Trigonometric Fourier Series Research Articles

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.

On the generalized absolute convergence of Fourier series

Abstract In the present paper the sufficient conditions are obtained for the generalized r-absolute convergence ( 0 < r < 2 {0<r<2} ) of the single Fourier trigonometric series in terms of the modulus of δ-variation of a function. It is proved that these conditions are unimprovable in a certain sense. The classical results of Berstein, Szasz, Zygmund and others, related to the absolute convergence of single trigonometric Fourier series, were previously generalized by [L. Gogoladze and R. Meskhia, On the absolute convergence of trigonometric Fourier series, Proc. A. Razmadze Math. Inst. 141 2006, 29–40].

On Exponential Summability of Rectangular Partial Sums of Double Trigonometric Fourier Series

In this paper, we study the a.e. exponential strong summability problem for the rectangular partial sums of double trigonometric Fourier series of functions in L logL.

An abstract Kolmogorov theorem, and an application to metric spaces and topological groups

Using the method of averaging over the supports of -functions, we generalise Kolmogorov’s fundamental theorem on the existence of a trigonometric Fourier series that diverges almost everywhere to any bounded biorthonormal systems of complex-valued functions on an arbitrary measurable space. The abstract Kolmogorov theorem thus obtained is applied to construct divergent Fourier series on metric spaces and topological groups. We establish the existence of a Fourier series in the system of characters of an arbitrary compact Abelian group that is divergent almost everywhere. Bibliography: 37 titles.

Summability Methods for Elastic Local Fields in Periodic Heterogeneous Materials

In the past, Eshelby’s method for a single inclusion in an infinite medium has been extended to periodic heterogeneous materials for the evaluation of global properties. In this work, the extended Eshelby’s method is used for the evaluation of local fields such as strain and stress in periodic heterogeneous materials characterized by unit cells containing multiple fibres or voids with different geometric and mechanical properties. The proposed method provides Fourier coefficients that are used to construct partial sums of three-dimensional trigonometric Fourier series of local fields. These partial sums exhibit unwanted effects such as the Gibbs phenomenon. In order to attenuate these effects, the behaviour of iterated Fejér partial sums and means of the Riesz summability method is investigated. Extensive numerical examples on both a multiphase composite and a material with voids are provided: in the examples, partial sums, iterated Fejér partial sums, and Riesz means for the local stress are compared with FEM solutions. The numerical comparison shows Riesz means perform better than partial sums and iterated Fejér partial sums and are effective in approximating elastic local fields in periodic heterogeneous solids.

On Absolute Summability of Factored Infinite Series and Trigonometric Fourier Series

In this paper, a known result dealing with \(|\bar{N},p_{n};\theta _{n}|_{k}\) summability factors of infinite series, has been proved under weaker conditions. And then this result has been applied to the trigonometric Fourier series. Some new results have also been obtained.

Синхронизация временных интервалов в канале связи с многопутевым распространением сигналов

It is shown that the need for synchronization in frequency and time of the transmitter and receiver data transmission system with orthogonal signals is caused by signal representation by trigonometric Fourier series. It is suggested to transmit data by means of the signals, which are obtained as a result of displacement of the impulse response of the channel-forming equipment at multiple intervals. The algorithm, which allows determining the time delay of the required signal relative to the reference oscillation, is considered. The block diagram of the synchronization device, that implements the algorithm, has been worked out. The method of determining the arrival time of useful and reflected signals to the receiver input, which used to reduce the influence of multipath propagation of the signal on the synchronization device, has been worked out.

Cesàro Means of Subsequences of Partial Sums of Trigonometric Fourier Series

In 1936 Zygmunt Zalcwasser asked, with respect to the trigonometric system, how “rare” can a sequence of strictly monotone increasing integers \((n_j)\) be such that the almost everywhere relation \(\frac{1}{N}\sum _{j=1}^N S_{n_j}f \rightarrow f\) is fulfilled for each integrable function f. In this paper, we give an answer to this question. It follows from the main result that this a.e. relation holds for every integrable function f and lacunary sequence \((n_j)\) of natural numbers.

Local stress in periodic composites via the Riesz summability method

Abstract Different summation methods of multidimensional trigonometric Fourier series are adopted for approximating the local strain and stress in elastic periodic composites. In fact, approximating the exact solution of a problem with the original partial sums of Fourier series can produce unwanted effects, such as the Gibbs phenomenon at the jump discontinuities that may occur at the interface between the constituents of the composites. Nevertheless, the Fourier coefficients of the original partial sums contain enough information to provide better estimates via the summability methods. In the means provided by the summability methods, the Fourier coefficients are suitably weighted by reducing windows, involving a regularization of the numerical solution. First, a complete solution method is used in order to determine some Fourier coefficients of the local strain in periodic composites. Next, these Fourier coefficients are used to construct suitable summations and means exhibiting better convergence properties than those of the original partial sums of Fourier series. The proposed method, valid for a great variety of the geometry of the constituents embedded in the matrix, is here applied to unidirectional composites with cylindrical fibers. The behavior of the iterated Fejer partial sums, Fejer and Riesz means is investigated in order to improve the convergence, observing that the Riesz means have better convergence properties than those of the original partial sums of Fourier series.

Теорема равносходимости для интегрального оператора с кусочно-постоянным ядром

Работа посвящена теореме равносходимости разложений в тригонометрические ряды Фурье и по собственным и присоединенным функциям одного интегрального оператора $A$, ядро которого терпит скачки на сторонах квадрата, вписанного в единичный квадрат. Рассматривается эквивалентный ему интегральный оператор в пространстве вектор-функций размерности 4. Этот оператор замечателен тем, что компоненты его ядра терпят разрывы лишь на линии $t=x$. Находятся необходимые и достаточные условия для обращения оператора $A$. Это условие есть отличие от нуля одного определителя четвертого порядка. Изучается резольвента Фредгольма оператора $A$. Найдена формула для резольвенты. Ее нахождение сводится к решению краевой задачи для дифференциальной системы первого порядка в пространстве вектор-функций размерности четыре. Проводится преобразование этой краевой задачи, которое помогает справиться с трудностями, возникающими при ее решении. Получены также условия, аналогичные условиям регулярности по Биркгофу. Они связаны с отличием от нуля некоторых легко считаемых определителей четвертого порядка. При выполнении этих условий имеет место некоторая оценка для определителя, нули которого являются собственными значениями рассматриваемой краевой задачи. Приводится теорема равносходимости для оператора $A$. Основной метод, применяемый при доказательстве теоремы, - метод Коши - Пуанкаре интегрирования резольвенты изучаемого оператора по расширяющимся контурам в комплексной плоскости спектрального параметра. В работе также приведен пример интегрального оператора с кусочно-постоянным ядром, которое удовлетворяет всем требованиям, полученным в ходе работы.

On Exponential Summability of Rectangular Partial Sums of Double Trigonometric Fourier Series

On Exponential Summability of Rectangular Partial Sums of Double Trigonometric Fourier Series

On the generalization of the Salem test

The assertion that the Salem test [5] for the uniform convergence of a trigonometric Fourier series is improvable, is proved. In particular, an example of a continuous function, which does not fulfill the condition of the Salem test but satisfies the condition of the generalized Salem test [10], is constructed. Besides, the theorem which improves Golubov’s [3,4] result for continuous functions of two variables, is given.

Sufficient conditions for convergence of multiple Fourier series with Jk-lacunary sequence of rectangular partial sums in terms of Weyl multipliers

We obtain sufficient conditions for convergence (almost everywhere) of multiple trigonometric Fourier series of functions $f$ in $L_2$ in terms of Weyl multipliers. We consider the case where rectangular partial sums of Fourier series $S_n(x;f)$ have indices $n=(n_1,\dots,n_N) \in \mathbb Z^N$, $N\ge 3$, in which $k$ $(1\leq k\leq N-2)$ components on the places $\{j_1,\dots,j_k\}=J_k \subset \{1,\dots,N\} = M$ are elements of (single) lacunary sequences (i.e., we consider the, so called, multiple Fourier series with $J_k$-lacunary sequence of partial sums). We prove that for any sample $J_k\subset M$ the Weyl multiplier for convergence of these series has the form $W(\nu)=\prod \limits_{j=1}^{N-k} \log(|\nu_{{\alpha}_j}|+2)$, where $\alpha_j\in M\setminus J_k $, $\nu=(\nu_1,\dots,\nu_N)\in{\mathbb Z}^N$. So, the "one-dimensional" Weyl multiplier -- $\log(|\cdot|+2)$ -- presents in $W(\nu)$ only on the places of "free" (nonlacunary) components of the vector $\nu$. Earlier, in the case where $N-1$ components of the index $n$ are elements of lacunary sequences, convergence almost everywhere for multiple Fourier series was obtained in 1977 by M.Kojima in the classes $L_p$, $p>1$, and by D.K.Sanadze, Sh.V.Kheladze in Orlizc class. Note, that presence of two or more "free" components in the index $n$ (as follows from the results by Ch.Fefferman (1971)) does not guarantee the convergence almost everywhere of $S_n(x;f)$ for $N\geq 3$ even in the class of continuous functions.

On absolute Riesz summability factors of infinite series and their application to Fourier series

Abstract In this paper, some known results on the absolute Riesz summability factors of infinite series and trigonometric Fourier series have been generalized for the | N ¯ , p n ; θ n | k {\lvert\bar{N},p_{n};\theta_{n}\rvert_{k}} summability method. Some new and known results are also obtained.

Generalized trigonometric Fourier-series method with automatic time step control for non-linear point kinetics equations

The trigonometric Fourier-series method (TFS) is generalized to provide approximate solutions for non-linear point kinetics equations with feedback using varying step sizes. This method can provide a very stable solution against the size of the discrete time step allowing much larger step sizes to be used. Systems of the point kinetics equations are solved using Fourier-series expansion over a partition of the total time interval. The approximate solution requires determining the series coefficients over each time step in that partition. These coefficients are determined using the high-order derivatives of the solution vector at the beginning of the time step introducing a system of linear algebraic equations to be solved at each step. This system is similar to the Vandermonde system. Two successive orders of the partial sums are used to estimate the local truncation error. This error and some other constrains are used to produce the largest step size allowable at each step while keeping the error within a specific tolerance. The process of calculating suitable step sizes should be automatic and inexpensive. Convergence and stability of the proposed method are discussed and a new formula is introduced to maintain stability. The proposed method solves the general linear and non-linear kinetics problems. The method has been applied to five different types of reactivities including step/ramp insertions with temperature feedback. The method is seemed to be valid for larger time intervals than those used in the conventional numerical integration, and is thus useful in reducing computing time. Computational results are found to be consistent with the analysis, they demonstrate that the convergence of the iteration scheme can be accelerated and the resulting computing times can be greatly reduced while maintain computational accuracy.

Approximation of the functions in weighted Lebesgue spaces with variable exponent

In the present work, we investigate the approximation problems of the functions by Fejér, and Zygmund means of Fourier trigonometric series in weighted Lebesgue spaces with variable exponents and of the functions by Fejér and Abel–Poisson sums of Faber series in weighted Smirnov classes with variable exponents defined on simply connected domains with a Dini-smooth boundary of the complex plane.

Convergence almost everywhere of multiple Fourier series over cubes

We study convergence almost everywhere of multiple trigonometric Fourier series over cubes defined on the d d -dimensional torus T d \mathbb {T}^d . We provide a new approach which allows us to prove the novel interpolation estimates for the Carleson maximal operators generated by the partial sums of the multiple Fourier series and all its conjugate series. Combining these estimates we show that these operators are bounded from a variant of the Arias-de-Reyna space Q A d Q\!A^d to the weak L 1 L^1 -space on T d \mathbb {T}^d . This implies that the multiple Fourier series of every function f ∈ Q A d f\in Q\!A^d and all its conjugate series converge over cubes almost everywhere. By a close analysis of the space Q A d Q\!A^d we prove that it contains a Lorentz space that strictly contains the Orlicz space L ( log L ) d log ⁡ log ⁡ log L ( T d ) L(\log \,L)^{d} \log \log \log \,L(\mathbb {T}^d) . This yields a significant improvement of a deep theorem proved by Antonov which was the best known result on the convergence of multiple Fourier series over cubes.

Addition to V. P. Zastavnyi’s paper “Estimates for sums of moduli of blocks in trigonometric Fourier series”

Results on the integrability and integrability with power weight for sums of moduli of blocks from the series Ʃ1/k sin kx are extended to the class of series Ʃb k sin kx with coefficients b k of a more general form.

On the divergence of trigonometric Fourier series in classes ϕ(L) close to L

We show the unimprovability of a theorem on sufficient convergence conditions for the trigonometric Fourier series of a function in classes ϕ(L) in the case when the class ϕ(L) is “close” to L.

Dynamic Analysis of Composite Rods Under Thermal and Force Loads

We propose a general approach to the solution of direct dynamic problems of the numerical analysis of composite rods subjected to thermal and force loads for a large variety of their materials and loading conditions. We take into account the shear effects and interactions between the elements of structures and supporting media. The proposed inhomogeneous rods are regarded as elements of rod systems and characterized by higher parameters of strength and stiffness and lower costs of their production as compared with homogeneous elements. In the equations of motion and physical relations, we introduce four functional characteristics of stiffness and viscosity and three mass functional characteristics equivalently reflecting the dynamic deformation of composite rods with the help of a one-dimensional model. With the help of trigonometric Fourier series, the dynamic loads, displacements, and temperature are represented as products of functions of the coordinate and time. The solution of the homogeneous problem is obtained via the matrizant of the system of equations of the first order. Partial solutions for loads of various types and generalized temperature loads are obtained on the basis of approximation of given quantities and the required displacements by trigonometric series.