Let tn(x) be any real trigonometric polynomial of degreen n such that , Here we are concerned with obtaining the best possible upper estimate ofwhere q>2. In addition, we shall obtain the estimate of in terms of and
Let an, n≥ 1 be a sequence of independent standard normal random variables. Consider the randomtrigonometric polynomial Tn（θ）=∑^n_i=1 aj cos（j θ）, 0≤θ≤π and let Nn be the number of real roots of Tn（θ）in...Let an, n≥ 1 be a sequence of independent standard normal random variables. Consider the randomtrigonometric polynomial Tn（θ）=∑^n_i=1 aj cos（j θ）, 0≤θ≤π and let Nn be the number of real roots of Tn（θ）in （0, 2π）. In this paper it is proved that limn→∞ Var（Nn）/n=co,where 0 〈 co〈 ∞.展开更多
A new family of trigonometric summation polynomials, Gn,r（f; θ）, of Bernstein type is constructed. In contrast to other trigonometric summation polynomials, the convergence properties of the new polynomials are sup...A new family of trigonometric summation polynomials, Gn,r（f; θ）, of Bernstein type is constructed. In contrast to other trigonometric summation polynomials, the convergence properties of the new polynomials are superior to others. It is proved that Gn,r（f; θ） converges to arbitrary continuous functions with period 2π uniformly on （-∞ ＋∞） as n→ ∞. In particular, Gn,r（f; θ） has the best convergence order, and its saturation order is 1/n^2r＋4.展开更多
A class of cubic trigonometric interpolation spline curves with two parameters is presented in this paper. The spline curves can automatically interpolate the given data points and become C^2 interpolation curves with...A class of cubic trigonometric interpolation spline curves with two parameters is presented in this paper. The spline curves can automatically interpolate the given data points and become C^2 interpolation curves without solving equations system even if the interpolation conditions are fixed. Moreover, shape of the interpolation spline curves can be globally adjusted by the two parameters. By selecting proper values of the two parameters,the optimal interpolation spline curves can be obtained.展开更多
In many fields of science and engineering, it is needed to find all solutions of mixed trigonometric polynomial systems. Commonly, mixed trigonometric polynomial systems are transformed into polynomial systems by vari...In many fields of science and engineering, it is needed to find all solutions of mixed trigonometric polynomial systems. Commonly, mixed trigonometric polynomial systems are transformed into polynomial systems by variable substitution and adding some quadratic equations, and then solved by some numerical methods. However, transformation of a mixed trigonometric polynomial system into a polynomial system will increase the dimension of the system and hence induces extra computational work. In this paper, we consider to solve the mixed trigonometric polynomial. systems by homotopy method directly. Homotopy with the start system constructed by GBQ-algorithm is presented and homotopy theorems are proved. Preliminary numerical results show that our constructed direct homotopy method is more efficient than the existent direct homotopy methods.展开更多
A class of quasi-cubic B-spline base functions by trigonometric polynomials are established which inherit properties similar to those of cubic B-spline bases. The corresponding curves with a shape parameter a, defined...A class of quasi-cubic B-spline base functions by trigonometric polynomials are established which inherit properties similar to those of cubic B-spline bases. The corresponding curves with a shape parameter a, defined by the introduced base functions, include the B-spline curves and can approximate the B-spline curves from both sides. The curves can be adjusted easily by using the shape parameter a, where dpi（a,t） is linear with respect to da for the fixed t. With the shape parameter chosen properly, the defined curves can be used to precisely represent straight line segments, parabola segments, circular arcs and some transcendental curves, and the corresponding tensor product surfaces can also represent spherical surfaces, cylindrical surfaces and some transcendental surfaces exactly. By abandoning positive property, this paper proposes a new C^2 continuous blended interpolation spline based on piecewise trigonometric polynomials associated with a sequence of local parameters. Illustration showed that the curves and surfaces constructed by the blended spline can be adjusted easily and freely. The blended interpolation spline curves can be shape-preserving with proper local parameters since these local parameters can be considered to be the magnification ratio to the length of tangent vectors at the interpolating points. The idea is extended to produce blended spline surfaces.展开更多
Suppose that a continuous 2re-periodic function f on the real axis changes its monotonicity at points Yi In this paper, for each n _ N, a trigonometric polynomial Pn of order cn is found such that： Pn has the same ...Suppose that a continuous 2re-periodic function f on the real axis changes its monotonicity at points Yi In this paper, for each n _ N, a trigonometric polynomial Pn of order cn is found such that： Pn has the same monotonicity as f, everywhere except, perhaps, the small intervals.展开更多
The numerical computation of partial differential equations(PDEs)is highly important across numerous scientific and engineering disciplines.The accuracy and convergence of integration-based methods depend primarily on...The numerical computation of partial differential equations(PDEs)is highly important across numerous scientific and engineering disciplines.The accuracy and convergence of integration-based methods depend primarily on the ability to perform analytical integration over complex domains.Owing to the inherent challenges posed by the complexities of irregular integration domains and general integrands,this paper introduces an innovative analytical method for nonpolynomial integration over complex domains for the first time.This method is initially applied within the framework of the numerical manifold method(NMM)to address the inevitable trigonometric and exponential polynomial integrations encountered in the analysis of the Laplace equation problem.First,a comprehensive overview of the fundamentals of the NMM and the simplex integration(SI)method is provided in this paper.Subsequently,the NMM framework for solving the Laplace equation is elaborated upon,with a focus on deriving closed-form formulas for trigonometric and exponential polynomial integration.Finally,a series of rigorous numerical experiments is conducted,where the proposed method demonstrates improved accuracy and efficiency.In conclusion,this study innovatively enhances the NMM by introducing the SI method for nonpolynomial functions over complex domains,which is a promising approach for increasing accuracy and convergence across various integration-based methods.This groundbreaking achievement has not yet been reported in the publicly available literature.展开更多
This paper presents a new kind of uniform spline curve, named trigonometric polynomial B-splines, over space Ω = span{sini,cost, tk-3,tk-4, …,t, 1} of which k is an arbitrary integer larger than or equal to 3. We sh...This paper presents a new kind of uniform spline curve, named trigonometric polynomial B-splines, over space Ω = span{sini,cost, tk-3,tk-4, …,t, 1} of which k is an arbitrary integer larger than or equal to 3. We show that trigonometric polynomial B-spline curves have many similar properties to traditional B-splines. Based on the explicit representation of the curve we have also presented the subdivision formulae for this new kind of curve. Since the new spline can include both polynomial curves and trigonometric curves as special cases without rational form, it can be used as an efficient new model for geometric design in the fields of CAD/CAM.展开更多
In computer aided geometric design(CAGD),the Bernstein-Bézier system for polynomial space including the triangular domain is an important tool for modeling free form shapes.The Bernstein-like bases for other spac...In computer aided geometric design(CAGD),the Bernstein-Bézier system for polynomial space including the triangular domain is an important tool for modeling free form shapes.The Bernstein-like bases for other spaces(trigonometric polynomial,hyperbolic polynomial,or blended space) has also been studied.However,none of them was extended to the triangular domain.In this paper,we extend the linear trigonometric polynomial basis to the triangular domain and obtain a new Bernstein-like basis,which is linearly independent and satisfies positivity,partition of unity,symmetry,and boundary represen-tation.We prove some properties of the corresponding surfaces,including differentiation,subdivision,convex hull,and so forth.Some applications are shown.展开更多
This paper proposes a new non-intrusive trigonometric polynomial approximation interval method for the dynamic response analysis of nonlinear systems with uncertain-but-bounded parameters and/or initial conditions.Thi...This paper proposes a new non-intrusive trigonometric polynomial approximation interval method for the dynamic response analysis of nonlinear systems with uncertain-but-bounded parameters and/or initial conditions.This method provides tighter solution ranges compared to the existing approximation interval methods.We consider trigonometric approximation polynomials of three types:both cosine and sine functions,the sine function,and the cosine function.Thus,special interval arithmetic for trigonometric function without overestimation can be used to obtain interval results.The interval method using trigonometric approximation polynomials with a cosine functional form exhibits better performance than the existing Taylor interval method and Chebyshev interval method.Finally,two typical numerical examples with nonlinearity are applied to demonstrate the effectiveness of the proposed method.展开更多
Estimating the number of isolated roots of a polynomial system is not only a fundamental study theme in algebraic geometry but also an important subproblem of homotopy methods for solving polynomial systems. For the m...Estimating the number of isolated roots of a polynomial system is not only a fundamental study theme in algebraic geometry but also an important subproblem of homotopy methods for solving polynomial systems. For the mixed trigonometric polynomial systems, which are more general than polynomial systems and rather frequently occur in many applications, the classical B6zout number and the multihomogeneous Bezout number are the best known upper bounds on the number of isolated roots. However, for the deficient mixed trigonometric polynomial systems, these two upper bounds are far greater than the actual number of isolated roots. The BKK bound is known as the most accurate upper bound on the number of isolated roots of a polynomial system. However, the extension of the definition of the BKK bound allowing it to treat mixed trigonometric polynomial systems is very difficult due to the existence of sine and cosine functions. In this paper, two new upper bounds on the number of isolated roots of a mixed trigonometric polynomial system are defined and the corresponding efficient algorithms for calculating them are presented. Numerical tests are also given to show the accuracy of these two definitions, and numerically prove they can provide tighter upper bounds on the number of isolated roots of a mixed trigonometric polynomial system than the existing upper bounds, and also the authors compare the computational time for calculating these two upper bounds.展开更多
In this paper, we study the approximation of identity operator and the convolution inte- gral operator Bm by Fourier partial sum operators, Fejer operators, Vallee--Poussin operators, Ces^ro operators and Abel mean op...In this paper, we study the approximation of identity operator and the convolution inte- gral operator Bm by Fourier partial sum operators, Fejer operators, Vallee--Poussin operators, Ces^ro operators and Abel mean operators, respectively, on the periodic Wiener space （C1 （R）, W°） and obtaia the average error estimations.展开更多
In this paper,we consider the best EFET(entire functions of the exponential type) approximations of some convolution classes associated with Laplace operator on R d and obtain exact constants in the spaces L1(R2) and ...In this paper,we consider the best EFET(entire functions of the exponential type) approximations of some convolution classes associated with Laplace operator on R d and obtain exact constants in the spaces L1(R2) and L2(Rd).Moreover,the best constants of trigonometric approximations of their analogies on Td are also gained.展开更多
We present a self-contained proof of a uniform bound on multi-point correlations of trigonometric functions of a class of Gaussian random fields.It corresponds to a special case of the general situation considered in ...We present a self-contained proof of a uniform bound on multi-point correlations of trigonometric functions of a class of Gaussian random fields.It corresponds to a special case of the general situation considered in Hairer and Xu(large-scale limit of interface fluctuation models.ArXiv e-prints arXiv:1802.08192,2018),but with improved estimates.As a consequence,we establish convergence of a class of Gaussian fields composite with more general functions.These bounds and convergences are useful ingredients to establish weak universalities of several singular stochastic PDEs.展开更多
A new kind of spline with variable frequencies, called ωB-spline, is presented. It not only unifies B-splines, trigonometric and hyperbolic polynomial B-splines, but also produces more new types of splines, ωB-splin...A new kind of spline with variable frequencies, called ωB-spline, is presented. It not only unifies B-splines, trigonometric and hyperbolic polynomial B-splines, but also produces more new types of splines, ωB-spline bases are defined in the space spanned by {coso） t, sino）t, ], t, ..., t^n, ...} with the sequence of frequencies m where n is an arbitrary nonnegative integer, ωB-splines persist all desirable properties of B-splines. Furthermore, they have some special properties advantageous for modeling free form curves and surfaces.展开更多
文摘Let tn(x) be any real trigonometric polynomial of degreen n such that , Here we are concerned with obtaining the best possible upper estimate ofwhere q>2. In addition, we shall obtain the estimate of in terms of and
基金supported by National Natural Science Foundation of China (GrantNos. 10671176 and 11071213)Zhejiang Provincial Natural Science Foundation of China (Grant No. R6090034)+1 种基金Doctoral Programs Foundation of Ministry of Education of China (Grant No. J20110031)Competitive Earmarked Research Grant of Research Grants Council (Grant No. 602608)
文摘Let an, n≥ 1 be a sequence of independent standard normal random variables. Consider the randomtrigonometric polynomial Tn（θ）=∑^n_i=1 aj cos（j θ）, 0≤θ≤π and let Nn be the number of real roots of Tn（θ）in （0, 2π）. In this paper it is proved that limn→∞ Var（Nn）/n=co,where 0 〈 co〈 ∞.
文摘A new family of trigonometric summation polynomials, Gn,r（f; θ）, of Bernstein type is constructed. In contrast to other trigonometric summation polynomials, the convergence properties of the new polynomials are superior to others. It is proved that Gn,r（f; θ） converges to arbitrary continuous functions with period 2π uniformly on （-∞ ＋∞） as n→ ∞. In particular, Gn,r（f; θ） has the best convergence order, and its saturation order is 1/n^2r＋4.
基金supported by the National Natural Science Foundation of China(11171181)the Scientific Research Fund of Hunan Provincial Education Department of China(14B099)
文摘A class of cubic trigonometric interpolation spline curves with two parameters is presented in this paper. The spline curves can automatically interpolate the given data points and become C^2 interpolation curves without solving equations system even if the interpolation conditions are fixed. Moreover, shape of the interpolation spline curves can be globally adjusted by the two parameters. By selecting proper values of the two parameters,the optimal interpolation spline curves can be obtained.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1110106711171051)the Fundamental Research Funds for the Central Universities(Grant No.DUT16LK04)
文摘In many fields of science and engineering, it is needed to find all solutions of mixed trigonometric polynomial systems. Commonly, mixed trigonometric polynomial systems are transformed into polynomial systems by variable substitution and adding some quadratic equations, and then solved by some numerical methods. However, transformation of a mixed trigonometric polynomial system into a polynomial system will increase the dimension of the system and hence induces extra computational work. In this paper, we consider to solve the mixed trigonometric polynomial. systems by homotopy method directly. Homotopy with the start system constructed by GBQ-algorithm is presented and homotopy theorems are proved. Preliminary numerical results show that our constructed direct homotopy method is more efficient than the existent direct homotopy methods.
基金Project supported by the National Natural Science Foundation of China (Nos. 10171026 and 60473114), the Research Funds forYoung Innovation Group, Education Department of Anhui Prov-ince (No. 2005TD03) and the Natural Science Foundation of An-hui Provincial Education Department (No. 2006KJ252B), China
文摘A class of quasi-cubic B-spline base functions by trigonometric polynomials are established which inherit properties similar to those of cubic B-spline bases. The corresponding curves with a shape parameter a, defined by the introduced base functions, include the B-spline curves and can approximate the B-spline curves from both sides. The curves can be adjusted easily by using the shape parameter a, where dpi（a,t） is linear with respect to da for the fixed t. With the shape parameter chosen properly, the defined curves can be used to precisely represent straight line segments, parabola segments, circular arcs and some transcendental curves, and the corresponding tensor product surfaces can also represent spherical surfaces, cylindrical surfaces and some transcendental surfaces exactly. By abandoning positive property, this paper proposes a new C^2 continuous blended interpolation spline based on piecewise trigonometric polynomials associated with a sequence of local parameters. Illustration showed that the curves and surfaces constructed by the blended spline can be adjusted easily and freely. The blended interpolation spline curves can be shape-preserving with proper local parameters since these local parameters can be considered to be the magnification ratio to the length of tangent vectors at the interpolating points. The idea is extended to produce blended spline surfaces.
文摘Suppose that a continuous 2re-periodic function f on the real axis changes its monotonicity at points Yi In this paper, for each n _ N, a trigonometric polynomial Pn of order cn is found such that： Pn has the same monotonicity as f, everywhere except, perhaps, the small intervals.
基金supported by the Natural Science Foundation of Shanghai(Grant No.21ZR1468500)the Fundamental Research Funds for the Central Universities(Grant No.22120240299)。
文摘The numerical computation of partial differential equations(PDEs)is highly important across numerous scientific and engineering disciplines.The accuracy and convergence of integration-based methods depend primarily on the ability to perform analytical integration over complex domains.Owing to the inherent challenges posed by the complexities of irregular integration domains and general integrands,this paper introduces an innovative analytical method for nonpolynomial integration over complex domains for the first time.This method is initially applied within the framework of the numerical manifold method(NMM)to address the inevitable trigonometric and exponential polynomial integrations encountered in the analysis of the Laplace equation problem.First,a comprehensive overview of the fundamentals of the NMM and the simplex integration(SI)method is provided in this paper.Subsequently,the NMM framework for solving the Laplace equation is elaborated upon,with a focus on deriving closed-form formulas for trigonometric and exponential polynomial integration.Finally,a series of rigorous numerical experiments is conducted,where the proposed method demonstrates improved accuracy and efficiency.In conclusion,this study innovatively enhances the NMM by introducing the SI method for nonpolynomial functions over complex domains,which is a promising approach for increasing accuracy and convergence across various integration-based methods.This groundbreaking achievement has not yet been reported in the publicly available literature.
基金This work was partially supported by the National Natural Science Foundation of China (Grant No. 19971079) and Foundation of State Key Basic Research 973 Development Programming Item (Grant No. G1998030600).
文摘This paper presents a new kind of uniform spline curve, named trigonometric polynomial B-splines, over space Ω = span{sini,cost, tk-3,tk-4, …,t, 1} of which k is an arbitrary integer larger than or equal to 3. We show that trigonometric polynomial B-spline curves have many similar properties to traditional B-splines. Based on the explicit representation of the curve we have also presented the subdivision formulae for this new kind of curve. Since the new spline can include both polynomial curves and trigonometric curves as special cases without rational form, it can be used as an efficient new model for geometric design in the fields of CAD/CAM.
基金supported by the National Natural Science Foundation of China (Nos.60773179,60933008,and 60970079)the National Basic Research Program (973) of China (No.2004CB318000)the China Hungary Joint Project (No.CHN21/2006)
文摘In computer aided geometric design(CAGD),the Bernstein-Bézier system for polynomial space including the triangular domain is an important tool for modeling free form shapes.The Bernstein-like bases for other spaces(trigonometric polynomial,hyperbolic polynomial,or blended space) has also been studied.However,none of them was extended to the triangular domain.In this paper,we extend the linear trigonometric polynomial basis to the triangular domain and obtain a new Bernstein-like basis,which is linearly independent and satisfies positivity,partition of unity,symmetry,and boundary represen-tation.We prove some properties of the corresponding surfaces,including differentiation,subdivision,convex hull,and so forth.Some applications are shown.
文摘This paper proposes a new non-intrusive trigonometric polynomial approximation interval method for the dynamic response analysis of nonlinear systems with uncertain-but-bounded parameters and/or initial conditions.This method provides tighter solution ranges compared to the existing approximation interval methods.We consider trigonometric approximation polynomials of three types:both cosine and sine functions,the sine function,and the cosine function.Thus,special interval arithmetic for trigonometric function without overestimation can be used to obtain interval results.The interval method using trigonometric approximation polynomials with a cosine functional form exhibits better performance than the existing Taylor interval method and Chebyshev interval method.Finally,two typical numerical examples with nonlinearity are applied to demonstrate the effectiveness of the proposed method.
基金supported in part by the National Natural Science Foundation of China under Grant Nos.11101067 and 11571061Major Research Plan of the National Natural Science Foundation of China under Grant No.91230103the Fundamental Research Funds for the Central Universities
文摘Estimating the number of isolated roots of a polynomial system is not only a fundamental study theme in algebraic geometry but also an important subproblem of homotopy methods for solving polynomial systems. For the mixed trigonometric polynomial systems, which are more general than polynomial systems and rather frequently occur in many applications, the classical B6zout number and the multihomogeneous Bezout number are the best known upper bounds on the number of isolated roots. However, for the deficient mixed trigonometric polynomial systems, these two upper bounds are far greater than the actual number of isolated roots. The BKK bound is known as the most accurate upper bound on the number of isolated roots of a polynomial system. However, the extension of the definition of the BKK bound allowing it to treat mixed trigonometric polynomial systems is very difficult due to the existence of sine and cosine functions. In this paper, two new upper bounds on the number of isolated roots of a mixed trigonometric polynomial system are defined and the corresponding efficient algorithms for calculating them are presented. Numerical tests are also given to show the accuracy of these two definitions, and numerically prove they can provide tighter upper bounds on the number of isolated roots of a mixed trigonometric polynomial system than the existing upper bounds, and also the authors compare the computational time for calculating these two upper bounds.
文摘In this paper, we study the approximation of identity operator and the convolution inte- gral operator Bm by Fourier partial sum operators, Fejer operators, Vallee--Poussin operators, Ces^ro operators and Abel mean operators, respectively, on the periodic Wiener space （C1 （R）, W°） and obtaia the average error estimations.
基金supported partly by National Natural Science Foundation of China(GrantNo.11071019)Research Fund for the Doctoral Program of Higher Education and Beijing Natural Science Foundation(Grant No.1102011)
文摘In this paper,we consider the best EFET(entire functions of the exponential type) approximations of some convolution classes associated with Laplace operator on R d and obtain exact constants in the spaces L1(R2) and L2(Rd).Moreover,the best constants of trigonometric approximations of their analogies on Td are also gained.
基金the support from the Engineering and Physical Sciences Research Council through the fellowship EP/N021568/1.
文摘We present a self-contained proof of a uniform bound on multi-point correlations of trigonometric functions of a class of Gaussian random fields.It corresponds to a special case of the general situation considered in Hairer and Xu(large-scale limit of interface fluctuation models.ArXiv e-prints arXiv:1802.08192,2018),but with improved estimates.As a consequence,we establish convergence of a class of Gaussian fields composite with more general functions.These bounds and convergences are useful ingredients to establish weak universalities of several singular stochastic PDEs.
基金the National Natural Science Foundation of China(Grant No.60773179)Foundation of State Key Basic Research 973 Development Programming Item of China(Grant No.G2004CB318000)
文摘A new kind of spline with variable frequencies, called ωB-spline, is presented. It not only unifies B-splines, trigonometric and hyperbolic polynomial B-splines, but also produces more new types of splines, ωB-spline bases are defined in the space spanned by {coso） t, sino）t, ], t, ..., t^n, ...} with the sequence of frequencies m where n is an arbitrary nonnegative integer, ωB-splines persist all desirable properties of B-splines. Furthermore, they have some special properties advantageous for modeling free form curves and surfaces.
