Showing papers on "Trigonometric interpolation published in 2022"

Lagrange Multivariate Polynomial Interpolation: A Random Algorithmic Approach

Abstract: The problems of polynomial interpolation with several variables present more difficulties than those of one-dimensional interpolation. The first problem is to study the regularity of the interpolation schemes. In fact, it is well-known that, in contrast to the univariate case, there is no universal space of polynomials which admits unique Lagrange interpolation for all point sets of a given cardinality, and so the interpolation space will depend on the set Z of interpolation points. Techniques of univariate Newton interpolating polynomials are extended to multivariate data points by different generalizations and practical algorithms. The Newton basis format, with divided-difference algorithm for coefficients, generalizes in a straightforward way when interpolating at nodes on a grid within certain schemes. In this work, we propose a random algorithm for computing several interpolating multivariate Lagrange polynomials, called RLMVPIA (Random Lagrange Multivariate Polynomial Interpolation Algorithm), for any finite interpolation set. We will use a Newton-type polynomials basis, and we will introduce a new concept called Z , z -partition. All the given algorithms are tested on examples. RLMVPIA is easy to implement and requires no storage.

Perbandingan Metode Interpolasi Newton dan Lagrange dengan Bahasa Pemrograman C++

Abstract: Interpolation is defined as an estimate of a known value. Extensive interpolation is an attempt to determine the approximate value of an analytic function that is unknown or alternatively a complex function whose analytic equation cannot be obtained. You can combine the use of math and math to analyze the price of an item. This study describes Newton's method and the polynomial method. Therefore, interpolation with Newton's method has an error value that is smaller than the error value of the Lagrange interpolation. The results in this research case study when x = 2.5 using 10 data performed using Newton's polynomial interpolation method have a result of 1.70956 where this value is lower than the value of the analysis using the Lagrange method which is 3.2163.

Построение оптимальной интерполяционной формулы методом Соболева точных для тригонометрических функций

Abstract: The paper is devoted to derivation of the optimal interpolation formula in W2(0,2)(0,1) Hilbert space by Sobolev’s method. Here the interpolation formula consists of a linear combination ΣNβ=0Cβφ(xβ) of the given values of a function φ from the space W2(0,2)(0,1). The difference between functions and the interpolation formula is considered as a linear functional called the error functional. The error of the interpolation formula is estimated by the norm of the error functional. We obtain the optimal interpolation formula by minimizing the norm of the error functional by coefficients Cβ(z) of the interpolation formula. The obtained optimal interpolation formula is exact for trigonometric functions sinx and cosx. At the end of the paper we give some numerical results which confirm the numerical convergence of the optimal interpolation formula. Работа посвящена построению оптимальной интерполяционной формулы методом Соболева в гильбертовом пространстве W2(0,2)(0,1). Здесь интерполяционная формула состоит из линейной комбинации ΣNβ=0Cβφ(xβ) заданных значений функции φ из пространство W2(0,2)(0,1). Отличие функций от интерполяционной формулы рассматривается как линейный функционал, называемый функционалом погрешности. Погрешность интерполяционной формулы оценивается нормой функционала погрешности. Мы получаем оптимальной интерполяционной формулы путем минимизации нормы функционала погрешности на коэффициенты Cβ(z) интерполяционной формулы. Полученная оптимальная интерполяция формула точна для тригонометрических функций sinx и cosx. В конце статьи мы приводим некоторые численные результаты, которые подтверждают наши теоретические результаты.

Convergence of Lagrange interpolation on the non uniform nodes on the unit circle

Abstract: This research article aims to staunchly study the approximation using Lagrange interpolation on the unit circle. Nodal system constitutes the vertically projected zeros of Jacobi polynomial onto the unit circle with boundary points at $ \pm $1. Moreover, convergence is obtained by considering analytic functions on a suitable domain accompanied by some numerical experiments.

Application of the Fast Expansion Method in Space–Related Problems

Abstract: In the paper, numerical and approximate analytical solutions for the problem of the motion of a spacecraft from a starting point to a final point during a certain time are obtained. The unpowered and powered portions of the flight are considered. For a numerical solution, a finite-difference scheme of the second order of accuracy is constructed. The space-related problem considered in the study is essentially nonlinear, which necessitates the use of trigonometric interpolation methods to replace the task of calculating the Fourier coefficients with the integral formulas by solving the interpolation system. One of the simplest options for trigonometric sine interpolation on a semi-closed segment [–a, a), where the right end is not included in the general system of interpolation points, is considered. In order to maintain the conditions of orthogonality of sines, an even number of 2M calculation points is uniformly applied to the segment. The sine interpolation theorem is proved and a compact formula is given for calculating the interpolation coefficients. A general theory of fast sine expansion is given. It is shown that in this case, the Fourier coefficients decrease much faster with the increase in serial number compared to the Fourier coefficients in the classical case. This property allows reducing the number of terms taken into account in the Fourier series, as well as the amount of computer calculations, and increasing the accuracy of calculations. The analysis of the obtained solutions is carried out, and their comparison with the exact solution of the test problem is proposed. With the same calculation error, the time spent on a computer using the fast expansion method is hundreds of times less than the time spent on classical finite-difference method.

Lasso trigonometric polynomial approximation for periodic function recovery in equidistant points

Abstract: In this paper, we propose a fully discrete soft thresholding trigonometric polynomial approximation on $[-\pi,\pi],$ named Lasso trigonometric interpolation. This approximation is an $\ell_1$-regularized discrete least squares approximation under the same conditions of classical trigonometric interpolation on an equidistant grid. Lasso trigonometric interpolation is sparse and meanwhile it is an efficient tool to deal with noisy data. We theoretically analyze Lasso trigonometric interpolation for continuous periodic function. The principal results show that the $L_2$ error bound of Lasso trigonometric interpolation is less than that of classical trigonometric interpolation, which improved the robustness of trigonometric interpolation. This paper also presents numerical results on Lasso trigonometric interpolation on $[-\pi,\pi]$, with or without the presence of data errors.

Mean convergence of Lagrange interpolation

Abstract: In this note we prove mean convergence of Lagrange interpolation at the zeros of para-orthogonal polynomials for measures in the unit circle which does not belong to Szeg\H{o}'s class in the unit circle. When the measure is in Szeg\H{o}'s class mean convergence of Lagrange interpolation is proved for functions in the disk algebra.

Determination of Derivative - and Regularized Derivative Interpolation for Band-limited Functions

Abstract: The ill-posedness of derivative interpolation is examined in this chapter, as well as a regularised derivative interpolation for band-limited signals. The Shannon Sampling Theorem is used to examine the ill-posedness. The convergence of the regularized derivative interpolation is studied by the combination of a regularized Fourier transform and the Shannon Sampling Theorem. The error estimation is given, and high order derivatives are also considered. The regularised derivative interpolation algorithm is compared to derivative interpolation with other algorithms. Some examples are used to prove and test the convergence property. The numerical findings reveal that for band-limited signals, the regularised sampling algorithm's derivative interpolation is more effective in decreasing noise.  

On some quasi-periodic approximations

Abstract: Trigonometric approximation or interpolation of a non-smooth function on a finite interval has poor convergence properties. This is especially true for discontinuous functions. The case of infinitely differentiable but non-periodic functions with discontinuous periodic extensions onto the real axis has attracted interest from many researchers. In a series of works, we discussed an approach based on quasi-periodic trigonometric basis functions whose periods are slightly bigger than the length of the approximation interval. We proved validness of the approach for trigonometric interpolations. In this paper, we apply those ideas to classical Fourier expansions.

Trigonometric Polynomial Solutions of Bernoulli Trigonometric Polynomial Differential Equations

Abstract: We consider real trigonometric polynomial Bernoulli equations of the form A(θ)y′=B1(θ)+Bn(θ)yn where n≥2, with A,B1,Bn being trigonometric polynomials of degree at most μ≥1 in variables θ and Bn(θ)≢0. We also consider trigonometric polynomials of the form A(θ)yn−1y′=B0(θ)+Bn(θ)yn where n≥2, being A,B0,Bn trigonometric polynomials of degree at most μ≥1 in the variable θ and Bn(θ)≢0. For the first equation, we show that when n≥4, it has at most 3 real trigonometric polynomial solutions when n is even and 5 real trigonometric polynomial solutions when n is odd. For the second equation, we show that when n≥3, it has at most 3 real trigonometric polynomial solutions when n is odd and 5 real trigonometric polynomial solutions when n is even. We also provide trigonometric polynomial equations of the two types mentioned above where the maximum number of trigonometric polynomial solutions is achieved. The proof method will be to apply extended Fermat problems to polynomial equations.

Using NUFFT in nonuniform sampling Fourier transform spectrometer and the comparison with conventional interpolation FFT method

Abstract: Resampling by interpolation is the traditional method to process sample in nonunform sampling Fourier transform spectrometer. Nonuniform discrete Fourier transform is an alternative to interpolation that has not been overlooked before. With the aid of experiment, we systematically compare the NUFFT method with resampling by interpolation FFT method in nonuniform sampling Fourier transform spectrometer. We found that NUFFT is comparable to interpolation in spectral profile and spectral noise levels and is better in spectral amplitudes. We also found that It has significant advantage in under-sampling and anti-aliasing property which is offered by the unique non-periodic nature of nonuniform sampling. It is faster and consumes less computer memory in our python implementation. Overall, we found that NUFFT is superior to traditional interpolation method with mostly better performances as well as additional capabilities.

Signals Recovery by Means of Three-Harmonic Equations Solutions

Abstract: An intensive development of signal theory has been taking place for the last decades. Therefore, the range of such problems has expanded considerably. Problems closely related to the mathematical methods have become especially important. We can include here problems related to image processing, filtering audio and other signals, etc. It should be noted that these problems are often related to the methods of systems analysis and approximation theory. In particular, in many digital signal processing problems, interpolation is one of the classic methods of their solution. When we talk about the usage of interpolation polynomials in signal theory, we are used to using standard methods of spline-type interpolation, Lagrange interpolation polynomials or Hermite polynomials. However, signals are known to be approximated the best by periodic functions. Trigonometric analogues of Lagrange interpolation polynomials are the standard examples of interpolation that could be used in practice. But such methods do not always give good results or are not easy to use. To solve this problem, we should keep in mind that signals are harmonic functions. Therefore, the discrete Fourier transformation has been used in many scientific works. However, this approach does not always give the desired effect, so the question of constructing alternative interpolation polynomials arises. In this paper, we propose a new approach to signal recovery, which allows getting better signal recovery than that given by the most common methods of signal recovery. The usage of interpolation polynomials proposed in this paper provides an improvement in signal recovery compared to their classical counterparts.

On the Problem of the Effectiveness of the Use of Trigonometric Polynomials in Signal Theory

Abstract: At the stage of analysis of the efficiency of complex systems of automated signal processing there is a need to select parameters that would optimize a number of processes. The paper considers the effectiveness of the use of trigonometric polynomials for digital signal processing. We investigate matrix summation methods with rectangular matrices and their efficiency in comparison with methods based on the use of trigonometric polynomials. We propose a new approach to the construction of the trigonometric series of the function. The question of Lebesgue convergence of the constructed trigonometric series for an arbitrary summable periodic function in terms of a system of numerical coefficients is also investigated. The obtained results can be used in digital signal processing systems.

On the theory of operator interpolation in spaces of rectangular matrixes

Abstract: The problem of constructing and studying interpolation operator polynomials of an arbitrary fixed degree, defined in spaces of rectangular matrices, which would be generalisations of the corresponding interpolation formulas in the case of square matrices, is considered. Linear interpolation formulas of various structures are constructed for rectangular matrices. Matrix polynomials, with respect to which the resulting interpolation formulas are invariant, are indicated. As a generalisation of linear formulas, formulas for quadratic interpolation and interpolation by polynomials of arbitrary fixed degree in the space of rectangular matrices are constructed. Particular cases of the obtained formulas are considered: when square matrices are chosen as nodes or when the values of the interpolated function are square matrices, as well as the case when both of these conditions are satisfied. For the last variant, the possibilities of different and identical matrix orders for nodes and function values are explored. The obtained results are based on the application of some well-known provisions of the theory of matrices and the theory of interpolation of scalar functions. The presentation of the material is illustrated by a number of examples.

Approximation and Interpolation of Singular Measures by Trigonometric Polynomials

Abstract: Complex signed measures of finite total variation are a powerful signal model in many applications. Restricting to the $d$-dimensional torus, finitely supported measures allow for exact recovery if the trigonometric moments up to some order are known. Here, we consider the approximation of general measures, e.g., supported on a curve, by trigonometric polynomials of fixed degree with respect to the Wasserstein-1 distance. We prove sharp lower bounds for their best approximation and (almost) matching upper bounds for effectively computable approximations when the trigonometric moments of the measure are known. A second class of sum of squares polynomials is shown to interpolate the characteristic function on the support of the measure and to converge to zero outside.

Interpolation of tabular functions from one independent variable using the taylor polynomial

Abstract: A method of local interpolation of tabular functions from one independent variable using the Taylor polynomial of the nth degree in arbitrarily located interpolation nodes has been developed. This makes it possible to calculate intermediate values of tabular functions between interpolation nodes. The conducted analysis of the latest research and publications in the field of interpolation of tabular functions showed that the main part of the research is a strict theory of interpolation, i.e. clarification of its fundamental mathematical provisions. Some features of the interpolation of tabular functions from one independent variable using the Taylor polynomial of the nth degree are considered, namely: the solution algorithm and mathematical formulation of the interpolation problem are given; its formalized notation is given, as well as the matrix notation of interpolation procedures for certain values of the argument. A scalar algorithm for solving the problem of interpolation of tabular functions from one independent variable using the Taylor polynomial of the 2nd, 3rd, and 4th degrees has been developed. The simplicity and clarity of this algorithm is one of its advantages, but the algorithm is inconvenient for software implementation. The mathematical formulation of the problem of interpolation of tabular functions in terms of matrix algebra is given. The interpolation task is reduced to performing the following actions: based on the values of nodal points known from the table, it is necessary to calculate the Taylor matrix of the nth degree; based on the function values specified in the table a column vector of interpolation nodes should be formed; solve a linear system of algebraic equations, the root of which is the numerical coefficients of the Taylor polynomial of the nth degree. A method of calculating the coefficients of the interpolant, given by the Taylor polynomial of the nth degree for one independent variable has been developed. The essence of the method reduces to the product of the matrix, inverse of the Taylor matrix, which is determined by the nodal points of the tabular function, by a column vector containing the values of the interpolation nodes. Specific examples demonstrate the peculiarities of calculating the interpolant coefficients of the 2nd, 3rd and 4th degrees for one independent variable, and for each of them the interpolated value of the function at a given point is calculated. Calculations were performed in the Excel environment, which by analogy can be successfully implemented in any other computing environment.