Showing papers on "Trigonometric interpolation published in 2019"

Bernstein Fractal Trigonometric Approximation

Abstract: Fractal interpolation and approximation received a lot of attention in the last thirty years. The main aim of the current article is to study a fractal trigonometric approximants which converge to the given continuous function even if the magnitude of the scaling factors does not approach zero. In this paper, we first introduce a new class of fractal approximants, namely, Bernstein $\alpha $ -fractal functions using the theory of fractal approximation and Bernstein polynomial. Using the proposed class of fractal approximants and imposing no condition on corresponding scaling factors, we establish that the set of Bernstein $\alpha $ -fractal trigonometric functions is fundamental in the space of continuous periodic functions. Fractal version of Gauss formula of trigonometric interpolation is obtained by means of Bernstein trigonometric fractal polynomials. We study the Bernstein fractal Fourier series of a continuous periodic function $f$ defined on $[-l,l]$ . The Bernstein fractal Fourier series converges to $f$ even if the magnitude of the scaling factors does not approach zero. Existence of the $\mathcal{C}^{r}$ -Bernstein fractal functions is investigated, and Bernstein cubic spline fractal interpolation functions are proposed based on the theory of $\mathcal{C}^{r}$ -Bernstein fractal functions.

Bounding Multivariate Trigonometric Polynomials

Abstract: The extremal values of multivariate trigonometric polynomials are of interest in fields ranging from control theory to filter design, but finding the extremal values of such a polynomial is generally NP-Hard. In this paper, we develop simple and efficiently computable estimates of the extremal values of a multivariate trigonometric polynomial directly from its samples. We provide an upper bound on the modulus of a complex trigonometric polynomial, and develop upper and lower bounds for real trigonometric polynomials. For a univariate polynomial, these bounds are tighter than existing bounds, and the extension to multivariate polynomials is new. As an application, the lower bound provides a sufficient condition to certify global positivity of a real trigonometric polynomial.

About one method of construction of interpolation trigonometric splines

Abstract: The method of constructing spline classes in the form of trigonometric Fourier series whose coefficients have a certain decreasing order are considered. in turn, this decrement determines the number of continuous derivatives of sum of this series. By grouping members of this series according with the effect of overlaying and introducing a multiplier that provides the interpolation properties of the sum of these series on even grids, we obtain classes of trigonometric interpolation splines. Depending on the types of convergence factors with a certain decreasing order, different classes of such splines are obtained. The classes of trigonometric splines include classes of periodic polynomial splines of even and odd power; At the same time, there exist trigonometric splines that do not have polynomial analogues. An example of the construction of trigonometric splines is given.

One-Second GPS Orbits: A Comparison Between Numerical Integration and Interpolation

Abstract: Precise positioning using the Global Positioning System (GPS) requires accurate knowledge of the satellite orbits. The International GNSS Service (IGS) distributes post-processed GPS satellite orbits that give the satellite positions at 15 minutes interval. For GPS applications involving high-rate (1 Hz) GPS, it is necessary to know the satellite positions at one second intervals. One approach of doing this is to interpolate the IGS precise orbit using a polynomial or trigonometric function. An alternative approach is to use the precise ephemeris distributed by the IGS to obtain the satellite position and velocity at the initial epoch and then perform numerical integration to determine the satellite position at one second intervals. JPL's GPS Inferred Positioning System/Orbit Analysis and Simulation Software (GIPSY/OASIS) is used to numerically integrate the orbit of a single GPS satellite in order to determine the position at one second intervals. A comparison of the numerically integrated positions and the interpolated positions shows that the numerically integrated satellite positions more closely match the IGS orbits than an orbit constructed using a trigonometric interpolation.

Approximating Periodic Potential Energy Surfaces with Sparse Trigonometric Interpolation.

Abstract: The potential energy surface (PES) describes the energy of a chemical system as a function of its geometry and is a fundamental concept in computational chemistry. A PES provides much useful information about the system, including the structures and energies of various stationary points, such as local minima, maxima, and transition states. Construction of full-dimensional PESs for molecules with more than 10 atoms is computationally expensive and often not feasible. Previous work in our group used sparse interpolation with polynomial basis functions to construct a surrogate reduced-dimensional PESs along chemically significant reaction coordinates, such as bond lengths, bond angles, and torsion angles. However, polynomial interpolation does not preserve the periodicity of the PES gradient with respect to angular components of geometry, such as torsion angles, which can lead to nonphysical phenomena. In this work, we construct a surrogate PES using trigonometric basis functions, for a system where the selected reaction coordinates all correspond to the torsion angles, resulting in a periodically repeating PES. We find that a trigonometric interpolation basis not only guarantees periodicity of the gradient but also results in slightly lower approximation error than polynomial interpolation.

A Method for Dimensionally Adaptive Sparse Trigonometric Interpolation of Periodic Functions

Abstract: We present a method for dimensionally adaptive sparse trigonometric interpolation of multidimensional periodic functions belonging to a smoothness class of finite order. This method targets applications where periodicity must be preserved and the precise anisotropy is not known a priori. To the authors' knowledge, this is the first instance of a dimensionally adaptive sparse interpolation algorithm that uses a trigonometric interpolation basis. The motivating application behind this work is the adaptive approximation of a multi-input model for a molecular potential energy surface (PES) where each input represents an angle of rotation. Our method is based on an anisotropic quasi-optimal estimate for the decay rate of the Fourier coefficients of the model; a least-squares fit to the coefficients of the interpolant is used to estimate the anisotropy. Thus, our adaptive approximation strategy begins with a coarse isotropic interpolant, which is gradually refined using the estimated anisotropic rates. The procedure takes several iterations where ever-more accurate interpolants are used to generate ever-improving anisotropy rates. We present several numerical examples of our algorithm where the adaptive procedure successfully recovers the theoretical "best" convergence rate, including an application to a periodic PES approximation. An open-source implementation of our algorithm resides in the Tasmanian UQ library developed at Oak Ridge National Laboratory.

Fundamental trigonometric interpolation and approximating polynomials and splines.

Abstract: The paper deals with two fundamental types of trigonometric polynomials and splines on uniform grids, which allow us to construct interpolation approximations that depend linearly on the values of the interpolated function. Fundamental on the same grids are trigonometric LS polynomials and LS splines, which allow us to construct the approximation of functions by the least squares method and also depend linearly on the values of the approximate function. Trigonometric splines were considered as splines with polynomial analogues in some cases. The material is illustrated in graphs. The considered fundamental trigonometric polynomials and splines can be recommended for use in many problems related to approximation of functions, in particular, mathematical modeling of signals and processes.

About a Non-Standard Interpolation Problem

Abstract: Using algebraic methods, and motivated by the one variable case, we study a multipoint interpolation problem in the setting of several complex variables. The duality realized by the residue generator associated with an underlying Gorenstein algebra, using the Lagrange interpolation polynomial, plays a key role in the arguments.

Trigonometric Interpolation Beamforming for a Circular Microphone Array

Abstract: Polynomial beamforming has previously been proposed for addressing the non-trivial problem of integrating acoustic echo cancellation with adaptive microphone beamforming. This paper demonstrates a design example for a circular array where traditional polynomial beamforming approaches exhibit severe (over 10 dB) directivity index (DI) oscillations at the edges of the design interval, leading to severe DI degradation for certain look directions. A solution, based on trigonometric interpolation, is proposed that stabilizes the oscillations significantly, resulting in a DI that deviates only about 1 dB from that of a fixed beamformer over all look directions.

Domain decomposition for quasi-periodic scattering by layered media via robust boundary-integral equations at all frequencies

Abstract: We develop a non-overlapping domain decomposition method (DDM) for the solution of quasi-periodic scalar transmission problems in layered media. Our approach relies on robust boundary-integral equation formulations of Robin-to-Robin (RtR) maps throughout the frequency spectrum, including at Wood, or cutoff, frequencies. We overcome the obstacle of non-convergent quasi-periodic Green functions at these frequencies by incorporating newly introduced shifted quasi-periodic Green functions. Using the latter in the definition of our quasi-periodic boundary-integral operators leads to rigorously stable computations of RtR operators. We develop Nystr\"om discretizations of the RtR maps that rely on trigonometric interpolation, singularity resolution, and fast convergent windowed quasi-periodic Green functions. We solve the tridiagonal DDM system via recursive Schur complements and we establish rigorously that this procedure is always completed successfully. We present a variety of numerical results concerning Wood frequencies in two and three dimensions as well as large numbers of layers.

Trefftz approximations in complex media: Accuracy and applications

Abstract: Approximations by Trefftz functions are rapidly gaining popularity in the numerical solution of boundary value problems of mathematical physics. By definition, these functions satisfy locally, in weak form, the underlying differential equations of the problem, which often results in high-order or even exponential accuracy with respect to the size of the basis set. We highlight two separate examples in applied electromagnetics and photonics: (i) homogenization of periodic structures, and (ii) numerical simulation of electromagnetic waves in slab geometries. Extensive numerical evidence and theoretical considerations show that Trefftz approximations can be applied much more broadly than is traditionally done: they are effective not only in physically homogeneous regions but also in complex inhomogeneous ones. Two mechanisms underlying the high accuracy of Trefftz approximations in such complex cases are pointed out. The first one is related to trigonometric interpolation and the second one – somewhat surprisingly – to well-posedness of random matrices.

About the classification of trigonometric splines

Abstract: One of the possible variants of the classification of trigonometric interpolation splines is considered, depending on the chosen convergence factors, the distribution of signs of the basis functions and the interpolation factors. The concept of crosslinking and interpolation grids is introduced; these grids can either match or not match. The proposed classification is illustrated by an example.

On Features of Reconstruction of Finite-Length Discrete-Time Signal with Nonzero Constant Component Using Sinc Interpolation

Abstract: In practice, one of the most important question is how accurate the method used for the finite-length discrete-time signal reconstruction. In the paper, research results of influence of the constant component as a zero-frequency component of signal spectrum on accuracy of finite-length signal reconstruction using truncated Whittaker–Kotelnikov sampling series and trigonometric interpolation are discussed. It was found that, in contrary to the truncated Fourier series, truncated Whittaker–Kotelnikov sampling series reconstruction error of model signals vs constant component values is nonmonotonic. It was also pointed out that mean-square error of constant signal interpolation by means of truncated Whittaker-Kotelnikov sampling series is much greater than mean-square error of trigonometric interpolation. The results are explained. The obtained results can be used in designing of the finite-length discrete-time signal reconstruction by means of sinc-function interpolation.

Algorithms Based on Trigonometric Interpolation for Signal Reconstruction with Even Number of Sampling Points

Abstract: Research results of the finite-length discrete-time signal reconstruction accuracy by means of the trigonometric interpolation are discussed. The research is performed for even number of sampling points by means of the well-known algorithm as well as the new one. We demonstrate that the discrete-time signal reconstruction error is the sum of the monotonically decreasing function (the one the number of samples of increases) and a specific periodic component. The periodic component occurrence reasons are described.