Fourier Sums Research Articles

Fast τ-p transforms by chirp modulation

We have developed simple, fast, and accurate algorithms for the linear Radon ([Formula: see text]-[Formula: see text]) transform and its inverse. The algorithms have an [Formula: see text] computational complexity in contrast to the [Formula: see text] cost of a direct implementation in 2D and an [Formula: see text] computational complexity compared to the [Formula: see text] cost of a direct implementation in 3D. The methods use Bluestein’s algorithm to evaluate discrete nonstandard Fourier sums, and they need, apart from the fast Fourier transform (FFT), only multiplication of chirp functions and their Fourier transforms. The computational cost and accuracy are thus reduced to that inherited by the FFT. Fully working algorithms can be implemented in a couple of lines of code. Moreover, we find that efficient graphics processing unit (GPU) implementations could achieve processing speeds of approximately [Formula: see text], implying that the algorithms are I/O bound rather than compute bound.

Приближение кусочно гладких функций тригонометрическими суммами Фурье

We obtain exact order-of-magnitude estimates of piecewise smooth functions approximation by trigonometric Fourier sums. It is shown that in continuity points Fourier series of piecewise Lipschitz function converges with rate $\ln n/n$. If function $f$ has a piecewise absolutely continuous derivative then it is proven that in continuity points decay order of Fourier series remainder $R_n(f,x)$ for such function is equal to $1/n$. We also obtain exact order-of-magnitude estimates for $q$-times differentiable functions with piecewise smooth $q$-th derivative. In particular, if $f^{(q)}(x)$ is piecewise Lipschitz then $|R_n(f,x)| \le c(x)\frac{\ln n}{n^{q+1}}$ in continuity points of $f^{(q)}(x)$ and $\sup_{x \in [0,2\pi]}|R_n(f,x)| \le \frac{c}{n^q}$. In case when $f^{(q)}(x)$ has piecewise absolutely continuous derivative it is shown that $|R_n(f,x)| \le \frac{c(x)}{n^{q+1}}$ in continuity points of $f^{(q)}(x)$. As a consequence of the last result convergence rate estimate of Fourier series to continuous piecewise linear functions is obtained.

Приближение непрерывных 2π-периодических кусочно-гладких функций дискретными суммами Фурье

Приближение непрерывных 2π-периодических кусочно-гладких функций дискретными суммами Фурье

Integral presentation of deviations of rectangular linear means of Fourier series on classes of periodic differentiable functions

The paper deals with the problems of approximation in a uniform metric of periodic functions of many variables by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. Questions of asymptotic behavior of the upper bounds of deviations of linear operators generated by the use of linear methods of summation of Fourier series on the classes of periodic differentiable functions are studied in many works. Methods of investigation of integral representations of deviations of polynomials on the classes of periodic differentiable functions of real variable originated and received its development through the works of S.M. Nikol'skii, S.B. Stechkin, N.P.Korneichuk, V.K. Dzadik, A.I. Stepanets, etc. Along with the study of approximation by linear methods of classes of functions of one variable, are studied similar problems of approximation by linear methods of classes of functions of many variables. In addition to the approximative properties of rectangular Fourier sums, are studied approximative properties of other approximation methods: the rectangular sums of Valle Poussin, Zigmund, Rogozinsky, Favar. In this paper we consider the classes of \(\overline{\psi}\)-differentiable periodic functions of many variables, allowing separately to take into account the properties of partial and mixed \(\overline{\psi}\)-derivatives, and given by analogy with the classes of \(\overline{\psi}\)-differentiable periodic functions of one variable. Integral representations of rectangular linear means of Fourier series on classes of \(\overline{\psi}\)-differentiable periodic functions of many variables are obtained. The obtained formulas can be useful for further investigation of the approximative properties of various linear rectangular methods on the classes \(\overline{\psi}\)-differentiable periodic functions of many variables in order to obtain a solution to the corresponding Kolmogorov-Nikolsky problems.

Uniform Approximation of the Curvature of Smooth Plane Curves with the Use of Partial Fourier Sums

An error bound for the approximation of the curvature of graphs of periodic functions from the class Wr for r ≥ 3 in the uniform metric is obtained with the use of the simplest approximation technique for smooth periodic functions, which is approximation by partial sums of their trigonometric Fourier series. From the mathematical point of view, the interest in this problem is connected with the specific nonlinearity of the graph curvature operator on the class of smooth functions Wr on a period or a closed interval for r ≥ 2. There are several papers on curvature approximation for plane curves in the mean-square and Chebyshev norms. In previous works, the approximation was performed by partial sums of trigonometric series (in the L2 norm), interpolation splines with uniform knots, Fejer means of partial sums of trigonometric series, and orthogonal interpolating wavelets based on Meyer wavelets (in the C∞ norm). The technique of this paper, based on the lemma, can possibly be generalized to the Lp metric and other approximation methods.

Exploring one loop amplitude at four-points vertices by the OPP method

We construct the general formula of one-loop amplitude at four-point vertices using Ossola, Papadopoulos, and Pittau (OPP) method. The incoming and outgoing particles are defined as arbitrary massless particles, and the intermediate state contains arbitrary particles inside. In this works, the amplitude is reconstructed via finding four-type rational coefficients. First, box coefficient is extracted using the four-cut technique with linear algebra. We found that triangle and bubble coefficients can be extracted using three-cut and two-cut technique with Cauchy residue theorem instead of with discrete Fourier sum like the original version of the OPP. Tadpole coefficient can be dropped out, because its scalar integral, which contains only UV divergence, is completely absorbed by renormalization.

Over-the-Air Computation for Cooperative Wideband Spectrum Sensing and Performance Analysis

For sensor network aided cognitive radio, cooperative wideband spectrum sensing can distribute the sampling and computing pressure of spectrum sensing to multiple sensor nodes (SNs). However, this may incur high latency due to distributed data fusion, especially when the number of SNs is large. In this paper, we propose a novel cooperative wideband spectrum sensing scheme using over-the-air computation. Its key idea is to utilize the superposition property of wireless channel to implement the summation of Fourier transform. This avoids distributed data fusion by computing the target function directly. The performance of the proposed scheme is analyzed with imperfect synchronization between different SNs considered. Furthermore, a synchronization phase offset estimation and equalization method is proposed. The corresponding performance after equalization is also derived. Both a working prototype based on universal software radio periphera and Monte Carlo simulations are provided to verify the performance of the proposed scheme.

Computation of Adaptive Fourier Series by Sparse Approximation of Exponential Sums

In this paper, we study the convergence of adaptive Fourier sums for real-valued $$2\pi $$ -periodic functions. For this purpose, we approximate the sequence of classical Fourier coefficients by a short exponential sum with a pre-defined number of $$N+1$$ terms. The obtained approximation can be interpreted as an adaptive N-th Fourier sum with respect to the orthogonal Takenaka-Malmquist basis. Using the theoretical results on rational approximation in Hardy spaces and on the decay of singular values of special infinite Hankel matrices, we show that adaptive Fourier sums can converge essentially faster than classical Fourier sums for a large class of functions. Further, we derive an algorithm to compute almost optimal adaptive Fourier sums. Our numerical results show that the significantly better convergence behavior of adaptive Fourier sums for optimally chosen basis elements can also be achieved in practice. For comparison, we also provide a greedy algorithm to determine an adaptive Fourier sum. This algorithm requires less computational effort but yields essentially slower convergence.

Two-Sided Estimates of Fourier Sums Lebesgue Functions with Respect to Polynomials Orthogonal on Nonuniform Grids

Let Ω = {t0, t1, …, tN} and ΩN = {x0, x1, …, xN–1}, where xj = (tj + tj + 1)/2, j = 0, 1, …, N–1 be arbitrary systems of distinct points of the segment [–1, 1]. For each function f(x) continuous on the segment [–1, 1], we construct discrete Fourier sums Sn, N( f, x) with respect to the system of polynomials {pk,N(x)} =0 –1 , forming an orthonormal system on nonuniform point systems ΩN consisting of finite number N of points from the segment [–1, 1] with weight Δtj = tj + 1–tj. We find the growth order for the Lebesgue function Ln,N (x) of the considered partial discrete Fourier sums Sn,N ( f, x) as n = O(δ −2/7 ), δN = max0≤ j≤N−1 Δtj More exactly, we have a two-sided pointwise estimate for the Lebesgue function Ln, N(x), depending on n and the position of the point x from [–1, 1].

Uniform Approximations by Fourier Sums in Classes of Generalized Poisson Integrals

We find asymptotic equalities for exact upper bounds of approximations by Fourier sums in uniform metric on classes of $2\pi$-periodic functions, representable in the form of convolutions of functions $\varphi$, which belong to unit balls of spaces $L_{p}$, with generalized Poisson kernels. For obtained asymptotic equalities we introduce the estimates of remainder, which are expressed in the explicit form via the parameters of the problem.

On a System of Rational Chebyshev–Markov Fractions

In the present paper an orthogonal system of Chebyshev–Markov rational fractions is considered. We introduce the corresponding Fourier series and find the Dirichlet integral. We obtain the decomposition of the function |x| into Fourier series with respect to the considered system in explicit form and an asymptotic estimate of the uniform approximation of this function by partial sums of the rational Fourier–Chebyshev series.

Sobolev-orthogonal systems of functions associated with an orthogonal system

For every system of functions which is orthonormal on with weight and every positive integer we construct a new associated system of functions which is orthonormal with respect to a Sobolev-type inner product of the form We study the convergence of Fourier series in the systems . In the important particular cases of such systems generated by the Haar functions and the Chebyshev polynomials , we obtain explicit representations for the that can be used to study their asymptotic properties as and the approximation properties of Fourier sums in the system . Special attention is paid to the study of approximation properties of Fourier series in systems of type generated by Haar functions and Chebyshev polynomials.

АППРОКСИМАТИВНЫЕ СВОЙСТВА ДИСКРЕТНЫХ СУММ ФУРЬЕ ДЛЯ НЕКОТОРЫХ КУСОЧНО-ЛИНЕЙНЫХ ФУНКЦИЙ

АППРОКСИМАТИВНЫЕ СВОЙСТВА ДИСКРЕТНЫХ СУММ ФУРЬЕ ДЛЯ НЕКОТОРЫХ КУСОЧНО-ЛИНЕЙНЫХ ФУНКЦИЙ

Double-sided to Lebesgue function of Fourier sums by polynomials orthogonal on non-uniform grids

Double-sided to Lebesgue function of Fourier sums by polynomials orthogonal on non-uniform grids

Main beam modeling for large irregular arrays

Large radio telescopes in the 21st century such as the Low-Frequency Array (LOFAR) or the Murchison Widefield Array (MWA) make use of phased aperture arrays of antennas to achieve superb survey speeds. The Square Kilometer Array low frequency instrument (SKA1-LOW) will consist of a collection of non-regular phased array systems. The prediction of the main beam of these arrays using a few coefficients is crucial for the calibration of the telescope. An effective approach to model the main beam and first few sidelobes for large non-regular arrays is presented. The approach exploits Zernike polynomials to represent the array pattern. Starting from the current defined on an equivalence plane located just above the array, the pattern is expressed as a sum of Fourier transforms of Zernike functions of different orders. The coefficients for Zernike polynomials are derived by two different means: least-squares and analytical approaches. The analysis shows that both approaches provide a similar performance for representing the main beam and first few sidelobes. Moreover, numerical results for different array configurations are provided, which demonstrate the performance of the proposed method, also for arrays with shapes far from circular.

On sharp estimates of the convergence of double Fourier–Bessel series

The problem of approximation of a differentiable function of two variables by partial sums of a double Fourier–Bessel series is considered. Sharp estimates of the rate of convergence of the double Fourier–Bessel series on the class of differentiable functions of two variables characterized by a generalized modulus of continuity are obtained. The proofs of four theorems on this issue, which can be directly applied to solving particular problems of mathematical physics, approximation theory, etc., are presented.

Approximation of the Classes of Generalized Poisson Integrals by Fourier Sums in Metrics of the Spaces L S

In metrics of the spaces L s , 1 ≤ s ≤ 1, we establish asymptotic equalities for the upper bounds of approximations by Fourier sums in the classes of generalized Poisson integrals of periodic functions that belong to the unit ball in the space L 1 .

Ab initio velocity-field curves in monoclinic β-Ga2O3

We investigate the high-field transport in monoclinic β-Ga2O3 using a combination of ab initio calculations and full band Monte Carlo (FBMC) simulation. Scattering rate calculation and the final state selection in the FBMC simulation use complete wave-vector (both electron and phonon) and crystal direction dependent electron phonon interaction (EPI) elements. We propose and implement a semi-coarse version of the Wannier-Fourier interpolation method [Giustino et al., Phys. Rev. B 76, 165108 (2007)] for short-range non-polar optical phonon (EPI) elements in order to ease the computational requirement in FBMC simulation. During the interpolation of the EPI, the inverse Fourier sum over the real-space electronic grids is done on a coarse mesh while the unitary rotations are done on a fine mesh. This paper reports the high field transport in monoclinic β-Ga2O3 with deep insight into the contribution of electron-phonon interactions and velocity-field characteristics for electric fields ranging up to 450 kV/cm in different crystal directions. A peak velocity of 2 × 107 cm/s is estimated at an electric field of 200 kV/cm.

Effects of aligned α-helix peptide dipoles on experimental electrostatic potentials.

Aligned protein α-helix dipoles have been implicated in protein function and structure. The recent breakthroughs in high-resolution electron microscopy (EM) of macromolecules makes it possible to explore fundamental aspects of structural biology at the detailed molecular level. The electrostatic potential (ESP) generated by aligned protein α-helix dipole should be observable in high-resolution EM maps despite the fact that the effect may be partially screened by induced electric fields. Here, we show that aligned backbone dipoles in protein α-helices account for long-range features in the protein ESP functions. Our results are consistent with experimental EM maps and density functional theory calculations, including direct Fourier summation for proper calculation of the ESP due to the nonlocal nature of the ESP function from aligned dipoles and other partial atomic charges.

A higher order shear deformation model of a periodically sectioned plate

This talk develops a higher order shear deformation model of a periodically sectioned plate. A parabolic deformation expression is used with periodic analysis methods to calculate the displacement field as a function of plate spatial location. The problem is formulated by writing the transverse displacement field and the in-plane rotations as a series solution of unknown wave propagation coefficients multiplied by an exponential indexed wavenumber term in the direction of varying structural properties multiplied by an exponential constant term in the direction of constant structural properties. These expansions, along with various structural properties written using Fourier summations, are inserted into the governing differential equations that were derived using Hamilton’s principle. The equations are now algebraic expressions that can be orthogonalized and written in a global matrix format whose solution is the wave propagation coefficients, thus yielding the transverse and in-plane displacements of the...