Because of the features involved with their varied kernels,differential operators relying on convolution formulations have been acknowledged as effective mathematical resources for modeling real-world issues.In this p...Because of the features involved with their varied kernels,differential operators relying on convolution formulations have been acknowledged as effective mathematical resources for modeling real-world issues.In this paper,we constructed a stochastic fractional framework of measles spreading mechanisms with dual medication immunization considering the exponential decay and Mittag-Leffler kernels.In this approach,the overall population was separated into five cohorts.Furthermore,the descriptive behavior of the system was investigated,including prerequisites for the positivity of solutions,invariant domain of the solution,presence and stability of equilibrium points,and sensitivity analysis.We included a stochastic element in every cohort and employed linear growth and Lipschitz criteria to show the existence and uniqueness of solutions.Several numerical simulations for various fractional orders and randomization intensities are illustrated.展开更多
In this paper we investigate simultaneous approximation for arbitrary system of nodes on smooth domain in complex plane. Some results which are better than those of known theorems are obtained.
This paper addresses the extremal problem of the null subcarriers based Doppler scale estimation in underwater acoustic （UWA） orthogonal frequency division multiplexing （OFDM） communication. The cost function cons...This paper addresses the extremal problem of the null subcarriers based Doppler scale estimation in underwater acoustic （UWA） orthogonal frequency division multiplexing （OFDM） communication. The cost function constructed of the total energy of null subcarriers through discrete Fourier transform （DFT） is proposed. The frequencies of null subcarriers are identified from non-uniform Doppler shift at each tentative scaling factor. Then it is proved that the cost function can be fitted as a quadratic polynomial near the global minimum. An accurate Doppler scale estimation is achieved by the location of the global scarifying precision and increasing the computation minimum through polynomial interpolation, without complexity. A shallow water experiment is conducted to demonstrate the performance of the proposed method. Excellent performance results are obtained in ultrawideband UWA channels with a relative bandwidth of 67%, when the transmitter and the receiver are moving at a relative speed of 5 kn, which validates the proposed method.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
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
This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of...This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodes X_nUY_n(see the definition of the next).展开更多
Based on the polynomial interpolation, a new finite difference （FD） method in solving the full-vectorial guidedmodes for step-index optical waveguides is proposed. The discontinuities of the normal components of the...Based on the polynomial interpolation, a new finite difference （FD） method in solving the full-vectorial guidedmodes for step-index optical waveguides is proposed. The discontinuities of the normal components of the electric field across abrupt dielectric interfaces are considered in the absence of the limitations of scalar and semivectorial approximation, and the present PD scheme can be applied to both uniform and non-uniform mesh grids. The modal propagation constants and field distributions for buried rectangular waveguides and optical rib waveguides are presented. The hybrid nature of the vectorial modes is demonstrated and the singular behaviours of the minor field components in the corners are observed. Moreover, solutions are in good agreement with those published early, which tests the validity of the present approach.展开更多
In this note, we seek for functions f which are approximated by the sequence of interpolation polynomials of f obtained by any prescribed system of nodes.
The tempered fractional calculus has been successfully applied for depicting the time evolution of a system describing non-Markovian diffusion particles.The related governing equations are a series of partial differen...The tempered fractional calculus has been successfully applied for depicting the time evolution of a system describing non-Markovian diffusion particles.The related governing equations are a series of partial differential equations with tempered fractional derivatives.Using the polynomial interpolation technique,in this paper,we present three efficient numerical formulas,namely the tempered L1 formula,the tempered L1-2 formula,and the tempered L2-1_(σ)formula,to approximate the Caputo-tempered fractional derivative of orderα∈(0,1).The truncation error of the tempered L1 formula is of order 2-α,and the tempered L1-2 formula and L2-1_(σ)formula are of order 3-α.As an application,we construct implicit schemes and implicit ADI schemes for one-dimensional and two-dimensional time-tempered fractional diffusion equations,respectively.Furthermore,the unconditional stability and convergence of two developed difference schemes with tempered L1 and L2-1_(σ)formulas are proved by the Fourier analysis method.Finally,we provide several numerical examples to demonstrate the correctness and effectiveness of the theoretical analysis.展开更多
In this paper,a new third type S.N.Bernstein interpolation polynomial H n(f;x,r) with zeros of the Chebyshev ploynomial of the second kind is constructed. H n(f;x,r) converge uniformly on [-1,1] for any continuous fun...In this paper,a new third type S.N.Bernstein interpolation polynomial H n(f;x,r) with zeros of the Chebyshev ploynomial of the second kind is constructed. H n(f;x,r) converge uniformly on [-1,1] for any continuous function f(x) . The convergence order is the best order if \{f(x)∈C j[-1,1],\}0jr, where r is an odd natural number.展开更多
Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P&...Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P'n-1(x). By the theory of the inverse Pal-Type interpolation, for a function f(x) ∈ C[-1 1], there exists a unique polynomial Rn(x) of degree 2n - 2 (if n is even) satisfying conditions Rn(f,ξk) = f(∈ek)(1≤ k≤ n - 1) ;R'n(f,xk) = f'(xk)(1≤ k≤ n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {Rn(f,x)} (n is even) and the main result of this paper is that if f ∈ C'[1,1], r≥2, n≥ + 2> and n is even thenholds uniformly for all x ∈ [- 1,1], where h(x) = 1 +展开更多
The multivariate splines which were first presented by deBooor as a complete theoretical system have intrigued many mathematicians who have devoted many works in this field which is still in the process of development...The multivariate splines which were first presented by deBooor as a complete theoretical system have intrigued many mathematicians who have devoted many works in this field which is still in the process of development.The author of this paper is interested in the area of inter- polation with special emphasis on the interpolation methods and their approximation orders. But such B-splines(both univariate and multivariate)do not interpolated directly,so I ap- proached this problem in another way which is to extend my interpolating spline of degree 2n-1 in univariate case(See[7])to multivariate case.I selected triangulated region which is inspired by other mathematicians'works(e.g.[2]and[3])and extend the interpolating polynomials from univariate to m-variate case(See[10])In this paper some results in the case m=2 are discussed and proved in more concrete details.Based on these polynomials,the interpolating splines(it is defined by me as piecewise polynomials in which the unknown par- tial derivatives are determined under certain continuous conditions)are also discussed.The approximation orders of interpolating polynomials and of cubic interpolating splines are inverstigated.We lunited our discussion on the rectangular domain which is partitioned into equal right triangles.As to the case in which the rectangular domain is partitioned into unequal right triangles as well as the case of more complicated domains,we will discuss in the next pa- per.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. 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
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
The Fourier p-element method is an improvement to the finite element method,and is particularly suitable for vibration analysis due to the well-behaved Fourier series.In this paper,an iteration procedure is presented ...The Fourier p-element method is an improvement to the finite element method,and is particularly suitable for vibration analysis due to the well-behaved Fourier series.In this paper,an iteration procedure is presented for solving the resulting nonlinear eigenvalue problem.Three types of Fourier version shape functions are constructed for analyzing the circular shaft torsional vibration,the plate in-plane vibration and annular plate flexural vibration modes,respectively. The numerical results show that this method can achieve higher accuracy and converge much faster than the FEM based on polynomial interpolation,especially for higher mode analysis.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.
Subpixel accuracy for V-groove center in robot welding is researched and a software measure to increase the accuracy of seam tracking by laser is presented. LOG （ Laplacian of Gaussian ） operator is adopted to detec...Subpixel accuracy for V-groove center in robot welding is researched and a software measure to increase the accuracy of seam tracking by laser is presented. LOG （ Laplacian of Gaussian ） operator is adopted to detect image edge. Vgroove center is extracted by corner detection of extremum curvature. Subpixel position is obtained by Lagarange polynomial interpolation algorithm. Experiment results show that the method is brief and applied, and is sufficient for the real time of robot welding by laser sensors.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.
文摘Because of the features involved with their varied kernels,differential operators relying on convolution formulations have been acknowledged as effective mathematical resources for modeling real-world issues.In this paper,we constructed a stochastic fractional framework of measles spreading mechanisms with dual medication immunization considering the exponential decay and Mittag-Leffler kernels.In this approach,the overall population was separated into five cohorts.Furthermore,the descriptive behavior of the system was investigated,including prerequisites for the positivity of solutions,invariant domain of the solution,presence and stability of equilibrium points,and sensitivity analysis.We included a stochastic element in every cohort and employed linear growth and Lipschitz criteria to show the existence and uniqueness of solutions.Several numerical simulations for various fractional orders and randomization intensities are illustrated.
文摘In this paper we investigate simultaneous approximation for arbitrary system of nodes on smooth domain in complex plane. Some results which are better than those of known theorems are obtained.
基金supported by the National Natural Science Foundation of China(6120109661471137+4 种基金61501061)the Qing Lan Project of Jiangsu Province,the Science and Technology Program of Changzhou City(CJ20130026CE20135060CE20145055)the State Key Laboratory of Ocean Engineering(Shanghai Jiao Tong University)(1316)
文摘This paper addresses the extremal problem of the null subcarriers based Doppler scale estimation in underwater acoustic （UWA） orthogonal frequency division multiplexing （OFDM） communication. The cost function constructed of the total energy of null subcarriers through discrete Fourier transform （DFT） is proposed. The frequencies of null subcarriers are identified from non-uniform Doppler shift at each tentative scaling factor. Then it is proved that the cost function can be fitted as a quadratic polynomial near the global minimum. An accurate Doppler scale estimation is achieved by the location of the global scarifying precision and increasing the computation minimum through polynomial interpolation, without complexity. A shallow water experiment is conducted to demonstrate the performance of the proposed method. Excellent performance results are obtained in ultrawideband UWA channels with a relative bandwidth of 67%, when the transmitter and the receiver are moving at a relative speed of 5 kn, which validates the proposed method.
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.
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.
基金The second named author was supported in part by an NSERC Postdoctoral Fellowship,Canada and a CR F Grant,University of Alberta
文摘This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodes X_nUY_n(see the definition of the next).
文摘Based on the polynomial interpolation, a new finite difference （FD） method in solving the full-vectorial guidedmodes for step-index optical waveguides is proposed. The discontinuities of the normal components of the electric field across abrupt dielectric interfaces are considered in the absence of the limitations of scalar and semivectorial approximation, and the present PD scheme can be applied to both uniform and non-uniform mesh grids. The modal propagation constants and field distributions for buried rectangular waveguides and optical rib waveguides are presented. The hybrid nature of the vectorial modes is demonstrated and the singular behaviours of the minor field components in the corners are observed. Moreover, solutions are in good agreement with those published early, which tests the validity of the present approach.
文摘In this note, we seek for functions f which are approximated by the sequence of interpolation polynomials of f obtained by any prescribed system of nodes.
文摘The tempered fractional calculus has been successfully applied for depicting the time evolution of a system describing non-Markovian diffusion particles.The related governing equations are a series of partial differential equations with tempered fractional derivatives.Using the polynomial interpolation technique,in this paper,we present three efficient numerical formulas,namely the tempered L1 formula,the tempered L1-2 formula,and the tempered L2-1_(σ)formula,to approximate the Caputo-tempered fractional derivative of orderα∈(0,1).The truncation error of the tempered L1 formula is of order 2-α,and the tempered L1-2 formula and L2-1_(σ)formula are of order 3-α.As an application,we construct implicit schemes and implicit ADI schemes for one-dimensional and two-dimensional time-tempered fractional diffusion equations,respectively.Furthermore,the unconditional stability and convergence of two developed difference schemes with tempered L1 and L2-1_(σ)formulas are proved by the Fourier analysis method.Finally,we provide several numerical examples to demonstrate the correctness and effectiveness of the theoretical analysis.
文摘In this paper,a new third type S.N.Bernstein interpolation polynomial H n(f;x,r) with zeros of the Chebyshev ploynomial of the second kind is constructed. H n(f;x,r) converge uniformly on [-1,1] for any continuous function f(x) . The convergence order is the best order if \{f(x)∈C j[-1,1],\}0jr, where r is an odd natural number.
文摘Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P'n-1(x). By the theory of the inverse Pal-Type interpolation, for a function f(x) ∈ C[-1 1], there exists a unique polynomial Rn(x) of degree 2n - 2 (if n is even) satisfying conditions Rn(f,ξk) = f(∈ek)(1≤ k≤ n - 1) ;R'n(f,xk) = f'(xk)(1≤ k≤ n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {Rn(f,x)} (n is even) and the main result of this paper is that if f ∈ C'[1,1], r≥2, n≥ + 2> and n is even thenholds uniformly for all x ∈ [- 1,1], where h(x) = 1 +
文摘The multivariate splines which were first presented by deBooor as a complete theoretical system have intrigued many mathematicians who have devoted many works in this field which is still in the process of development.The author of this paper is interested in the area of inter- polation with special emphasis on the interpolation methods and their approximation orders. But such B-splines(both univariate and multivariate)do not interpolated directly,so I ap- proached this problem in another way which is to extend my interpolating spline of degree 2n-1 in univariate case(See[7])to multivariate case.I selected triangulated region which is inspired by other mathematicians'works(e.g.[2]and[3])and extend the interpolating polynomials from univariate to m-variate case(See[10])In this paper some results in the case m=2 are discussed and proved in more concrete details.Based on these polynomials,the interpolating splines(it is defined by me as piecewise polynomials in which the unknown par- tial derivatives are determined under certain continuous conditions)are also discussed.The approximation orders of interpolating polynomials and of cubic interpolating splines are inverstigated.We lunited our discussion on the rectangular domain which is partitioned into equal right triangles.As to the case in which the rectangular domain is partitioned into unequal right triangles as well as the case of more complicated domains,we will discuss in the next pa- per.
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.
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.
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The Fourier p-element method is an improvement to the finite element method,and is particularly suitable for vibration analysis due to the well-behaved Fourier series.In this paper,an iteration procedure is presented for solving the resulting nonlinear eigenvalue problem.Three types of Fourier version shape functions are constructed for analyzing the circular shaft torsional vibration,the plate in-plane vibration and annular plate flexural vibration modes,respectively. The numerical results show that this method can achieve higher accuracy and converge much faster than the FEM based on polynomial interpolation,especially for higher mode analysis.
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.
基金This work is financially supported by National Nature Science Foundation of China (Grant No. 50175027).
文摘Subpixel accuracy for V-groove center in robot welding is researched and a software measure to increase the accuracy of seam tracking by laser is presented. LOG （ Laplacian of Gaussian ） operator is adopted to detect image edge. Vgroove center is extracted by corner detection of extremum curvature. Subpixel position is obtained by Lagarange polynomial interpolation algorithm. Experiment results show that the method is brief and applied, and is sufficient for the real time of robot welding by laser sensors.
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.
