## Trigonometric

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### Xuli Han - One of the best experts on this subject based on the ideXlab platform.

• ##### A Practical Method for Generating Trigonometric Polynomial Surfaces over Triangular Domains
Mediterranean Journal of Mathematics, 2015
Co-Authors: Xuli Han, Yuanpeng Zhu
Abstract:

A class of Trigonometric polynomial basis functions over triangular domain with three shape parameters is constructed in this paper. Based on these new basis functions, a kind of Trigonometric polynomial patch over triangular domain, which can be used to construct some surfaces whose boundaries are arcs of ellipse or parabola, is proposed. Without changing the control points, the shape of the Trigonometric polynomial patch can be adjusted flexibly in a foreseeable way using the shape parameters. For computing the proposed Trigonometric polynomial patch stably and efficiently, a practical de Casteljau-type algorithm is developed. Moveover, the conditions for G 1 continuous smooth joining two Trigonometric polynomial patches are deduced.

• ##### Piecewise Trigonometric Hermite interpolation
Applied Mathematics and Computation, 2015
Co-Authors: Xuli Han
Abstract:

Based on the symmetric, nonnegative and normalized basis of the Trigonometric polynomial space, the piecewise Trigonometric Hermite interpolation methods are presented. The C n - 1 and Cn continuous piecewise Trigonometric Hermite interpolants of degree n are constructed and the interpolation methods are local. The integral and the differential representations of the errors of the Trigonometric Hermite interpolants are given. Several examples are supplied to support the practical value of the given interpolation methods.

• ##### The Trigonometric polynomial like Bernstein polynomial.
TheScientificWorldJournal, 2014
Co-Authors: Xuli Han
Abstract:

A symmetric basis of Trigonometric polynomial space is presented. Based on the basis, symmetric Trigonometric polynomial approximants like Bernstein polynomials are constructed. Two kinds of nodes are given to show that the Trigonometric polynomial sequence is uniformly convergent. The convergence of the derivative of the Trigonometric polynomials is shown. Trigonometric quasi-interpolants of reproducing one degree of Trigonometric polynomials are constructed. Some interesting properties of the Trigonometric polynomials are given.

• ##### Curve construction based on five Trigonometric blending functions
BIT Numerical Mathematics, 2012
Co-Authors: Xuli Han, Yuanpeng Zhu
Abstract:

Five new Trigonometric blending functions with two exponential shape parameters are given in this paper. Based on these blending functions, Trigonometric Bezier curves analogous to the quartic Bezier curves, with two exponential shape parameters, are presented. The ellipses and parabolas can be represented exactly by using the Trigonometric Bezier curves. Based on the blending functions, Trigonometric B-spline curves with three local shape parameters and a global shape parameter are also constructed. The obtained spline curves can be C 2∩FC 2k+3 (k∈ℤ+) continuous by fixing some values of the shape parameters. Without solving a linear system, the spline curves can be also used to interpolate sets of points with C 2 continuity partly or entirely.

• ##### Quartic Trigonometric B ezier Curves and Shape Preserving Interpolation Curves
2012
Co-Authors: Yuanpeng Zhu, Xuli Han, Jing Han
Abstract:

In this paper, we firstly construct quartic Trigonometric polynomial blending functions which possess the properties analogous to the quintic Bernstein basis functions, and then we give a kind of quartic Trigonometric Bezier curves. Basing on the blending functions and using the geometrical information of four adjacent interpolation points, we introduce shape parameters which are used to generate the quartic Trigonometric Bezier control points. We give the shape preserving conditions of the shape parameters. Thereby, shape preserving quartic Trigonometric interpolation spline curves with shape parameters are constructed. For any shape parameters satisfying the shape preserving conditions, the obtained spline curves are F 3 continuous. There is no need to solve a linear system, and the changes of a local shape parameter will only affect two curve segments.

### Hanxuli - One of the best experts on this subject based on the ideXlab platform.

• ##### Cubic Trigonometric polynomial curves with a shape parameter
Computer Aided Geometric Design, 2004
Co-Authors: Hanxuli
Abstract:

Cubic Trigonometric polynomial curves with a shape parameter are presented in this paper. The Trigonometric polynomial curves are C2 continuous and G3 continuous with a non-uniform knot vector. Wit...

• ##### Piecewise quadratic Trigonometric polynomial curves
Mathematics of Computation, 2003
Co-Authors: Hanxuli
Abstract:

Analogous to the quadratic B-spline curve, a piecewise quadratic Trigonometric polynomial curve is presented in this paper. The quadratic Trigonometric polynomial curve has C2 continuity, while the...

• ##### Quadratic Trigonometric polynomial curves with a shape parameter
Computer Aided Geometric Design, 2002
Co-Authors: Hanxuli
Abstract:

Quadratic Trigonometric polynomial curves with a shape parameter are presented in this paper. Analogous to the quadratic B-spline curves, the Trigonometric polynomial curves are constructed with th...

### Yuanpeng Zhu - One of the best experts on this subject based on the ideXlab platform.

• ##### Quasi-quintic Trigonometric Bézier curves with two shape parameters
Computational and Applied Mathematics, 2019
Co-Authors: Xuewen Tan, Yuanpeng Zhu
Abstract:

In this work, we propose a family of six new quasi-quintic Trigonometric blending functions with two shape parameters. Based on these blending functions, a class of quasi-quintic Trigonometric Bezier curve is proposed, which has some properties analogous to the classical quintic Bezier curves. For the same control points, the resulting quasi-quintic Trigonometric Bezier curves can be closer to the control polygon than the classical quintic Bezier curves. The shape of the quasi-quintic Trigonometric Bezier curves can be flexibly adjusted by altering the values of the two shape parameters without changing their control points. Under the $${C^2}$$ smooth connection conditions, the resulting composite quasi-quintic Trigonometric Bezier curves can automatically reach $${C^2} \cap F{C^3}$$ continuity.

• ##### A Practical Method for Generating Trigonometric Polynomial Surfaces over Triangular Domains
Mediterranean Journal of Mathematics, 2015
Co-Authors: Xuli Han, Yuanpeng Zhu
Abstract:

A class of Trigonometric polynomial basis functions over triangular domain with three shape parameters is constructed in this paper. Based on these new basis functions, a kind of Trigonometric polynomial patch over triangular domain, which can be used to construct some surfaces whose boundaries are arcs of ellipse or parabola, is proposed. Without changing the control points, the shape of the Trigonometric polynomial patch can be adjusted flexibly in a foreseeable way using the shape parameters. For computing the proposed Trigonometric polynomial patch stably and efficiently, a practical de Casteljau-type algorithm is developed. Moveover, the conditions for G 1 continuous smooth joining two Trigonometric polynomial patches are deduced.

• ##### Curve construction based on five Trigonometric blending functions
BIT Numerical Mathematics, 2012
Co-Authors: Xuli Han, Yuanpeng Zhu
Abstract:

Five new Trigonometric blending functions with two exponential shape parameters are given in this paper. Based on these blending functions, Trigonometric Bezier curves analogous to the quartic Bezier curves, with two exponential shape parameters, are presented. The ellipses and parabolas can be represented exactly by using the Trigonometric Bezier curves. Based on the blending functions, Trigonometric B-spline curves with three local shape parameters and a global shape parameter are also constructed. The obtained spline curves can be C 2∩FC 2k+3 (k∈ℤ+) continuous by fixing some values of the shape parameters. Without solving a linear system, the spline curves can be also used to interpolate sets of points with C 2 continuity partly or entirely.

• ##### Quartic Trigonometric B ezier Curves and Shape Preserving Interpolation Curves
2012
Co-Authors: Yuanpeng Zhu, Xuli Han, Jing Han
Abstract:

In this paper, we firstly construct quartic Trigonometric polynomial blending functions which possess the properties analogous to the quintic Bernstein basis functions, and then we give a kind of quartic Trigonometric Bezier curves. Basing on the blending functions and using the geometrical information of four adjacent interpolation points, we introduce shape parameters which are used to generate the quartic Trigonometric Bezier control points. We give the shape preserving conditions of the shape parameters. Thereby, shape preserving quartic Trigonometric interpolation spline curves with shape parameters are constructed. For any shape parameters satisfying the shape preserving conditions, the obtained spline curves are F 3 continuous. There is no need to solve a linear system, and the changes of a local shape parameter will only affect two curve segments.

### Mohsin Javed - One of the best experts on this subject based on the ideXlab platform.

• ##### Algorithms for Trigonometric polynomial and rational approximation
2016
Co-Authors: Mohsin Javed
Abstract:

This thesis presents new numerical algorithms for approximating functions by Trigonometric polynomials and Trigonometric rational functions. We begin by reviewing Trigonometric polynomial interpolation and the barycentric formula for Trigonometric polynomial interpolation in Chapter 1. Another feature of this chapter is the use of the complex plane, contour integrals and phase portraits for visualising various properties and relationships between periodic functions and their Laurent and Trigonometric series. We also derive a periodic analogue of the Hermite integral formula which enables us to analyze interpolation error using contour integrals. We have not been able to find such a formula in the literature. Chapter 2 discusses Trigonometric rational interpolation and Trigonometric linearized rational least-squares approximations. To our knowledge, this is the first attempt to numerically solve these problems. The contribution of this chapter is presented in the form of a robust algorithm for computing Trigonometric rational interpolants of prescribed numerator and denominator degrees at an arbitrary grid of interpolation points. The algorithm can also be used to compute Trigonometric linearized rational least-squares and Trigonometric polynomial least-squares approximations. Chapter 3 deals with the problem of Trigonometric minimax approximation of functions, first in a space of Trigonometric polynomials and then in a set of Trigonometric rational functions. The contribution of this chapter is presented in the form of an algorithm, which to our knowledge, is the first description of a Remez-like algorithm to numerically compute Trigonometric minimax polynomial and rational approximations. Our algorithm also uses Trigonometric barycentric interpolation and Chebyshev-eigenvalue based root finding. Chapter 4 discusses the Fourier-Padé (called Trigonometric Padé) approximation of a function. We review two existing approaches to the problem, both of which are based on rational approximations of a Laurent series. We present a numerical algorithm with examples and compute various type (m, n) Trigonometric Padé approximants.

• ##### Algorithms for Trigonometric polynomial and rational approximation
2016
Co-Authors: Mohsin Javed
Abstract:

This thesis presents new numerical algorithms for approximating functions by Trigonometric polynomials and Trigonometric rational functions. We begin by reviewing Trigonometric polynomial interpolation and the barycentric formula for Trigonometric polynomial interpolation in Chapter 1. Another feature of this chapter is the use of the complex plane, contour integrals and phase portraits for visualising various properties and relationships between periodic functions and their Laurent and Trigonometric series. We also derive a periodic analogue of the Hermite integral formula which enables us to analyze interpolation error using contour integrals. We have not been able to find such a formula in the literature. Chapter 2 discusses Trigonometric rational interpolation and Trigonometric linearized rational least-squares approximations. To our knowledge, this is the first attempt to numerically solve these problems. The contribution of this chapter is presented in the form of a robust algorithm for computing Trigonometric rational interpolants of prescribed numerator and denominator degrees at an arbitrary grid of interpolation points. The algorithm can also be used to compute Trigonometric linearized rational least-squares and Trigonometric polynomial least-squares approximations. Chapter 3 deals with the problem of Trigonometric minimax approximation of functions, first in a space of Trigonometric polynomials and then in a set of Trigonometric rational functions. The contribution of this chapter is presented in the form of an algorithm, which to our knowledge, is the first description of a Remez-like algorithm to numerically compute Trigonometric minimax polynomial and rational approximations. Our algorithm also uses Trigonometric barycentric interpolation and Chebyshev-eigenvalue based root finding. Chapter 4 discusses the Fourier-Pade (called Trigonometric Pade) approximation of a function. We review two existing approaches to the problem, both of which are based on rational approximations of a Laurent series. We present a numerical algorithm with examples and compute various type (m, n) Trigonometric Pade approximants.

### Xiang Zhang - One of the best experts on this subject based on the ideXlab platform.

• ##### The number of polynomial solutions of polynomial Riccati equations
arXiv: Classical Analysis and ODEs, 2016
Co-Authors: Armengol Gasull, Joan Torregrosa, Xiang Zhang
Abstract:

Consider real or complex polynomial Riccati differential equations $a(x) \dot y=b_0(x)+b_1(x)y+b_2(x)y^2$ with all the involved functions being polynomials of degree at most $\eta$. We prove that the maximum number of polynomial solutions is $\eta+1$ (resp. 2) when $\eta\ge 1$ (resp. $\eta=0$) and that these bounds are sharp. For real Trigonometric polynomial Riccati differential equations with all the functions being Trigonometric polynomials of degree at most $\eta\ge 1$ we prove a similar result. In this case, the maximum number of Trigonometric polynomial solutions is $2\eta$ (resp. $3$) when $\eta\ge 2$ (resp. $\eta=1$) and, again, these bounds are sharp. Although the proof of both results has the same starting point, the classical result that asserts that the cross ratio of four different solutions of a Riccati differential equation is constant, the Trigonometric case is much more involved. The main reason is that the ring of Trigonometric polynomials is not a unique factorization domain.

• ##### The number of polynomial solutions of polynomial Riccati equations
Journal of Differential Equations, 2016
Co-Authors: Armengol Gasull, Joan Torregrosa, Xiang Zhang
Abstract:

Consider real or complex polynomial Riccati differential equations a(x) y=b_0(x) b_1(x)y b_2(x)y^2 with all the involved functions being polynomials of degree at most . We prove that the maximum number of polynomial solutions is 1 (resp. 2) when 1 (resp. =0) and that these bounds are sharp. For real Trigonometric polynomial Riccati differential equations with all the functions being Trigonometric polynomials of degree at most 1 we prove a similar result. In this case, the maximum number of Trigonometric polynomial solutions is 2 (resp. 3) when 2 (resp. =1) and, again, these bounds are sharp. Although the proof of both results has the same starting point, the classical result that asserts that the cross ratio of four different solutions of a Riccati differential equation is constant, the Trigonometric case is much more involved. The main reason is that the ring of Trigonometric polynomials is not a unique factorization domain.