bernstein that the sequence of Lagrange interpolation polumomials to |x| at equally spaced nodes in [-1, 1] diverges everywhere, except at zero and the end-points. In the present paper, t...It is a classical result of Bernstein that the sequence of Lagrange interpolation polumomials to |x| at equally spaced nodes in [-1, 1] diverges everywhere, except at zero and the end-points. In the present paper, toe prove that the sequence of Lagrange interpolation polynomials corresponding to |x|^α (2 〈 α 〈 4) on equidistant nodes in [-1, 1] diverges everywhere, except at zero and the end-points.展开更多
In this paper we present a generalized quantitative version of a result the exact convergence rate at zero of Lagrange interpolation polynomial to spaced nodes in [-1,1] due to M.Revers concerning f(x) = |x|α wit...In this paper we present a generalized quantitative version of a result the exact convergence rate at zero of Lagrange interpolation polynomial to spaced nodes in [-1,1] due to M.Revers concerning f(x) = |x|α with on equally展开更多
This paper shows that the sequence of Lagrange interpolation polynomials corresponding to the rune tion f(z) =|x|^α(1〈α〈2) on [-1,1] can diverge everywhere in the interval except at zero and the end-points.
Properties of Lebesgue function for Lagrange interpolation on equidistant nodes are investigated. It is proved that Lebesgue function can be formulated both in terms of a hypergeometric function 2F1 and Jacobi polynom...Properties of Lebesgue function for Lagrange interpolation on equidistant nodes are investigated. It is proved that Lebesgue function can be formulated both in terms of a hypergeometric function 2F1 and Jacobi polynomials. Moreover, an integral expression of Lebesgue function is also obtained and the asymptotic behavior of Lebesgue constant is studied.展开更多
We study the optimal order of approximation for |x|α (0 < α < 1) by Lagrange interpolation polynomials based on Chebyshev nodes of the first kind. It is proved that the Jackson order of approximation is attained.
In this paper, we investigate the negative extremums of fundamental functions of Lagrange interpolation based on Chebyshev nodes. Moreover, we establish some companion results to the theorem of J. Szabados on the posi...In this paper, we investigate the negative extremums of fundamental functions of Lagrange interpolation based on Chebyshev nodes. Moreover, we establish some companion results to the theorem of J. Szabados on the positive extremum.展开更多
This paper offers a general formula for surface subdivision rules for quad meshes by using 2-D Lagrange interpolating polynomial [1]. We also see that the result obtained is equivalent to the tensor product of (2N + 4...This paper offers a general formula for surface subdivision rules for quad meshes by using 2-D Lagrange interpolating polynomial [1]. We also see that the result obtained is equivalent to the tensor product of (2N + 4)-point n-ary interpolating curve scheme for N ≥ 0 and n ≥ 2. The simple interpolatory subdivision scheme for quadrilateral nets with arbitrary topology is presented by L. Kobbelt [2], which can be directly calculated from the proposed formula. Furthermore, some characteristics and applications of the proposed work are also discussed.展开更多
We study some approximation properties of Lagrange interpolation polynomial based on the zeros of (1-x^2)cosnarccosx. By using a decomposition for f(x) ∈ C^τC^τ+1 we obtain an estimate of ‖f(x) -Ln+2(f, ...We study some approximation properties of Lagrange interpolation polynomial based on the zeros of (1-x^2)cosnarccosx. By using a decomposition for f(x) ∈ C^τC^τ+1 we obtain an estimate of ‖f(x) -Ln+2(f, x)‖ which reflects the influence of the position of the x's and ω(f^(r+1),δ)j,j = 0, 1,... , s,on the error of approximation.展开更多
In this paper, we obtain a properly posed set of nodes for interpolation on a sphere. Moreover it is applied to construct properly posed set of nodes for Lagrange interpolation on the trivariate polynomial space of to...In this paper, we obtain a properly posed set of nodes for interpolation on a sphere. Moreover it is applied to construct properly posed set of nodes for Lagrange interpolation on the trivariate polynomial space of total degree n.展开更多
In this paper, we discuss the average errors of multivariate Lagrange interpolation based on the Chebyshev nodes of the first kind. The average errors of the interpolation sequence are determined on the multivariate W...In this paper, we discuss the average errors of multivariate Lagrange interpolation based on the Chebyshev nodes of the first kind. The average errors of the interpolation sequence are determined on the multivariate Wiener space.展开更多
In this paper,Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation.First of all,the interpolation basis function is appli...In this paper,Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation.First of all,the interpolation basis function is applied to treat the spatial variables and their partial derivatives,and the collocation method for solving the second order differential equations is established.Secondly,the differential matrix is used to simplify the given differential equations on a given test node.Finally,based on three kinds of test nodes,numerical experiments show that the present scheme can not only calculate the high wave numbers problems,but also calculate the variable wave numbers problems.In addition,the algorithm has the advantages of high calculation accuracy,good numerical stability and less time consuming.展开更多
In the paper, a result of Walsh and Sharma on least square convergence of Lagrange interpolation polynomials based on the n-th roots of unity is extended to Lagrange interpolation on the sets obtained by projecting ve...In the paper, a result of Walsh and Sharma on least square convergence of Lagrange interpolation polynomials based on the n-th roots of unity is extended to Lagrange interpolation on the sets obtained by projecting vertically the zeros of (1-x)2=P (a,β) n(x),a>0,β>0,(1-x)P(a,β) n(x),a>0,β>-1,(1+x)P P(a,β) n(x),a>-1,β0 and P(a,β) n(x),a>-1,β>-1, respectively, onto the unit circle, where P(a,β) n(x),a>-1,β>-1, stands for the n-th Jacobi polynomial. Moreover, a result of Saff and Walsh is also extended.展开更多
The following theorem is proved Theorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying in the interval[-1,1] and △'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}. If polynomial pP_n satisfies ...The following theorem is proved Theorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying in the interval[-1,1] and △'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}. If polynomial pP_n satisfies the inequality then for each k=1,n and any x[-1,1]its k-th derivative satisfies the inequality 丨p^(k)(x)丨≤max{丨q^((k))(x)丨,丨1/k(x^2-1)q^(k+1)(x)+xq^((k))(x)丨}. This estimate leads to the Markov inequality for the higher order derivatives of polynomials if we set q=T_n,where Tn is Chebyshev polynomial least deviated from zero. Some other results are established which gives evidence to the conjecture that under the conditions of Theorem 1 the inequality ‖p^((k))‖≤‖q^(k)‖holds.展开更多
We propose new hybrid Lagrange neural networks called LaNets to predict the numerical solutions of partial differential equations.That is,we embed Lagrange interpolation and small sample learning into deep neural netw...We propose new hybrid Lagrange neural networks called LaNets to predict the numerical solutions of partial differential equations.That is,we embed Lagrange interpolation and small sample learning into deep neural network frameworks.Concretely,we first perform Lagrange interpolation in front of the deep feedforward neural network.The Lagrange basis function has a neat structure and a strong expression ability,which is suitable to be a preprocessing tool for pre-fitting and feature extraction.Second,we introduce small sample learning into training,which is beneficial to guide themodel to be corrected quickly.Taking advantages of the theoretical support of traditional numerical method and the efficient allocation of modern machine learning,LaNets achieve higher predictive accuracy compared to the state-of-the-artwork.The stability and accuracy of the proposed algorithmare demonstrated through a series of classical numerical examples,including one-dimensional Burgers equation,onedimensional carburizing diffusion equations,two-dimensional Helmholtz equation and two-dimensional Burgers equation.Experimental results validate the robustness,effectiveness and flexibility of the proposed algorithm.展开更多
This paper considers to replace △_m(x)=(1-x^2)~2(1/2)/n +1/n^2 in the following result for simultaneous Lagrange interpolating approximation with (1-x^2)~2(1/2)/n: Let f∈C_(-1.1)~0 and r=[(q+2)/2],then |f^(k)(x)-P_^...This paper considers to replace △_m(x)=(1-x^2)~2(1/2)/n +1/n^2 in the following result for simultaneous Lagrange interpolating approximation with (1-x^2)~2(1/2)/n: Let f∈C_(-1.1)~0 and r=[(q+2)/2],then |f^(k)(x)-P_^(k)(f,x)|=O(1)△_(n)^(a-k)(x)ω(f^(a),△(x))(‖L_n-‖+‖L_n‖),0≤k≤q, where P_n( f ,x)is the Lagrange interpolating polynomial of degree n+ 2r-1 of f on the nodes X_n U Y_n(see the definition of the text), and thus give a problem raised in [XiZh] a complete answer.展开更多
文摘S.M.Lozinskii proved the exact convergence rate at the zero of Lagrange interpolation polynomials to |x| based on equidistant nodes in [-1,1]. In 2000, M. Rever generalized S.M.Lozinskii's result to |x|α(0 <≤ α≤ 1). In this paper we will present the exact rate of convergence at the point zero for the interpolants of |x|α1(1 < α < 2)..
文摘It is a classical result of Bernstein that the sequence of Lagrange interpolation polynomials to \x\ at e-qually spaced nodes in [-1.1] diverges everywhere. except at zero and the end-points. In this paper we show that the sequence of Lagrange interpolation polynomials corresponding to the functions which possess better smoothness on equidistant nodes in [-1.1] still diverges every -where in the interval except at zero and the end-points.
文摘It is a classical result of Bernstein that the sequence of Lagrange interpolation polumomials to |x| at equally spaced nodes in [-1, 1] diverges everywhere, except at zero and the end-points. In the present paper, toe prove that the sequence of Lagrange interpolation polynomials corresponding to |x|^α (2 〈 α 〈 4) on equidistant nodes in [-1, 1] diverges everywhere, except at zero and the end-points.
文摘In this paper we present a generalized quantitative version of a result the exact convergence rate at zero of Lagrange interpolation polynomial to spaced nodes in [-1,1] due to M.Revers concerning f(x) = |x|α with on equally
文摘This paper shows that the sequence of Lagrange interpolation polynomials corresponding to the rune tion f(z) =|x|^α(1〈α〈2) on [-1,1] can diverge everywhere in the interval except at zero and the end-points.
文摘Properties of Lebesgue function for Lagrange interpolation on equidistant nodes are investigated. It is proved that Lebesgue function can be formulated both in terms of a hypergeometric function 2F1 and Jacobi polynomials. Moreover, an integral expression of Lebesgue function is also obtained and the asymptotic behavior of Lebesgue constant is studied.
文摘We study the optimal order of approximation for |x|α (0 < α < 1) by Lagrange interpolation polynomials based on Chebyshev nodes of the first kind. It is proved that the Jackson order of approximation is attained.
文摘In this paper, we investigate the negative extremums of fundamental functions of Lagrange interpolation based on Chebyshev nodes. Moreover, we establish some companion results to the theorem of J. Szabados on the positive extremum.
文摘This paper offers a general formula for surface subdivision rules for quad meshes by using 2-D Lagrange interpolating polynomial [1]. We also see that the result obtained is equivalent to the tensor product of (2N + 4)-point n-ary interpolating curve scheme for N ≥ 0 and n ≥ 2. The simple interpolatory subdivision scheme for quadrilateral nets with arbitrary topology is presented by L. Kobbelt [2], which can be directly calculated from the proposed formula. Furthermore, some characteristics and applications of the proposed work are also discussed.
基金Supported by the National Nature Science Foundation.
文摘We study some approximation properties of Lagrange interpolation polynomial based on the zeros of (1-x^2)cosnarccosx. By using a decomposition for f(x) ∈ C^τC^τ+1 we obtain an estimate of ‖f(x) -Ln+2(f, x)‖ which reflects the influence of the position of the x's and ω(f^(r+1),δ)j,j = 0, 1,... , s,on the error of approximation.
文摘In this paper, we obtain a properly posed set of nodes for interpolation on a sphere. Moreover it is applied to construct properly posed set of nodes for Lagrange interpolation on the trivariate polynomial space of total degree n.
文摘In this paper, we discuss the average errors of multivariate Lagrange interpolation based on the Chebyshev nodes of the first kind. The average errors of the interpolation sequence are determined on the multivariate Wiener space.
基金partially supported by National Natural Science Foundation of China(11772165,11961054,11902170)Key Research and Development Program of Ningxia(2018BEE03007)+1 种基金National Natural Science Foundation of Ningxia(2018AAC02003,2020AAC03059)Major Innovation Projects for Building First-class Universities in China’s Western Region(Grant No.ZKZD2017009).
文摘In this paper,Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation.First of all,the interpolation basis function is applied to treat the spatial variables and their partial derivatives,and the collocation method for solving the second order differential equations is established.Secondly,the differential matrix is used to simplify the given differential equations on a given test node.Finally,based on three kinds of test nodes,numerical experiments show that the present scheme can not only calculate the high wave numbers problems,but also calculate the variable wave numbers problems.In addition,the algorithm has the advantages of high calculation accuracy,good numerical stability and less time consuming.
基金NSFC under grant1 0 0 71 0 3 9and by Education Committee of Jiangsu Province under grant0 0 KJB1 1 0 0 0 5 .
文摘In the paper, a result of Walsh and Sharma on least square convergence of Lagrange interpolation polynomials based on the n-th roots of unity is extended to Lagrange interpolation on the sets obtained by projecting vertically the zeros of (1-x)2=P (a,β) n(x),a>0,β>0,(1-x)P(a,β) n(x),a>0,β>-1,(1+x)P P(a,β) n(x),a>-1,β0 and P(a,β) n(x),a>-1,β>-1, respectively, onto the unit circle, where P(a,β) n(x),a>-1,β>-1, stands for the n-th Jacobi polynomial. Moreover, a result of Saff and Walsh is also extended.
文摘The following theorem is proved Theorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying in the interval[-1,1] and △'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}. If polynomial pP_n satisfies the inequality then for each k=1,n and any x[-1,1]its k-th derivative satisfies the inequality 丨p^(k)(x)丨≤max{丨q^((k))(x)丨,丨1/k(x^2-1)q^(k+1)(x)+xq^((k))(x)丨}. This estimate leads to the Markov inequality for the higher order derivatives of polynomials if we set q=T_n,where Tn is Chebyshev polynomial least deviated from zero. Some other results are established which gives evidence to the conjecture that under the conditions of Theorem 1 the inequality ‖p^((k))‖≤‖q^(k)‖holds.
基金supported by NSFC(No.11971296)National Key Research and Development Program of China(No.2021YFA1003004).
文摘We propose new hybrid Lagrange neural networks called LaNets to predict the numerical solutions of partial differential equations.That is,we embed Lagrange interpolation and small sample learning into deep neural network frameworks.Concretely,we first perform Lagrange interpolation in front of the deep feedforward neural network.The Lagrange basis function has a neat structure and a strong expression ability,which is suitable to be a preprocessing tool for pre-fitting and feature extraction.Second,we introduce small sample learning into training,which is beneficial to guide themodel to be corrected quickly.Taking advantages of the theoretical support of traditional numerical method and the efficient allocation of modern machine learning,LaNets achieve higher predictive accuracy compared to the state-of-the-artwork.The stability and accuracy of the proposed algorithmare demonstrated through a series of classical numerical examples,including one-dimensional Burgers equation,onedimensional carburizing diffusion equations,two-dimensional Helmholtz equation and two-dimensional Burgers equation.Experimental results validate the robustness,effectiveness and flexibility of the proposed algorithm.
文摘This paper considers to replace △_m(x)=(1-x^2)~2(1/2)/n +1/n^2 in the following result for simultaneous Lagrange interpolating approximation with (1-x^2)~2(1/2)/n: Let f∈C_(-1.1)~0 and r=[(q+2)/2],then |f^(k)(x)-P_^(k)(f,x)|=O(1)△_(n)^(a-k)(x)ω(f^(a),△(x))(‖L_n-‖+‖L_n‖),0≤k≤q, where P_n( f ,x)is the Lagrange interpolating polynomial of degree n+ 2r-1 of f on the nodes X_n U Y_n(see the definition of the text), and thus give a problem raised in [XiZh] a complete answer.