Fourier Sums Research Articles

Asymptotics of approximation of continuous periodic functions by linear means of their Fourier series

We establish an asymptotic formula for the rate of approximation of Fourier series of individual periodic functions by linear averages with an error , . This formula is applicable to the means of Riesz, Gauss–Weierstrass, Picard and others. The result is new even for the arithmetic means of partial Fourier sums. We use the formula to determine the asymptotic behaviour of functions in a certain class. Separately, we consider the case of positive integral convolution operators.

Modelling Dengue Fever Epidemics in Jakarta

This article thematizes the qualitative estimation of transmission dynamics of Dengue fever. At first, a single-compartment vector-host model for the total infective cases in one homogeneous area has been set up and simplified using a steady-state approximation. Seasonality has been considered in the transmission parameter which is modelled by a Fourier sum. The equilibria of the model and their stability as well as the computation of the basic reproductive number are presented. As a modification, models considering a segmenting of the area in separate districts and their inter-district mobility have been set up, both with and without dependence of the disease transmission parameters on the district. Those have also been analysed in terms of equilibria and stability. Parameter estimation on available Dengue data from Jakarta in the time interval of 2008–2016 using the Metropolis algorithm has been done. $${\mathscr {L}}_1$$ and $${\mathscr {L}}_2$$ comparisons show that using the multi-patch model with district-dependent parameters a decent approximation to the infection data is possible.

A continuous function with universal Fourier series on a given closed set of Lebesgue measure zero

Given a closed set E of Lebesgue measure zero on the unit circle T there is a continuous function f on T such that for every continuous function g on E there is a subsequence of partial Fourier sums Sn+(f,ζ)=∑k=0nfˆ(k)ζkof f, which converges to g uniformly on E. This result completes one result in a recent paper by C. Papachristodoulos and M. Papadimitrakis (2019), see Papachristodoulos and Papadimitrakis (2019). They proved that for a classical one third Cantor set C there is no universal function in the disk algebra. They also proved that for a symmetric Cantor set C∗ on T there is no universal continuous function for the classical symmetric Fourier sums. See also [2].

Приближения сопряженных функций частичными суммами сопряженных рядов Фурье по одной системе алгебраических дробей Чебышева-Маркова

In this paper, we study approximative properties of partial sums of a conjugate Fourier series with respect to a certain system of Chebyshev – Markov algebraic fractions. We cite the main results obtained in known works devoted to studying approximations of conjugate functions in polynomial and rational cases. We introduce a system of Chebyshev – Markov algebraic fractions and construct the corresponding conjugate rational Fourier – Chebyshev series. We obtain an integral representation for approximations of the conjugate function by partial sums of the constructed conjugate series. Moreover, we study approximations of the function, which is conjugate to $|x|^s, 1 < s < 2,$ on the segment $[-1,1],$ by partial sums of the conjugate rational Fourier – Chebyshev series. We obtain an integral representation of approximations, establish their estimates, using the considered method, in dependence of the location of the point x on the segment, and find their asymptotic forms with $n \to \infty$ . We also calculate the optimal value of the parameter that makes deviations of partial sums of the conjugate rational Fourier – Chebyshev series from the conjugate function $|x|^s, 1 < s < 2,$ tend to zero on the segment $[-1,1]$ at the highest possible rate. The obtained results have allowed us to thoroughly study properties of approximations of the function conjugate to $|x|^s, s > 1,$ by partial sums of the conjugate Fourier series with respect to a system of Chebyshev polynomials of the first kind.

APPROXIMATION OF CLASSES OF POISSON INTEGRALS BY REPEATED FEJER SUMS

The paper is devoted to the approximation by arithmetic means of Fourier sums of classes of periodic functions of high smoothness. The simplest example of a linear approximation of continuous periodic functions of a real variable is the approximation by partial sums of the Fourier series. The sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. A significant number of works is devoted to the study of other approximation methods, which are generated by transformations of Fourier sums and allow us to construct trigonometrical polynomials that would be uniformly convergent for each continuous function. Over the past decades, Fejer sums and de la Vallee Poussin sums have been widely studied. One of the most important direction in this field is the study of the asymptotic behavior of upper bounds of deviations of linear means of Fourier sums on different classes of periodic functions. Methods of investigation of integral representations of deviations of trigonometric polynomials generated by linear methods of summation of Fourier series, were originated and developed in the works of S.M. Nikolsky, S.B. Stechkin, N.P. Korneichuk, V.K. Dzadyk and others. The aim of the work systematizes known results related to the approximation of classes of Poisson integrals by arithmetic means of Fourier sums, and presents new facts obtained for particular cases. In the paper is studied the approximative properties of repeated Fejer sums on the classes of periodic analytic functions of real variable. Under certain conditions, we obtained asymptotic formulas for upper bounds of deviations of repeated Fejer sums on classes of Poisson integrals. The obtained formulas provide a solution of the corresponding Kolmogorov-Nikolsky problem without any additional conditions.

Приближение функций комплексного переменного суммами Фурье по ортогональным системам в L2

We obtain sharp inequalities of Jackson-Stechkin type between the best approximations $${E_{n - s - 1}}\left( {{f^{\left( s \right)}}} \right)\left( {s = \overline {0,r} ,r \inℕ } \right)$$ of successive derivatives $${f^{\left( s \right)}}\left( {s = \overline {0,r} ,r \inℕ } \right)$$ of analytic in the disk U functions f ∈ L2(U). Two the following restrictions on f are considered: $${{\rm{\Omega }}_m}{\left( {{f^{\left( r \right)}},t} \right)_2} \le {\rm{\Phi }}\left( t \right),0 < t < 1,$$ with a special module of continuity Ωm of mth order, and $${{\cal K}_m}\left( {{f^{\left( r \right)}},{t^m}} \right) \le {\rm{\Phi }}\left( {{t^m}} \right),0 < t < 1,$$ connected with Peetre $${\cal K}$$ -functional; here Φ is a given majorant.

МЕТОДЫ ПОЛИНОМИЗАЦИИ ФУНКЦИЙ И ИХ ПРИЛОЖЕНИЯ

A class of semicontinuous quasiconvex methods of summation of Fourier – Chebyshev series is studied. Upper bounds are obtained for the norms of the corresponding operators in the space of continuous functions. The convergence of means in the metric of space is established. The summability at break points of the first kind is also considered. Processes for restoring functions from a given sequence of power moments are proposed. Ways of generalizing the results and extending them to the case of summability at Lebesgue points are indicated.

Approximation of Gaussians by spherical Gauss-Laguerre basis in the weighted Hilbert space

This paper is devoted to the study of approximation of Gaussian functions bytheir partial Fourier sums of degree $N \in \mathbb{N}$ with respect to thespherical Gauss-Laguerre (SGL) basis in the weighted Hilbert space$L_2(\mathbb{R}^3, \omega_\lambda)$, where$\omega_\lambda(|\boldsymbol{x}|)=\exp({-|\boldsymbol{x}|^2/\lambda})$,$\lambda>0$. We investigate the behavior of the corresponding error ofapproximation with respect to the scale factor $\lambda$ and order ofexpansion $N$. As interim results we obtained formulas for the Fouriercoefficients of Gaussians with respect to SGL basis in the space$L_2(\mathbb{R}^3, \omega_\lambda)$. Possible application of obtained resultsto the docking problem are described.

Приближение функций частными суммами ряда Фурье по многочленам, ортогональным на произвольных сетках

for arbitrary continuous function f(t) on the segment [−1,1] we construct discrete Fourier sums Sn,N(f,t) on a system of polynomials forming an orthonormal system on non-uniform grids $${T_N} = \left\{{{t_j}} \right\}\matrix{{N - 1} \cr {j = 0} \cr}$$ of N points from segment [−1,1] with weight Δtj=tj+1−tj. Approximation properties of the constructed partial sums Sn,N(f,t) of order n ≤ N−1 are investigated. A two-sided pointwise estimate is obtained for the Lebesgue function Ln,N(t) of considered discrete Fourier sums for $$n = O\left({\delta \matrix{{- 1/5} \cr N \cr}} \right),{\delta _N} = {\max _0} \le j \le N - 1\Delta {t_j}$$. The question of the convergence of Sn,N(f,t) to f(t) is also investigated. In particular, we obtain the deflection estimation of a partial sum Sn,N(f,t) from f(t) for $$n = O\left({\delta \matrix{{- 1/5} \cr N \cr}} \right)$$, which depends on n and the position of a point t ∊ [−1,1].

The Abel summation of the inverse Fourier transform of the homogeneous functions in $R^n$

The Abel summation of the inverse Fourier transform of the homogeneous functions in $R^n$

Approximation properties of some discrete Fourier sums for piecewise smooth discontinuous functions

Approximation properties of some discrete Fourier sums for piecewise smooth discontinuous functions

Estimate of the Lebesgue Function of Fourier Sums in Terms of Modified Meixner Polynomials

The paper is devoted to the study of the approximation properties of Fourier sums in terms of the modified Meixner polynomials m (x), n = 0,1,..., which generate, for α > -1, an orthonormal system on the grid Ωδ = {0, δ, 2δ,...} with weight $${\rho _N}(x) = {e^{ - x}}\frac{{\Gamma (Nx + \alpha + 1)}}{{\Gamma (Nx + 1)}}{(1 - {e^{ - \delta }})^{\alpha + 1}},\;\;\;\;\text{where}\;\;\delta = \frac{1}{N},\;N \geq 1.$$ The main attention is paid to the derivation of a pointwise estimate for the Lebesgue function λ (x) of Fourier sums in terms of the modified Meixner polynomials for x ∈ [θn/2, ∞) and θn = 4n + 2α + 2.

Conditioning of polynomial Fourier sums

In order to measure the stability properties of Fourier sums, we introduce a conditioning of a representation of the solution of the least squares problem. We relate the conditioning of the discrete polynomial Fourier sums with the corresponding continuous one. We study the asymptotic growth of the conditioning in terms of the degree n. For Fourier–Legendre sums, it is $$O(n^{3/2})$$ . In the case of Fourier–Chebyshev sums, it is linear in the degree. For Fourier sums with Chebyshev polynomials of the second kind, it is $$O(n^2)$$ .

Polynomial approximation with Pollaczek–type weights. A survey

The paper deals with weighted polynomial approximation for functions defined on (−1,1), which can grow exponentially both at −1 and at 1. We summarize recent results on function spaces with new moduli of smoothness, estimates for the best approximation, Lagrange interpolation, Fourier sums and Gaussian rules with respect to weights of the form w(x)=(1−x2)βe−(1−x2)−α.

Approximation Properties of Repeated de la Vallée-Poussin Means for Piecewise Smooth Functions

Basing on Fourier’s trigonometric sums and the classical de la Vallee-Poussin means, we introduce the repeated de la Vallee-Poussin means. Under study are the approximation properties of the repeated means for piecewise smooth functions. We prove that the repeated means achieve the rate of approximation for the discontinuous piecewise smooth functions which is one or two order higher than the classical de la Vallee-Poussin means and the partial Fourier sums respectively.

Estimating the Structure of a Discontinuous Layer by Tomographic Methods

The method of constructing a mathematical model of the internal structure of the geological environment is considered for the case, where the function that describes this model has a discontinuity of the first kind. The model is proposed for use in mine seismic tomography. The results of the computing experiment show that even for small orders of Fourier sums, the Fourier coefficients found by means of information about the first times of the arrival of the seismic signal from the sources at the observation points are close to the Fourier coefficients found for the test function, which describes the given terrain image with tectonic damage. The described approaches can be used to improve the mathematical model of the distribution of the slowness of propagation of seismic waves in a given section of the geological environment.

Theory of diffraction of vortex beams from structured apertures and generation of elegant elliptical vortex Hermite-Gaussian beams.

In this work, a comprehensive analytic study of the diffraction of vortex beams from structured apertures is presented. We formulate the near- and far-field diffraction of a vortex beam from an aperture having an arbitrary functionality in the Cartesian coordinates by two general and different approaches. We show that each of the resulting diffraction patterns can be determined by a number of successive derivatives of the 2D Fourier transform of the corresponding hypothetical aperture function or equally can be obtained by a summation of 2D Fourier transforms of the corresponding modified aperture function. We implement both introduced analytic approaches in predicting the diffraction of a vortex beam from an elliptic Gaussian aperture, an elliptic Gaussian phase mask, and a hyperbolic Gaussian phase mask in the near- and far-field regimes. It is shown that the predicted diffraction patterns by both these approaches are exactly the same. It is shown that the diffraction of a vortex beam from an elliptic Gaussian aperture at the far-field regime forms a light beam that belongs to a family of light beams we call elegant elliptical vortex Hermite-Gaussian beams. In addition, the diffractions of a vortex beam from a Fresnel zone plate in general form for the on- and off-axis situations are formulated, and sinusoidal and binary zone plates are investigated in detail. Our general analytic formula can be used for a large variety of apertures including off-center situations and asymmetrical cases.

Menshov Type Correction Theorems for Sequences of Compact Operators

We prove Menshov type "correction" theorems for sequences of compact operators, recovering several results of Fourier series in trigonometric and Walsh systems. The paper clarifies the main ingredient, which is important in the study of such "correction" theorems. That is the weak-$L^1$ estimate for the maximal Fourier sums of indicator functions of some specific sets.

The Unique Solvability of Boundary Value Problems for a 3D Elliptic Equation With Three Singular Coefficients

We state and study several boundary value problems for an elliptic equation with three singular coefficients in a rectangular parallelepiped. Using the method of energy integrals, we prove the uniqueness of solutions of the stated problems. For proving the existence of solutions, we use the spectral Fourier method based on separation of variables. We construct solutions to the stated problems in the form of the sum of the double Fourier—Bessel series. We justify the uniform convergence of the constructed series with the help of asymptotic bounds for Bessel functions of real and imaginary arguments. Using these bounds, we estimate each term of the series, which allows us to prove the convergence of the series and its derivatives up to the second order inclusive and to justify the existence theorem in the class of regular solutions.

Uniform Convergence of Trigonometric Series with General Monotone Coefficients

AbstractWe study criteria for the uniform convergence of trigonometric series with general monotone coefficients. We also obtain necessary and sufficient conditions for a given rate of convergence of partial Fourier sums of such series.