Trigonometric Splines Research Articles

On uniform Lebesgue constants of third-order local trigonometric splines

On uniform Lebesgue constants of third-order local trigonometric splines

A unified approach to non-polynomial B-spline curves based on a novel variant of the polar form

We develop a general, unified theory of splines for a wide collection of spline spaces, including trigonometric splines, hyperbolic splines, and special Muntz spaces of splines by invoking a novel variant of the homogeneous polar form where we alter the diagonal property. Using this polar form, we derive de Boor type recursive algorithms for evaluation and differentiation. We also show that standard knot insertion procedures such as Boehm's algorithm and the Oslo algorithm readily extend to these general spline spaces. In addition, for these spaces we construct compactly supported B-spline basis functions with simple two term recurrences for evaluation and differentiation, and we show that these B-spline basis functions form a partition of unity, have curvilinear precision, and satisfy a dual functional property and a Marsden identity.

Gabor frames with trigonometric spline dual windows

A dual Gabor window pair has two functions that can reconstruct any function in [Formula: see text] using certain systems of their modulated and translated forms. Few explicit examples of dual Gabor window pairs are known. This paper constructs explicit examples with trigonometric form in one dimension as well as higher dimensions. Also, in the discrete time domain, the trigonometric form allows us to evaluate the Gabor coefficients efficiently using the Discrete Fourier Transform. The windows have fixed support and arbitrary smoothness.

A method for local interpolation with tension trigonometric spline curves and surfaces

A method for local interpolation with tension trigonometric spline curves and surfaces

C1Rational Quadratic Trigonometric Interpolation Spline for Data Visualization

A newC1piecewise rational quadratic trigonometric spline with four local positive shape parameters in each subinterval is constructed to visualize the given planar data. Constraints are derived on these free shape parameters to generate shape preserving interpolation curves for positive and/or monotonic data sets. Two of these shape parameters are constrained while the other two can be set free to interactively control the shape of the curves. Moreover, the order of approximation of developed interpolant is investigated asO(h3). Numeric experiments demonstrate that our method can construct nice shape preserving interpolation curves efficiently.

Application of integrodifferential splines to solving an interpolation problem

This paper deals with cases when the values of derivatives of a function are given at grid nodes or the values of integrals of a function over grid intervals are known. Polynomial and trigonometric integrodifferential splines for computing the value of a function from given values of its nodal derivatives and/or from its integrals over grid intervals are constructed. Error estimates are obtained, and numerical results are presented.

Spectral Bounds on the Solution of Linear Time‐Periodic Systems

AbstractLinear time‐periodic systems arise whenever a nonlinear system is linearized about a periodic trajectory. Stability of the solution may be proven by rigorous bounds on the solution. The key idea of this paper is to derive Chebyshev projection bounds on the original system by solving an approximated system. Depending on the smoothness of the original function, we formulate two upper bounds. The theoretical results are illustrated and compared to trigonometric spline bounds by means of two examples which include an anisotropic rotor‐bearing system. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An upper bound of the Bezout number for piecewise algebraic curves

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. Based on the discussion of the number of the zeros of homogeneous trigonometric splines with different smoothness and the common points of two piecewise algebraic curves over a star partition, a better upper bound of Bezout number of two piecewise algebraic curves over arbitrary triangulation is found. Moreover, upper bounds of the Bezout number BN(m,r;n,r;Δ) for piecewise algebraic curves over several special partitions such as rectangular partition, type-1 triangulation and type-2 triangulation are obtained.

A new trigonometric spline approach to numerical solution of generalized nonlinear Klien-Gordon equation.

The generalized nonlinear Klien-Gordon equation plays an important role in quantum mechanics. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline is presented for the approximate solution of this equation with Dirichlet boundary conditions. The usual finite difference approach is used to discretize the time derivative while cubic trigonometric B-spline is applied as an interpolating function in the space dimension. Several examples are discussed to exhibit the feasibility and capability of the approach. The absolute errors and error norms are also computed at different times to assess the performance of the proposed approach and the results were found to be in good agreement with known solutions and with existing schemes in literature.

The Applications of Cardinal Trigonometric Splines in Solving Nonlinear Integral Equations

The cardinal trigonometric splines on small compact supports are employed to solve integral equations. The unknown function is expressed as a linear combination of cardinal trigonometric splines functions. Then a simple system of equations on the coefficients is deducted. When solving the Volterra integral equations, the system is triangular, so it is relatively straight forward to solve the nonlinear system of the coefficients and a good approximation of the original solution is obtained. The sufficient condition for the existence of the solution is discussed and the convergence rate is investigated.

Simple filter design for first and second order derivatives by a double filtering approach

Spline filters are usually implemented in two steps, where in the first step the basis coefficients are computed by deconvolving the sampled function with a factorized filter and the second step reconstructs the sampled function. It will be shown how separable spline filters using different splines can be constructed with fixed kernels, requiring no inverse filtering. Especially, it is discussed how first and second order derivatives can be computed correctly using cubic or trigonometric splines by a double filtering approach giving filters of length 7.

Blending of the Trigonometric Polynomial Spline Curve with Arbitrary Continuous Orders

In order to develop the theory of trigonometric polynomial spline curve, the representation of trigonometric polynomial spline curve is blended to a general form based on blending of trigonometric polynomial curves. Moreover, some properties of the blending curve are discussed in details. The research shows that the basis of the trigonometric polynomial curve is relative simple, and the blending curve includes the original trigonometric polynomial spline curve and shows much better shape-control capability than the original curve. Meanwhile, the blending curve keeps the same degree of original one. It is easy to find that the curve can be reshaping by adjusting the shape factor. At the same time, a new method of the representation of closed curve is given and can also accurately represent the whole ellipse arc and other conic curve.

Ternary approximating non-stationary subdivision schemes for curve design

AbstractIn this paper, an algorithm has been introduced to produce ternary 2m-point (for any integer m ≥ 1) approximating non-stationary subdivision schemes which can generate the linear spaces spanned by {1; cos(α.); sin(α.)}. The theory of asymptotic equivalence is being used to analyze the convergence and smoothness of the schemes. The proposed algorithm can be consider as the non-stationary counter part of the 2-point and 4-point existing ternary stationary approximating schemes, for different values of m. Moreover, the proposed algorithm has the ability to reproduce or regenerate the conic sections, trigonometric polynomials and trigonometric splines.

A symmetric non-stationary subdivision scheme

AbstractThis paper proposes a new family of symmetric $4$-point ternary non-stationary subdivision schemes that can generate the limit curves of $C^3$ continuity. The continuity of this scheme is higher than the existing 4-point ternary approximating schemes. The proposed scheme has been developed using trigonometric B-spline basis functions and analyzed using the theory of asymptotic equivalence. It has the ability to reproduce or regenerate the conic sections, trigonometric polynomials and trigonometric splines as well. Some graphical and numerical examples are being considered, by choosing an appropriate tension parameter $0<\alpha <\pi /3 $, to show the usefulness of the proposed scheme. Moreover, the Hölder regularity and the reproduction property are also being calculated.

Positivity Preserving Interpolation of Positive Data by Rational Quadratic Trigonometric Spline

Positivity Preserving Interpolation of Positive Data by Rational Quadratic Trigonometric Spline

A scheme for interpolation with trigonometric spline curves

We present a method for the interpolation of a given sequence of data points with Cn continuous trigonometric spline curves of order n+1 (n≥1) that are produced by blending elliptical arcs. Ready to use explicit formulae for the control points of the interpolating arcs are also provided. Each interpolating arc depends on a global parameter α∈(0,π) that can be used for global shape modification. Associating non-negative weights with data points, rational trigonometric interpolating spline curves can be obtained, where weights can be used for local shape modification. The proposed interpolation scheme is a generalization of the Overhauser spline, and it includes a Cn Bézier spline interpolation method as the limiting case α→0.

Method of phantom nodes in the interpolation problem trigonometric splines

Method of phantom nodes in the interpolation problem trigonometric splines

Bounds on the Solution of Linear Time‐Periodic Systems

AbstractLinear time‐periodic systems have been an active area of research in the last decades. They arise in various applications such as anisotropic rotor‐bearing systems and nonlinear systems linearized about a periodic trajectory. Rigorous bounds support the transient analysis of these systems. Optimal constants are determined by the differential calculus for norms of matrix functions. Bounds based on trigonometric spline approximations of the solution are introduced and convergence results for the approximations are stated. Bounds are illustrated by means of an anisotropic rotor‐bearing system. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Approximation of planar offset curves using quadratic trigonometric splines with shape parameter

This paper presents a new method for generating the approximation offset of freeform planar curves using quadratic trigonometric splines with shape parameter. Although the input is higher-degree NURBS curves, only knot insertion at the inflexion and end points of the self-intersecting loop are required for initial offsetting. Based on the obtained new control polygon and its offsets, the quadratic trigonometric spline was constructed to approximate the exact offset curves. Because of the great controllability, the proposed method could generate a completely overestimating offset curve by adjusting the value of the shape parameter, which guarantees no overcutting in NC machining. Furthermore, it also produces the lowest number of control points compared with other works. Experimental results demonstrated the usefulness and effectiveness of this method.

C1 rational quadratic trigonometric spline

A Bézier like C1 rational quadratic trigonometric polynomial spline is developed. It defines two shape parameters in each subinterval. The approximation and geometric properties are investigated. The curvature continuity is established. The developed rational quadratic trigonometric polynomial spline is extended to C1 piecewise rational bi-quadratic function with four shape parameters in each rectangular patch. Data dependent constraints are developed on the shape parameters in the description of piecewise rational quadratic and bi-quadratic trigonometric polynomial spline for shape preservation of curve and regular surface data. The developed shape preserving schemes provide tangent continuity in quadratic form and does not restrict interval length, derivatives or data.