Search

The foundations of fast universal trigonometric interpolation for nonperiodic functions used to obtain high-accuracy approximate solutions are described. High-accuracy formulas are derived for computing the launch point coordinates of a flight vehicle by applying universal fast trigonometric interpolation combined with extrapolation at the endpoints of a given interval. Numerical experiments show that, after launching the first vehicle, the launch point coordinates can be determined in 7 s with accuracy of m. [ABSTRACT FROM AUTHOR]

This paper presents a new trigonometric composite Hermite interpolation method for solving Fredholm linear integral equations. This operator approximates locally both the function and its derivative, which is known on the subdivision nodes. Then we derive a class of quadrature rules with endpoint corrections based on integrating the composite Hermite interpolant. We also provide error estimation and numerical examples to illustrate that this new operator can provide highly accurate results. [ABSTRACT FROM AUTHOR]

This paper introduces a fully discrete soft thresholding trigonometric polynomial approximation on [ − π , π ] , named Lasso trigonometric interpolation. This approximation is an ℓ 1 -regularized discrete least squares approximation under the same conditions of classical trigonometric interpolation on an equidistant grid. Lasso trigonometric interpolation is a sparse scheme which is efficient in dealing with noisy data. We theoretically analyze Lasso trigonometric interpolation quality for continuous periodic function. The L 2 error bound of Lasso trigonometric interpolation is less than that of classical trigonometric interpolation, which improved the robustness of trigonometric interpolation. The performance of Lasso trigonometric interpolation for several testing functions (sin wave, triangular wave, sawtooth wave, square wave), is illustrated with numerical examples, with or without the presence of data errors. [ABSTRACT FROM AUTHOR]

In this work, a new type of interpolation formula is introduced. These formulas can be an extension of the Lagrange interpolation formula. The error of this new type of interpolation is calculated. In order to display efficiency of the proposed formulas, three numerical examples are presented.

Unevenly spaced samples from a periodic function are common in signal processing and can often be viewed as a perturbed equally spaced grid. In this paper, the question of how the uneven distribution of the samples impacts the quality of interpolation and quadrature is analyzed. Starting with equally spaced nodes on [ - π , π) with grid spacing h, suppose the unevenly spaced nodes are obtained by perturbing each uniform node by an arbitrary amount ≤ α h , where 0 ≤ α < 1 / 2 is a fixed constant. A discrete version of the Kadec-1/4 theorem is proved, which states that the nonuniform discrete Fourier transform associated with perturbed nodes has a bounded condition number independent of h, for any α < 1 / 4 . Then, it is shown that unevenly spaced quadrature rules converge for all continuous functions and interpolants converge uniformly for all differentiable functions whose derivative has bounded variation when 0 ≤ α < 1 / 4 . Though, quadrature rules at perturbed nodes can have negative weights for any α > 0 , a bound on the absolute sum of the quadrature weights is provided, which shows that perturbed equally spaced grids with small α can be used without numerical woes. While the proof techniques work primarily when 0 ≤ α < 1 / 4 , it is shown that a small amount of oversampling extends our results to the case when 1 / 4 ≤ α < 1 / 2 . [ABSTRACT FROM AUTHOR]

We consider the numerical computation of Hadamard Finite Part (HFP) integrals K m (t ; u) = ⨎ 0 T S m (π (x − t) T) u (x) d x , 0 < t < T , m ∈ { 1 , 2 , ... } , where u (x) is T -periodic and sufficiently differentiable and S 2 r − 1 (y) = cos ⁡ y sin 2 r − 1 ⁡ y , S 2 r (y) = 1 sin 2 r ⁡ y , r = 1 , 2 , 3 , .... For each m , we regularize the HFP integral K m (t ; u) and show that K m (t ; u) = K 0 (t ; U m) ≡ ∫ 0 T (log ⁡ | sin ⁡ π (x − t) T |) U m (x) d x , U m (x) being some linear combination of the first m derivatives of u (x). We then propose to approximate K m (t ; u) by the quadrature formula Q m , n (t ; u) ≡ K m (t ; ϕ n) , where ϕ n (x) is the n th -order balanced trigonometric polynomial that interpolates u (x) on [ 0 , T ] at the 2 n equidistant points x n , k = k T 2 n , k = 0 , 1 , ... , 2 n − 1. The implementation of Q m , n (t ; u) is simple, the only input needed for this being the 2 n function values u (x n , k) , k = 0 , 1 , ... , 2 n − 1. Using Fourier analysis techniques, we develop a complete convergence theory for Q m , n (t ; u) as n → ∞ and prove that it enjoys spectral convergence when u ∈ C ∞ (R). We illustrate the effectiveness of Q m , n (t ; u) with numerical examples for m = 0 , 1 , ... , 5. We also show that the HFP integral ⨎ 0 T f (x , t) d x of any T -periodic integrand f (x , t) that has m th order poles at x = t + k T , k = 0 , ± 1 , ± 2 , ... , but is sufficiently differentiable in x on R ∖ { t ± k T } k = 0 ∞ , can be expressed in terms of the K s (t ; u (⋅ , t)) , where u (x , t) is a T -periodic and sufficiently differentiable function in x on R that can be computed from f (x , t). Therefore, ⨎ 0 T f (x , t) d x can be computed efficiently using our new numerical quadrature formulas Q s , n (t ; u (⋅ , t)) on the individual K s (t ; u (⋅ , t)). Again, only 2 n function evaluations, namely, u (x n , k , t) , k = 0 , 1 , ... , 2 n − 1 , are needed for the whole process. [ABSTRACT FROM AUTHOR]

Cubic Hermite interpolation curve plays a very important role in interpolation curves modeling, but it has three shortcomings including low continuity, difficult shape adjustment, and the inability to accurately represent some common engineering curves. We construct a cubic trigonometric Hermite interpolation curve to make up the three shortcomings of cubic Hermite interpolation curve once and for all. The cubic trigonometric Hermite interpolation curve not only inherits the features of cubic Hermite interpolation curve but also achieves C2 continuity, has local and global adjustability, and can accurately represent elliptical arc, circular arc, quadratic parabolic arc, cubic parabolic arc, and astroid arc that often appear in engineering. In addition, we give the schemes for optimizing the shape of the cubic trigonometric Hermite interpolation curve based on internal energy minimization. The schemes include optimizing the shape of planar curve and spatial curve. Some modeling examples show that the proposed schemes are effective and the cubic trigonometric Hermite interpolation curve is more practical than cubic Hermite interpolation curve. [ABSTRACT FROM AUTHOR]

In this study, a triangular polynomial interpolation (TPI) scheme was developed to estimate the vertical eddy viscosity coefficient (VEVC) on the basis of the Ekman model with adjoint assimilation. In the twin experiments, the advantages and disadvantages of estimating the VEVC using the TPI scheme under different factors are discussed. The results indicated that (1) the TPI scheme proves to be better than the cubic spline interpolation (CSI) and Cressman interpolation (CI) schemes; (2) the inversion results are more sensitive to observations from upper ocean layers than those from lower layers, and the TPI scheme is less likely to be influenced by missing data; (3) for various boundary layer depths, the inversion results of the TPI scheme remain consistent with the given distributions; (4) the inversion results can be influenced considerably by observational errors, and the TPI scheme is more resistant to noise than the CSI and CI schemes; and (5) the inversion accuracy of the TPI scheme can be improved by selecting the temporal wind stress drag coefficients. In practical experiments, the adjoint method with the TPI scheme was developed to estimate the Ekman currents by assimilating the observations from a buoy stationed in the Yellow Sea. The results showed the successful estimation of the VEVC and demonstrated that more precise current velocities can be obtained with this estimation scheme. In summary, this study provides a useful approach for the effective estimation of the VEVC. [ABSTRACT FROM AUTHOR]

A trigonometrically fitted diagonally implicit two-derivative Runge–Kutta method (TFDITDRK) is used for the numerical integration of first-order delay differential equations (DDEs) which possesses oscillatory solutions. Using the trigonometrically fitted property, a three-stage fifth-order diagonally implicit two- derivative Runge–Kutta (DITDRK) method is derived. Here, we employed trigonometric interpolation for the approximation of the delay term. The curves of efficiency based on the log of maximum errors against the log of function evaluations and the CPU time spent to perform the integration are plotted, which then clearly illustrated the superiority of the trigonometrically fitted DITDRK method in comparison with its original method and other existing diagonally implicit Runge–Kutta (DIRK) methods. [ABSTRACT FROM AUTHOR]

Directional scanning measurement is the dominant strategy taken to measure channel spatial profiles for millimeter wave (mmWave) frequency bands. The focus of the paper is on wideband channel spatial profile reconstruction and angular sampling interval (ASI) selection for directional scanning measurement. We propose to employ the trigonometric interpolation technique to reconstruct the spatial channel profile and to use Nyquist sampling theorem and spherical wave mode theory to determine the ASI. Our analysis demonstrated that the half-power beam width (HPBW) of the antenna is a good approximation for the ASI setting to ensure accurate directional scanning measurements. The reconstructed power-angle-delay profile (PADP), composite power angular spectrum (PAS), delay and angle characteristics match well with those of the target in the measured data when the ASI is set to the antenna HPBW. [ABSTRACT FROM AUTHOR]

Proposed in the paper is a trigonometric interpolation method for efficient determination of turbomachinery blade aero-damping curves which are required in a flutter assessment. The trigonometric interpolation method was proposed to be incorporated with the widely used travelling wave method to replace the influence coefficient method. Through analyzing aero-damping/worksum at a few carefully chosen nodal diameters, trigonometric interpolation was applied through existing data points to get aero-damping/worksum at the rest nodal diameters. The proposed approach is much more efficient than the travelling wave method for determining the aero-damping curve of a blade. In principle, the method can be as efficient as the influence coefficient method. Unlike the influence coefficient method, the trigonometric interpolation method does not involve linear superposition, and it can include nonlinear effect and is expected to be more accurate. Two test cases were provided to validate the proposed method and demonstrate its effectiveness. The method is not only effective, but also very easy to be incorporated into existing widely used aero-damping/worksum analysis system using the travelling wave method. [ABSTRACT FROM AUTHOR]

In this paper, we propose two methods to compute the double Hilbert transform of periodic functions. First, we establish the quadratic formula of trigonometric interpolation type for double Hilbert transform and obtain an estimation of the remainder. We call this method 2D mechanical quadrature method (2D‐MQM). Numerical experiments show that 2D‐MQM outperforms the library function "hilbert" in Matlab when the values of the functions being handled are very large or approach to infinity. Second, we propose a complex analytic method to calculate the double Hilbert transform, which is based on the 2D adaptive Fourier decomposition, and the method is called as 2D‐HAFD. In contrast to the pointwise approximation, 2D‐HAFD provides explicit rational functional approximations and is valid for all signals of finite energy. [ABSTRACT FROM AUTHOR]

In this work we construct an Hermite interpolant starting from basis functions that satisfy a Lagrange property. In fact, we extend and generalise an iterative approach, introduced by Cirillo and Hormann (2018) for the Floater–Hormann family of interpolants. Secondly, we apply this scheme to produce an effective barycentric rational trigonometric Hermite interpolant at general ordered nodes using as basis functions the ones of the trigonometric interpolant introduced by Berrut (1988). For an easy computational construction, we calculate analytically the differentiation matrix. Finally, we conclude with various examples and a numerical study of the convergence at equidistant nodes and conformally mapped nodes. [ABSTRACT FROM AUTHOR]

We consider the numerical computation of Hadamard Finite Part (HFP) integrals H σ (t ; u) = ⨎ 0 T | sin π (x − t) T | σ u (x) d x , 0 < t < T , σ < − 1 , σ ⁄ ∈ Z , u (x) being sufficiently differentiable and T -periodic on R. Thus σ = − (m + δ) , m ∈ { 1 , 2 , 3 , ... } , 0 < δ < 1. For each such σ , we regularize H σ (t ; u) , and show that H σ (t ; u) = H σ + 2 r (t ; U σ) , r = ⌊ (m + 1) / 2 ⌋ , where U σ (x) = ∑ k = 0 r a k u (2 k) (x) for some constants a k , H σ + 2 r (t ; U σ) being a regular integral. We then propose to approximate H σ (t ; u) by the quadrature formula Q σ , n (t ; u) ≡ H σ (t ; ϕ n) , where ϕ n (x) is the n th -order balanced trigonometric polynomial that interpolates u (x) on [ 0 , T ] at the 2 n equidistant points x n , k = k T 2 n , k = 0 , 1 , ... , 2 n − 1. The implementation of Q σ , n (t ; u) is simple, the only input needed for this being the 2 n function values u (x n , k) , k = 0 , 1 , ... , 2 n − 1. Using Fourier analysis techniques, we develop a complete convergence theory for Q σ , n (t ; u) as n → ∞ and prove that it enjoys spectral convergence when u ∈ C ∞ (R). We also show that the theoretical developments and numerical quadrature formulas developed for the HFP integrals H σ (t ; u) with σ < − 1 and σ ⁄ ∈ Z apply, with no changes, to the regular singular integrals H σ (t ; u) with σ > − 1 and σ ⁄ ∈ Z. We illustrate the effectiveness of Q σ , n (t ; u) with numerical examples both for σ < − 1 and σ > − 1. Finally, we show that the HFP or regular integral ⨎ 0 T f (x) d x of any T -periodic integrand f (x) that has algebraic singularities of the form | x − t + k T | σ , 0 < t < T , k = 0 , ± 1 , ± 2 , ... , with σ ⁄ ∈ Z , but is sufficiently differentiable in x on R ∖ { t ± k T } k = 0 ∞ , can be expressed as H σ (t ; u) , where u (x) is a T -periodic and sufficiently differentiable function of x on R that can be computed from f (x). Therefore, ⨎ 0 T f (x) d x can be computed efficiently using our new numerical quadrature formulas Q σ , n (t ; u) on the individual H σ (t ; u). Again, only 2 n function evaluations, namely, u (x n , k) , k = 0 , 1 , ... , 2 n − 1 , are needed for the whole process. [ABSTRACT FROM AUTHOR]

We study some procedures for the approximation of three-dimensional data on a grid with a hypothesis of periodicity. The first part proposes a generalization of a discrete periodic approximation defined by Dunham Jackson. The functions used have the advantage of owning an analytical explicit expression in terms of the samples (specific values) of the original function or data. In the second part, we describe a continuous approximation function for the same problem, defined through an integral. Some results of the rate of convergence and bounds of the approximation error are presented, with the single hypothesis of Hölder continuity or continuity of the original function. [ABSTRACT FROM AUTHOR]

Abstract This article proposes a generalization of the Fourier interpolation formula, where a wider range of the basic trigonometric functions is considered. The extension of the procedure is done in two ways: adding an exponent to the maps involved, and considering a family of fractal functions that contains the standard case. The studied interpolation converges for every continuous function, for a large range of the nodal mappings chosen. The error of interpolation is bounded in two ways: one theorem studies the convergence for Hölder continuous functions and other develops the case of merely continuous maps. The stability of the approximation procedure is proved as well. [ABSTRACT FROM AUTHOR]

Due to the deficiency of sliding Lagrange polynomial interpolation, the author proposes a new interpolation method, which considers the physical character of satellite movement in coordinate transformation and reasonable selection of interpolation function. Precision of the two methods is compared by a numerical example. The result shows that the new method is superior to the sliding Lagrange polynomial interpolation in interpolation and extrapolation, especially in extrapolation that is over short time spans.

The Dynamic Finite Element (DFE) formulation is a superconvergent, semianalytical method used to perform vibration analysis of structural components during the early stages of design. It was presented as an alternative to analytical and numerical methods that exhibit various drawbacks, which limit their applicability during the preliminary design stages. The DFE method, originally developed by the second author, has been exploited heavily to study the modal behaviour of beams in the past. Results from these studies have shown that the DFE method is capable of arriving at highly accurate results with a coarse mesh, thus, making it an ideal choice for preliminary stage modal analysis and design of structural components. However, the DFE method has not yet been extended to study the vibration behaviour of plates. Thus, the aim of this study is to develop a set of frequency-dependent, trigonometric shape functions for a 4-noded, 4-DOF per node element as a basis for developing a DFE method for thin rectangular plates. To this end, the authors exploit a distinct quasi-exact solution to the plate governing equation and this solution is then used to derive the new, trigonometric basis and shape functions, based on which the DFE method would be developed. [ABSTRACT FROM AUTHOR]

Based on multiquadric trigonometric spline quasi-interpolation, the paper proposes a scheme for numerical differentiation of noisy data, which is a well-known ill-posed problem in practical applications. In addition, in the perspective of kernel regression, the paper studies its large sample properties including optimal bandwidth selection, convergence rate, almost sure convergence, and uniformly asymptotic normality. Simulations are provided at the end of the paper to demonstrate features of the scheme. Both theoretical results and simulations show that the scheme is simple, easy to compute, and efficient for numerical differentiation of noisy data. [ABSTRACT FROM AUTHOR]

The trigonometric interpolants to a periodic function f in equispaced points converge if f is Dini-continuous, and the associated quadrature formula, the trapezoidal rule, converges if f is continuous. What if the points are perturbed? With equispaced grid spacing h, let each point be perturbed by an arbitrary amount ≤ αh, where α ∊ [0,1/2) is a fixed constant. The Kadec 1/4 theorem of sampling theory suggests there may be trouble for α ≥ 1/4. We show that convergence of both the interpolants and the quadrature estimates is guaranteed for all α < 1/2 if f is twice continuously differentiable, with the convergence rate depend ing on the smoothness of f. More precisely, it is enough for f to have 4α derivatives in a certai n sense, and we conjecture that 2α derivatives are enough. Connections with the Fejer--Kalmar theorem are discussed. [ABSTRACT FROM AUTHOR]

We consider the numerical stability of the second barycentric formula for evaluation at points in [0, 2π] of trigonometric interpolants in an odd number of equispaced points in that interval.We show that, contrary to the prevailing view, which claims that this formula is always stable, it actually possesses a subtle instability that seems not to have been noticed before. This instability can be corrected by modifying the formula.We establish the forward stability of the resulting algorithm by using techniques that mimic those employed previously by Higham (2004, The numerical stability of barycentric Lagrange interpolation. IMA J. Numer. Anal., 24, 547-556) to analyse the second barycentric formula for polynomial interpolation.We show how these results can be extended to interpolation on other intervals of length-2π in many cases. Finally, we investigate the formula for an even number of points and show that, in addition to the instability that affects the odd-length formula, it possesses another instability that is more difficult to correct. [ABSTRACT FROM AUTHOR]

In this article, a new quadratic trigonometric B-spline with control parameters is constructed to address the problems related to two dimensional digital image interpolation. The newly constructed spline is then used to design an image interpolation scheme together with one of the soft computing techniques named as Genetic Algorithm (GA). The idea of GA has been formed to optimize the control parameters in the description of newly constructed spline. The Feature SIMilarity (FSIM), Structure SIMilarity (SSIM) and Multi-Scale Structure SIMilarity (MS-SSIM) indices along with traditional Peak Signal-to-Noise Ratio (PSNR) are employed as image quality metrics to analyze and compare the outcomes of approach offered in this work, with three of the present digital image interpolation schemes. The upshots show that the proposed scheme is better choice to deal with the problems associated to image interpolation. [ABSTRACT FROM AUTHOR]

A class of C¹ rational cubic/linear trigonometric interpolation spline with two local parameters is proposed. Simple sufficient conditions for constructing positivity-preserving interpolation curves are developed. By using the boolean sum of quadratic trigonometric interpolating operators to blend together the proposed rational cubic/linear trigonometric interpolation splines as four boundary functions, a kind of C¹ blending rational cubic/linear interpolation surface with four families of local parameters is constructed. Simple sufficient data dependent conditions are also deduced for generating C¹ positivity-preserving interpolation surfaces on rectangular grids. [ABSTRACT FROM AUTHOR]

An algorithm is presented for numerical computation of choreographies in spaces of constant negative curvature in a hyperbolic cotangent potential, extending the ideas given in a companion paper [14] for computing choreographies in the plane in a Newtonian potential and on a sphere in a cotangent potential. Following an idea of Diacu, Pérez-Chavela and Reyes Victoria [9], we apply stereographic projection and study the problem in the Poincaré disk. Using approximation by trigonometric polynomials and optimization methods with exact gradient and exact Hessian matrix, we find new choreographies, hyperbolic analogues of the ones presented in [14]. The algorithm proceeds in two phases: first BFGS quasi-Newton iteration to get close to a solution, then Newton iteration for high accuracy. [ABSTRACT FROM AUTHOR]

In this paper we construct non-aliasing interpolation spaces and Lagrange functions for lattice grids. We argue that lattice grids are good for trigonometric interpolation and support this claim by numerical experiments. A greedy algorithm allows us to embed hyperbolic crosses in our interpolation spaces, and numerical experiments indicate that lattice grids are at least as good as sparse grids for trigonometric interpolation. A straightforward FFT-algorithm for functions sampled on lattice grids allows for fast computation and good approximation. [ABSTRACT FROM AUTHOR]

An algorithm is presented for numerical computation of choreographies in the plane in a Newtonian potential and on the sphere in a cotangent potential. It is based on stereographic projection, approximation by trigonometric polynomials, and quasi-Newton and Newton optimization methods with exact gradient and exact Hessian matrix. New choreographies on the sphere are presented. [ABSTRACT FROM AUTHOR]