Trigonometric Interpolation Research Articles

Incompatible 3-node interpolation for gradient-dependent plasticity

In gradient-dependent plasticity theory, the yield strength depends on the Laplacian of an equivalent plastic strain measure (hardening parameter), and the consistency condition results in a differential equation with respect to the plastic multiplier. The plastic multiplier is then discretized in addition to the usual discretization of the displacements, and the consistency condition is solved simultaneously with the equilibrium equations. The disadvantage is that the plastic multiplier requires a Hermitian interpolation that has four degrees of freedom at each node. Instead of using a Hermitian interpolation, in this article, a 3-node incompatible (trigonometric) interpolation is proposed for the plastic multiplier. This incompatible interpolation uses only the function values of each node, but it is continuous across element boundaries and its second-order derivatives exist within the elements. It greatly reduces the degrees of freedom for a problem, and is shown through a numerical example on localization to yield good results.

Integral equation methods for scattering from an impedance crack

For the scattering problem for time-harmonic waves from an impedance crack in two dimensions, we give a uniqueness and existence analysis via a combined single- and double-layer potential approach in a Hölder space setting leading to a system of integral equations that contains a hypersingular operator. For its numerical solution we describe a fully discrete collocation method based on trigonometric interpolation and interpolatory quadrature rules including a convergence analysis and numerical examples.

A bootstrap algorithm for the two-sample problem using trigonometric Hermite spline interpolation

In this paper we study the two-sample problem using procedures based on the empirical characteristic function. We consider the L 2 norm of the difference between the empirical characteristic functions and use an interpolant to obtain a numerical integration formula to approximate the test statistic. For our approximation method we have used a trigonometric Hermite interpolant, and afterwards, to estimate the p-value of the resultant statistic, a bootstrap algorithm was implemented.

3D Fourier based discrete Radon transform

The Radon transform is a fundamental tool in many areas. For example, in reconstruction of an image from its projections (CT scanning). Recently A. Averbuch et al. [SIAM J. Sci. Comput., submitted for publication] developed a coherent discrete definition of the 2D discrete Radon transform for 2D discrete images. The definition in [SIAM J. Sci. Comput., submitted for publication] is shown to be algebraically exact, invertible, and rapidly computable. We define a notion of 3D Radon transform for discrete 3D images (volumes) which is based on summation over planes with absolute slopes less than 1 in each direction. Values at nongrid locations are defined using trigonometric interpolation on a zero-padded grid. The 3D discrete definition of the Radon transform is shown to be geometrically faithful as the planes used for summation exhibit no wraparound effects. There exists a special set of planes in the 3D case for which the transform is rapidly computable and invertible. We describe an algorithm that computes the 3D discrete Radon transform which uses O( Nlog N) operations, where N= n 3 is the number of pixels in the image. The algorithm relies on the 3D discrete slice theorem that associates the Radon transform with the pseudo-polar Fourier transform. The pseudo-polar Fourier transform evaluates the Fourier transform on a non-Cartesian pointset, which we call the pseudo-polar grid. The rapid exact evaluation of the Fourier transform at these non-Cartesian grid points is possible using the fractional Fourier transform.

Abschätzung der Normen von Operatoren analytischer Funktionen

This article investigates the norms of certain interpolation operators of analytic functions on the unit disc. In particular, it is shown that the norms of interpolation operators being the identical operator for all n -degree polynomials have a lower bound of order ln n . This result is compared with a recent result regarding trigonometric interpolation of continuous functions on the unit circle. It is shown that opposed to the operators of analytic functions on the unit disc, the method of oversampling can be applied in order to uniformly bound the interpolation operators. Moreover, some practical implications with regard to communication engineering are discussed. It is concluded that in practice the results lead to non-linear interpolation operators.

Two trigonometric quadrature formulae for evaluating hypersingular integrals

AbstractTwo trigonometric quadrature formulae, one of non‐interpolatory type and one of interpolatory type for computing the hypersingular integral ${\int\hskip-0.33cm=}_{-1}^{1} w(\tau)g(\tau)/(\tau-t)^{2} \,{\rm d}\tau$ are developed on the basis of trigonometric quadrature formulae for Cauchy principal value integrals. The formulae use the cosine change of variables and trigonometric polynomial interpolation at the practical abscissae. Fast three‐term recurrence relations for evaluating the quadrature weights are derived. Numerical tests are carried out using the current formula. As applications, two simple crack problems are considered. One is a semi‐infinite plane containing an internal crack perpendicular to its boundary and the other is a centre cracked panel subjected to both normal and shear tractions. It is found that the present method generally gives superior results. Copyright © 2002 John Wiley & Sons, Ltd.

Some results on linear rational trigonometric interpolation

We present here formulae for calculating the p th derivative of a linear rational trigonometric interpolant written in barycentric form. We give sets of interpolating points for which the interpolant converges exponentially towards the interpolated function.

Approximation of Periodic Analytic Functions by Interpolation Trigonometric Polynomials in the Metric of the Space L

We obtain asymptotic equalities for the upper bounds of approximations by interpolation trigonometric polynomials in the metric of the space L on classes of convolutions of periodic functions admitting a regular extension into a fixed strip of the complex plane.

On Convergence of Interpolation to Analytic Functions

In the present paper, both the perfect convergence for the Lagrange interpolation of analytic functions on [−1, 1] and the perfect convergence for the trigonometric interpolation of analytic functions on [− π , π ] with period 2 π are discussed.

A method for estimating light interception by a conifer shoot.

We present an operational method for estimating the amount of PAR intercepted by a coniferous shoot. Interception of PAR by a shoot is divided into three components: the amount of radiation coming from the sky, the transmission of radiation through the surrounding vegetation, and the shoot' s silhouette area facing the direction of the incoming radiation. All three components usually vary with direction. Radiation incident from the sky consists of direct and diffuse radiation. The well-known equation of motion for the sun and Beer' s Law for atmospheric transmittance are used to simulate the directional distribution of direct sunlight for any given period of time. The diffuse component is assumed to be uniform. Meteorological field measurements are used to calibrate the absolute amounts of the direct and diffuse components. The gap fraction (proportion of visible sky) in different directions around a shoot is measured by analyzing a hemispherical fish-eye photograph, taken at the location of the shoot, with an image processing program. Similarly, the shoot silhouette area (SSA) is measured by photographing the shoot from many different directions. The measurements of SSA are interpolated by a method called trigonometric interpolation to obtain the directional distribution of SSA over the entire hemisphere. This distribution is then rotated according to the shoot' s position in the canopy. Multiplying incoming PAR, canopy gap fraction and SSA in different directions, and summing over all directions, gives an estimate of PAR intercepted by the shoot during the chosen period of time. The method is described step by step, and applied, as an example, to a shoot from a Scots pine (Pinus sylvestris L.) stand in central Finland. Differences in radiation interception properties between sun and shade shoots and their relevance to canopy-scale models are discussed.

On the arclength of trigonometric interpolants

As pointed out recently by Strichartz [5], the arclength of the graph \(\Gamma(S_N(f))\) of the partial sums \(S_N(f)\) of the Fourier series of a jump function \(f\) grows with the order of \(\log N\).In this paper we discuss the behaviour of the arclengths of the graphs of trigonometric interpolants to a jump function. Here the boundedness of the arclengths depends essentially on the fact whether the jump discontinuity is at an interpolation point or not.
 In addition convergence results for the arclengths of interpolants to smoother functions are presented.

Approximation of Infinitely Differentiable Periodic Functions by Interpolation Trigonometric Polynomials in Integral Metric

We obtain asymptotic equalities for the upper bounds of approximations of periodic infinitely differentiable functions by interpolation trigonometric polynomials in the metric of L1 on the classes of convolutions.

On the Summability of Trigonometric Interpolation Processes

The aim of this paper is to show that several processes studied in trigonometric interpolation theory can be obtained by φ-sums of discrete Fourier series. We shall investigate the uniform convergence of the sequences of thesepolynomials. We show that the convergence of several processes can be seen immediately from suitable explicit forms of the corresponding polynomials. Error estimates for the approximation can be also obtained by certain general results.

A General Approach to Dyadic Periodic Interpolation Based on Discrete Sigma-Pi Orthogonality

Starting with some well known formulas of discrete sigma-pi orthogonality of bipolar vectors we introduce a quite general and easy strategy to obtain periodic interpolation schemes on dyadic grids including fast algorithms for explicit calculation. As special cases of our presented approach we mention dyadic trigonometric and Walsh-type interpolation.

A Padé-based algorithm for overcoming the Gibbs phenomenon

Truncated Fourier series and trigonometric interpolants converge slowly for functions with jumps in value or derivatives. The standard Fourier–Pade approximation, which is known to improve on the convergence of partial summation in the case of periodic, globally analytic functions, is here extended to functions with jumps. The resulting methods (given either expansion coefficients or function values) exhibit exponential convergence globally for piecewise analytic functions when the jump location(s) are known. Implementation requires just the solution of a linear system, as in standard Pade approximation. The new methods compare favorably in experiments with existing techniques.

Vibration of thick plates using finite strip-elements

This paper develops a finite strip-element method for the vibration analysis of thick plates. The method uses a combined polynomial and trigonometric interpolation scheme that enables all boundary conditions to be correctly treated. The global equations are derived in the usual manner of the finite element method, and natural frequencies of vibration can be found by solving a linear eigenvalue problem.

A quadrature rule for weighted Cauchy integrals

Using a variable transformation t= cos y for weighted Cauchy integrals, we develop an automatic quadrature method based on trigonometric interpolation. We show that error bounds are independent of the location of poles.

A quadrature rule of interpolatory type for Cauchy integrals

We construct a quadrature rule of interpolatory-type based on a trigonometric interpolation for Cauchy integrals. We obtain the same error bound O(| a N N |) as Hasegawa and Torii (Math. Comp. 56 (1991) 741–754). Numerical examples are included to substantiate the performance of the present method.

VIBRATION OF STIFFENED PLATES USING HIERARCHICAL TRIGONOMETRIC FUNCTIONS

The vibration analysis of stiffened plates using hierarchical finite elements with a set of local trigonometric interpolation functions is presented. The local functions extend on the plate domain comprised between consecutive stiffeners, thereby allowing a coarse discretization of the global structure. Convergence studies as well as comparison of the present approach with the literature and experimental results are presented. The great numerical stability of the trigonometric functions and their readiness for symbolic manipulations make them potentially attractive for vibration and sound radiation analysis in the mid-frequency range.

A closed algebraic interpolation curve

We introduce a new interpolation method employing a closed algebraic curve. This note describes a trigonometric interpolation method to connect a given point sequence in R d by an algebraic and therefore smooth closed curve. The interpolation formula is of Lagrangian type and seems to be new. In contrast to spline interpolation our method does not work piecewise but global. Nevertheless the curve connects the given points in a quite natural manner preserving the given order and without overshooting tendency. The curves are, like spline curves, stable in the following sense: changing the position of a single point of the given sequence only changes the curve in the neighbourhood of this point essentially. Besides its trigonometric parametrization the curve possesses a numerically efficient parametrization on the basis of Chebyshev polynomials of the second kind. To what extend the new interpolation method can replace the spline method, the experts may decide.