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

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.

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.

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.

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.

A family of quasi-cubic blended splines and applications

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.

Nearly Comonotone Approximation of Periodic Functions

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.

A trigonometric interval method for dynamic response analysis of uncertain nonlinear systems

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.

Average Error Bounds of Trigonometric Approximation on Periodic Wiener Spaces

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.

Best constants of harmonic approximation of some convolution classes associated with Laplace operator in d-variables

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.

Sharp Convergence of Nonlinear Functionals of a Class of Gaussian Random Fields

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.

ωB-splines

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.