Suppose that the outer mapping function of domain D has its second continuous derivatives. In this paper, the order proximation by (0,1,…,q) Hermite-Fejer interpolating polynomials at nearly Fejer's points of fun...Suppose that the outer mapping function of domain D has its second continuous derivatives. In this paper, the order proximation by (0,1,…,q) Hermite-Fejer interpolating polynomials at nearly Fejer's points of function of class A(D) are presented. Moreover in general the order of approximation is sharp.展开更多
The 'o' saturation theorem and the degree of Lwp, approximation by (0 - q' - q) type Hermite-Fejer interpolating polynomials for mean convergence are obtained.
A novel suppression method of the phase noise is proposed to reduce the negative impacts of phase noise in coherent optical orthogonal frequency division multiplexing(CO-OFDM)systems.The method integrates the sub-symb...A novel suppression method of the phase noise is proposed to reduce the negative impacts of phase noise in coherent optical orthogonal frequency division multiplexing(CO-OFDM)systems.The method integrates the sub-symbol second-order polynomial interpolation(SSPI)with cubature Kalman filter(CKF)to improve the precision and effectiveness of the data processing through using a three-stage processing approach of phase noise.First of all,the phase noise values in OFDM symbols are calculated by using pilot symbols.Then,second-order Newton interpolation(SNI)is used in second-order interpolation to acquire precise noise estimation.Afterwards,every OFDM symbol is partitioned into several sub-symbols,and second-order polynomial interpolation(SPI)is utilized in the time domain to enhance suppression accuracy and time resolution.Ultimately,CKF is employed to suppress the residual phase noise.The simulation results show that this method significantly suppresses the impact of the phase noise on the system,and the error floors can be decreased at the condition of 16 quadrature amplitude modulation(16QAM)and 32QAM.The proposed method can greatly improve the CO-OFDM system's ability to tolerate the wider laser linewidth.This method,compared to the linear interpolation sub-symbol common phase error compensation(LI-SCPEC)and Lagrange interpolation and extended Kalman filter(LRI-EKF)algorithms,has superior suppression effect.展开更多
In is paper, a necessary and sufficient condition of regularity of (0,2)_interpolation on the zeros of the Lascenov Polynomials R (α,β) n(x)(α,β>-1) in a manageable form is estabished. Meanwhile, the exp...In is paper, a necessary and sufficient condition of regularity of (0,2)_interpolation on the zeros of the Lascenov Polynomials R (α,β) n(x)(α,β>-1) in a manageable form is estabished. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given.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.
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
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.
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.
L' convergence of Hermite-Fejer interpolation and quasi-Hermite-Fejer interpolation based upon ze- ros of general orthogonal polynomials is investigated. This paper 'almost' characterizes such convergence ...L' convergence of Hermite-Fejer interpolation and quasi-Hermite-Fejer interpolation based upon ze- ros of general orthogonal polynomials is investigated. This paper 'almost' characterizes such convergence for all continuous functions.展开更多
For the nonpositive Hermite-Fejér interpolation based on the Laguerre abscissas, a pointwise two-sided estimate of the degree of approximation in the aleatoric interval [0, A] is first established.
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
In this paper we introduce a new kind of the mixed Hermite--Fejér interpolation with boundary condi- tions and obtain the mean approximation order.Our results include a new theorem of Varma and Prasad.Be- sides,w...In this paper we introduce a new kind of the mixed Hermite--Fejér interpolation with boundary condi- tions and obtain the mean approximation order.Our results include a new theorem of Varma and Prasad.Be- sides,we also get some other results about the mean approximation.展开更多
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
Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and repr...Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and representative numerical model of projectile flight requires a relatively good approximation of the aerodynamics.The aerodynamic coefficients of the projectile model should be described as a series of piecewise polynomial functions of the Mach number that ideally meet the following conditions:they are continuous,differentiable at least once,and have a relatively low degree.The paper provides the steps needed to generate such piecewise polynomial functions using readily available tools,and then compares Piecewise Cubic Hermite Interpolating Polynomial(PCHIP),cubic splines,and piecewise linear functions,and their variant,as potential curve fitting methods to approximate the aerodynamics of a generic small arms projectile.A key contribution of the paper is the application of PCHIP to the approximation of projectile aerodynamics,and its evaluation against a set of criteria.Finally,the paper provides a baseline assessment of the impact of the polynomial functions on flight trajectory predictions obtained with 6-degree-of-freedom simulations of a generic projectile.展开更多
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
We propose a k-domain spline interpolation method with constrained polynomial fit based on spectral phase in swept-source optical coherence tomography(SS-OCT).A Mach-Zehnder interferometer(MZI)unit is connected to.the...We propose a k-domain spline interpolation method with constrained polynomial fit based on spectral phase in swept-source optical coherence tomography(SS-OCT).A Mach-Zehnder interferometer(MZI)unit is connected to.the swept-source of the SS-OCT system to generate calibration signal in sync with the fetching of interference spectra.The spectral phase of the calibration signal is extracted by Hilbert transformation.The fitted phase-time relationship is obtained by polynomial fitting with the constraint of passing through the central spectral phase.The fitting curve is then adopted for k-domain uniform interpolation based on evenly spaced phase.In comparison with conventional k-domain spline interpolation,the proposed method leads to improved axial resolution and peak response of the axial point spread function(PSF)of the SS-OCT system.Enhanced performance resulting from the proposed method is further verified by OCT imaging of a home-constructed microspheres-agar sample and a fresh lemon.Besides SS-OCT,the proposed method is believed to be applicable to spectral domain OCT as well.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
Based on compressive sensing and fractional discrete cosine transform(DCT)via polynomial interpolation(PI-FrDCT),an image encryption algorithm is proposed,in which the compression and encryption of an image are accomp...Based on compressive sensing and fractional discrete cosine transform(DCT)via polynomial interpolation(PI-FrDCT),an image encryption algorithm is proposed,in which the compression and encryption of an image are accomplished simultaneously.It can keep information secret more effectively with low data transmission.Three-dimensional piecewise and nonlinear chaotic maps are employed to obtain a generating sequence and the exclusive OR(XOR)matrix,which greatly enlarge the key space of the encryption system.Unlike many other fractional transforms,the output of PI-FrDCT is real,which facilitates the storage,transmission and display of the encrypted image.Due to the introduction of a plain-image-dependent disturbance factor,the initial values and system parameters of the encryption scheme are determined by cipher keys and plain-image.Thus,the proposed encryption scheme is very sensitive to the plain-image,which makes the encryption system more secure.Experimental results demonstrate the validity and the reliability of the proposed encryption algorithm.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
文摘Suppose that the outer mapping function of domain D has its second continuous derivatives. In this paper, the order proximation by (0,1,…,q) Hermite-Fejer interpolating polynomials at nearly Fejer's points of function of class A(D) are presented. Moreover in general the order of approximation is sharp.
基金This work is supported by the Doctor Foundation (No:02.T20102-06) and the Post Doctor Foundation of Ningbo University.
文摘The 'o' saturation theorem and the degree of Lwp, approximation by (0 - q' - q) type Hermite-Fejer interpolating polynomials for mean convergence are obtained.
基金supported by the National Natural Science Foundation of China(Nos.U21A20447 and 61971079)。
文摘A novel suppression method of the phase noise is proposed to reduce the negative impacts of phase noise in coherent optical orthogonal frequency division multiplexing(CO-OFDM)systems.The method integrates the sub-symbol second-order polynomial interpolation(SSPI)with cubature Kalman filter(CKF)to improve the precision and effectiveness of the data processing through using a three-stage processing approach of phase noise.First of all,the phase noise values in OFDM symbols are calculated by using pilot symbols.Then,second-order Newton interpolation(SNI)is used in second-order interpolation to acquire precise noise estimation.Afterwards,every OFDM symbol is partitioned into several sub-symbols,and second-order polynomial interpolation(SPI)is utilized in the time domain to enhance suppression accuracy and time resolution.Ultimately,CKF is employed to suppress the residual phase noise.The simulation results show that this method significantly suppresses the impact of the phase noise on the system,and the error floors can be decreased at the condition of 16 quadrature amplitude modulation(16QAM)and 32QAM.The proposed method can greatly improve the CO-OFDM system's ability to tolerate the wider laser linewidth.This method,compared to the linear interpolation sub-symbol common phase error compensation(LI-SCPEC)and Lagrange interpolation and extended Kalman filter(LRI-EKF)algorithms,has superior suppression effect.
文摘In is paper, a necessary and sufficient condition of regularity of (0,2)_interpolation on the zeros of the Lascenov Polynomials R (α,β) n(x)(α,β>-1) in a manageable form is estabished. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given.
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.
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.
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Supported by the National Natural Science Foundation of China.
文摘L' convergence of Hermite-Fejer interpolation and quasi-Hermite-Fejer interpolation based upon ze- ros of general orthogonal polynomials is investigated. This paper 'almost' characterizes such convergence for all continuous functions.
基金Supported by Science and Research Fund Item of Education Department of Zhejiang Province(20050408).
文摘For the nonpositive Hermite-Fejér interpolation based on the Laguerre abscissas, a pointwise two-sided estimate of the degree of approximation in the aleatoric interval [0, A] is first established.
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In this paper we introduce a new kind of the mixed Hermite--Fejér interpolation with boundary condi- tions and obtain the mean approximation order.Our results include a new theorem of Varma and Prasad.Be- sides,we also get some other results about the mean approximation.
文摘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.
文摘Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and representative numerical model of projectile flight requires a relatively good approximation of the aerodynamics.The aerodynamic coefficients of the projectile model should be described as a series of piecewise polynomial functions of the Mach number that ideally meet the following conditions:they are continuous,differentiable at least once,and have a relatively low degree.The paper provides the steps needed to generate such piecewise polynomial functions using readily available tools,and then compares Piecewise Cubic Hermite Interpolating Polynomial(PCHIP),cubic splines,and piecewise linear functions,and their variant,as potential curve fitting methods to approximate the aerodynamics of a generic small arms projectile.A key contribution of the paper is the application of PCHIP to the approximation of projectile aerodynamics,and its evaluation against a set of criteria.Finally,the paper provides a baseline assessment of the impact of the polynomial functions on flight trajectory predictions obtained with 6-degree-of-freedom simulations of a generic projectile.
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.
基金The authors acknowledge funding from National Key Research and Development Program of China(2017FA0700501)National Natural Science Foundation of China(62035011,11974310,31927801,61905214)+1 种基金Natural Science Foundation of Zhejiang Province(LR20F050001)Fundamental Research Funds for the Central Universities.
文摘We propose a k-domain spline interpolation method with constrained polynomial fit based on spectral phase in swept-source optical coherence tomography(SS-OCT).A Mach-Zehnder interferometer(MZI)unit is connected to.the swept-source of the SS-OCT system to generate calibration signal in sync with the fetching of interference spectra.The spectral phase of the calibration signal is extracted by Hilbert transformation.The fitted phase-time relationship is obtained by polynomial fitting with the constraint of passing through the central spectral phase.The fitting curve is then adopted for k-domain uniform interpolation based on evenly spaced phase.In comparison with conventional k-domain spline interpolation,the proposed method leads to improved axial resolution and peak response of the axial point spread function(PSF)of the SS-OCT system.Enhanced performance resulting from the proposed method is further verified by OCT imaging of a home-constructed microspheres-agar sample and a fresh lemon.Besides SS-OCT,the proposed method is believed to be applicable to spectral domain OCT as well.
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supported by National Natural Science Foundation of China(Nos.61662047 and 61462061).
文摘Based on compressive sensing and fractional discrete cosine transform(DCT)via polynomial interpolation(PI-FrDCT),an image encryption algorithm is proposed,in which the compression and encryption of an image are accomplished simultaneously.It can keep information secret more effectively with low data transmission.Three-dimensional piecewise and nonlinear chaotic maps are employed to obtain a generating sequence and the exclusive OR(XOR)matrix,which greatly enlarge the key space of the encryption system.Unlike many other fractional transforms,the output of PI-FrDCT is real,which facilitates the storage,transmission and display of the encrypted image.Due to the introduction of a plain-image-dependent disturbance factor,the initial values and system parameters of the encryption scheme are determined by cipher keys and plain-image.Thus,the proposed encryption scheme is very sensitive to the plain-image,which makes the encryption system more secure.Experimental results demonstrate the validity and the reliability of the proposed encryption algorithm.
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.
