Uniform trigonometric polynomial B-spline curves

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.

INEQUALITIES FOR TRIGONOMETRIC POLYNOMIALS

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

The Best m-Term One-Sided Approximation of Besov Classes by the Trigonometric Polynomials

In this paper, we continue studying the so called best m-term one-sided approximation and Greedy-liked one-sided ap- proximation by the trigonometric polynomials. The asymptotic estimations of the best m-terms one-sided approximation by the trigonometric polynomials on some classes of Besov spaces in the metricLp(Td(1≤p≤∞ are given.

On a New Family of Trigonometric Summation Polynomials of Bernstein Type

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.

An efficient cubic trigonometric B-spline collocation scheme for the time-fractional telegraph equation

In this paper,a proficient numerical technique for the time-fractional telegraph equation(TFTE)is proposed.The chief aim of this paper is to utilize a relatively new type of B-spline called the cubic trigonometric B-spline for the proposed scheme.This technique is based on finite difference formulation for the Caputo time-fractional derivative and cubic trigonometric B-splines based technique for the derivatives in space.A stability analysis of the scheme is presented to confirm that the errors do not amplify.A convergence analysis is also presented.Computational experiments are carried out in addition to verify the theoretical analysis.Numerical results are contrasted with a few present techniques and it is concluded that the presented scheme is progressively right and more compelling.

The Cubic Trigonometric Automatic Interpolation Spline

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.

ORTHONORMAL POLYNOMIAL BASSES AND WAVELET-PACKETS

For any fixed ε > 0, an explicit construction of anorthonormal trigonometric polynomial basis {Tk}∞k=1 inL2 [0,1) with degTk≤ 0.5(1 +ε)k is presented. Thus weimprove the results obtained by D. Offin and K. Oskolkov in [4] and by Al. A. Privalov in [6]. and practically solve the open problemasked in [4], [8] and [9]. Moreover, as in [4], Fourier sums with respectto this polynomial basis are projectors onto subspaces of trigonometricpolynomials of high degree, which implies almost best approximation- properties.

ASYMPTOTIC BEHAVIOR OF THE ECKHOFF METHOD FOR CONVERGENCE ACCELERATION OF TRIGONOMETRIC INTERPOLATION

Convergence acceleration of the classical trigonometric interpolation by the Eckhoff method is considered, where the exact values of the ＂jumps＂ are approximated by solution of a system of linear equations. The accuracy of the ＂jump＂ approximation is explored and the corresponding asymptotic error of interpolation is derived. Numerical results validate theoretical estimates.

Generalized Exponential Euler Polynomials and Exponential Splines

Here presented is constructive generalization of exponential Euler polynomial and exponential splines based on the interrelationship between the set of concepts of Eulerian polynomials, Eulerian numbers, and Eulerian fractions and the set of concepts related to spline functions. The applications of generalized exponential Euler polynomials in series transformations and expansions are also given.

Triangular domain extension of linear Bernstein-like trigonometric polynomial basis

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.

Asymptotics of the variance of the number of real roots of random trigonometric polynomials

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〈 ∞.

Efficiently Counting Affine Roots of Mixed Trigonometric Polynomial Systems

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.

A Fourier Companion Matrix (Multiplication Matrix) with Real-Valued Elements: Finding the Roots of a Trigonometric Polynomial by Matrix Eigensolving

We show that the zeros of a trigonometric polynomial of degree N with the usual(2N+1)terms can be calculated by computing the eigenvalues of a matrix of dimension 2N with real-valued elements M_(jk).This matrix M is a multiplication matrix in the sense that,after first defining a vector φwhose elements are the first 2N basis functions,Mφ=2cos(t)φ.This relationship is the eigenproblem;the zeros tk are the arccosine function of λ_(k)/2 where theλk are the eigenvalues of M.We dub this the“Fourier Division Companion Matrix”,or FDCM for short,because it is derived using trigonometric polynomial division.We show through examples that the algorithm computes both real and complex-valued roots,even double roots,to near machine precision accuracy.

Knot Insertion Algorithms for ECT B-spline Curves

Knot insertion algorithm is one of the most important technologies of B-spline method. By inserting a knot the local prop- erties of B-spline curve and the control flexibility of its shape can be fiu＇ther improved, also the segmentation of the curve can be rea- lized. ECT spline curve is drew by the multi-knots spline curve with associated matrix in ECT spline space; Muehlbach G and Tang Y and many others have deduced the existence and uniqueness of the ECT spline function and developed many of its important properties .This paper mainly focuses on the knot insertion algorithm of ECT B-spline curve.It is the widest popularization of B-spline Behm algorithm and theory. Inspired by the Behm algorithm, in the ECT spline space, structure of generalized P61ya poly- nomials and generalized de Boor Fix dual functional, expressing new control points which are inserted after the knot by linear com- bination of original control vertex the single knot, and there are two cases, one is the single knot, the other is the double knot. Then finally comes the insertion algorithm of ECT spline curve knot. By application of the knot insertion algorithm, this paper also gives out the knot insertion algorithm of four order geometric continuous piecewise polynomial B-spline and algebraic trigonometric spline B-spline, which is consistent with previous results.

结构随机振动时域响应统计特征分析的多项式维数分解法

针对结构在随机激励作用下的动力响应统计特征分析问题,提出了结构平稳和非平稳随机振动时域响应分析的多项式维数分解PDD(polynomial dimensional decomposition)法,高效地实现了结构随机振动响应统计矩和概率密度的计算。首先,采用三角级数叠加法模拟随机激励,将其中的随机相位作为结构系统的随机输入,并将结构随机振动时域响应视为关于随机相位的函数。其次,将结构响应函数采用成员函数(Component function)进行维数分解,并对成员函数进行Fourier多项式展开,从而构造出结构响应预测的PDD展开模型。最后,为了解决构造PDD展开模型时面临的高维积分问题,采用降维积分方法降低积分维度,显著提高了计算效率。在数值算例中,进行了单自由度系统和20层框架结构的随机振动时域响应分析,并将本文方法的计算结果与Monte Carlo模拟结果进行对比,验证了建立方法的精确性和高效性。

两种IGS精密星历插值方法的比较分析

考虑到GPS卫星的设计轨道具有明显的周期性,用三角函数多项式方法进行精密星历插值,并与经典的拉格朗日插值方法进行了比较。结果表明,无论是内插还是外推,三角函数多项式插值都比传统的朗格朗日插值精度高;并在分析精密星历误差对定位影响的基础上,指出利用三角函数多项式方法进行精密星历插值时,不需要将前一天和后一天的数据拼接在一起处理就能满足大部分定位要求。

带形状参数的二次三角Bézier曲线

给出了二次三角多项式Bézier曲线,基函数由一组带形状参数的二次三角多项式组成。由四个控制顶点生成的曲线具有与三次Bézier曲线类似的性质,但具有比三次Bézier曲线更好的逼近性。形状参数有明确几何意义:参数越大,曲线越逼近控制多边形。曲线可精确表示椭圆弧,还给出了两段三角多项式曲线的G2和C3连续的拼接条件。

曲率连续的三角B样条曲线与曲面

给出了一种二次参数三角样条曲线 ,基函数由一组特殊的二次三角多项式组成 ;曲线的每一段由三个控制顶点生成 ,不仅具有二次均匀B样条曲线的端点性质 ,而且具有更好的逼近性、整体达到曲率连续。该曲线 (面 )可用于曲线曲面的造型。

带参的三次三角多项式样条曲线

给出了带有参数λ的三次三角多项式样条曲线。与二次B样条曲线类似,曲线的每一段由相继的三个控制顶点生成;对于等距节点,在一般情形下,曲线达到了C1连续;而当λ=1时,曲线达到了C3连续。λ有明确的几何意义,λ越大,曲线越逼近控制多边形。实例表明,所给曲线为曲线/曲面的设计提供了一种有效的方法。

多形状参数的三次非均匀三角多项式曲线

对于非均匀节点向量给出了一类带多个形状参数的三次三角多项式曲线,这类曲线具有三次多项式B样条的许多重要性质:对非重节点为C2-连续,对均匀节点则为C3-连续,能直接表示椭圆.根据形状参数的各种不同取值,人们既能整体、又能局部地调控这类曲线的形状.此外,还讨论了多形状参数的三角Bézier曲线的情况.