Trigonometric Fourier Series Research Articles

Approximation properties of trigonometric Fourier series in generalized variation classes

Approximation properties of trigonometric Fourier series in generalized variation classes

Estimate of the Convergence Rate in the Riemann Localization Principle for Trigonometric Fourier Series of Continuous Functions

Estimate of the Convergence Rate in the Riemann Localization Principle for Trigonometric Fourier Series of Continuous Functions

POLYNOMIAL SOLUTIONS FOR THE KOLMOGOROV–WIENER PREDICTION OF MODELED SMOOTHED HEAVY-TAIL PROCESS

Nowadays telecommunication traffic in systems with data packet transfer is considered as a heavy-tail random process. In a couple of rather simple models traffic is considered to be stationary one. In our recent papers we generated modeled heavy-tail data, which is based on the smoothing of the fractional Gaussian noise. In particular, the applicability if the continuous Kolmogorov–Wiener filter to the prediction of such data was investigated, the corresponding Wiener–Hopf integral equation was solved on the basis of the truncated Walsh function expansion. However, a question occurs – may another truncated orthogonal function expansion be applied to the problem under consideration? So, the corresponding investigation may be an actual question. In our recent papers we investigated theoretical fundamentals of the Kolmogorov–Wiener filter construction for different models, in particular, on the basis of the truncated polynomial expansion method and on the basis of the truncated trigonometric Fourier series expansion method. In this paper we restrict ourselves to the investigation of the applicability of the truncated polynomial expansion method to the problem under consideration, the corresponding method is based on the Сhebyshev polynomials of the first kind. The applicability of another polynomial or trigonometric expansions to the problem under consideration may be discussed in other papers. The aim of the work is to investigate the applicability of the Galerkin method based on the Chebyshev polynomials of the first kind to the Kolmogorov–Wiener prediction of smoothed heavy-tail data. The methodology consists in the solving of the Wiener–Hopf integral equation on the basis of the truncated polynomial expansion method which is based on the Chebyshev polynomials of the first kind. The scientific novelty consists in the proof of the fact that the Galerkin method based on the Chebyshev polynomials of the first kind may be applied to the Kolmogorov–Wiener prediction of smoothed heavy-tail data. The conclusions are as follows. The truncated polynomial expansion method based on the Chebyshev polynomials of the first kind may give reliable results in the framework of the Kolmogorov–Wiener prediction of smoothed heavy-tail data.

Optimal with Respect to Accuracy Recovery of Some Classes Functions by Fourier Series

Introduction. Function approximation (approximation or restoration) is widely used in data analysis, model building, and forecasting. The goal of function approximation is to find the function that best approximates the original function. This can be useful when the original function is too complex to analyze or when a model needs to be simplified for more efficient computation or interpretation. Function approximation is an important tool in science, engineering, economics, and other fields where data analysis and modeling are required. It allows you to simplify complex functions, identify patterns in the behavior of the object of study, and predict the value of a function beyond the available data. The purpose of the paper is consider the problems of approximation of a function, which on some interval is given by its values in some set of nodal points and belongs to some class of functions by trigonometric Fourier series with a given accuracy and at fulfillment of given constraints on its execution time. The main attention is paid to obtaining estimates of computational complexity (implementation time) and solving the problem of function approximation by Fourier series with a given or maximum possible accuracy using efficient algorithms for solving optimization problems. Results. The general formulation of the problem of approximation of functions by Fourier series in accordance with the technology of solving problems of computational and applied mathematics with specified values of quality characteristics is presented. Estimates of the error of the proposed approximation algorithms using for the computation of Fourier coefficients the optimal in accuracy and close to them quadrature formulas for the computation of integrals from rapidly oscillating functions of the classes of Helder and Lipschitz with given fixed values in the nodes of a fixed grid are given. The corresponding quadrature formulas and constructive estimates of the error of the method of approximation of functions of the specified classes are given. Estimates of computational complexity of the given algorithms are obtained, which allow us to set real constraints on the time of algorithm implementation with a given or maximum possible accuracy. Conclusions. A comprehensive analysis of the quality of the considered algorithms for the approximation of functions by Fourier series using the accuracy-optimal (or close to them) quadrature formulas for the computation of Fourier coefficients for the computation of integrals from rapidly oscillating functions is presented. The estimates of their main characteristics – accuracy and computational complexity – are obtained. Keywords: function approximation, Fourier series, Fourier series coefficients, approximation error, computational complexity.

Trigonometric approximation of signals (functions) belonging to certain Lipschitz spaces using C δ.C operator

Abstract In 2015, Srivastava and Singh [S. K. Srivastava and U. Singh, Trigonometric approximation of periodic functions belonging to weighted Lipschitz class W ⁢ ( L p , Ψ ⁢ ( t ) , β ) W(L^{p},\Psi(t),\beta) , Function Spaces in Analysis, Contemp. Math. 645, American Mathematical Society, Providence 2015, 283–291] determined the order of approximation of periodic functions belonging to W ⁢ ( L p , Ψ ⁢ ( u ) , β ) {W(L^{p},\Psi(u),\beta)} -class, which is a weighted version of Lip ⁢ ( ω ⁢ ( u ) , p ) {\mathrm{Lip}(\omega(u),p)} -class with weight function sin β ⁢ p ⁡ ( y 2 ) {\sin^{\beta p}(\frac{y}{2})} through matrix means of their trigonometric Fourier series. It is a well-known fact that the product summability methods are stronger than the single summability methods, and they can approximate a wider class of functions. Therefore, in this article, an effort is made to determine the degree of approximation for periodic functions from the same weighted Lipschitz class W ⁢ ( L p , Ψ ⁢ ( u ) , β ) {W(L^{p},\Psi(u),\beta)} , p ≥ 1 {p\geq 1} , using C δ . T {C^{\delta}.T} -means of their trigonometric Fourier series.

SOLUTION OF A MIXED BOUNDARY VALUE PROBLEM FOR THE LAPLACE EQUATION BY THE METHOD OF DIVIDING AN OBJECT IN A RECTANGULAR PARALLELEPIPED

The aim of the study is to solve the boundary value problem for the Laplace equation in a rectangular parallelepiped by the method of separation of variables and to estimate the obtained Fourier variable separation constants.
 
 Materials and methods. Methods of mathematical physics were used to solve the boundary value problem for the Laplace equation. The initial problem was divided into three standard ones, in which the inhomogeneous boundary conditions were given only on two parallel sides, for the rest of the problem they being assumed to be equal to 0.
 
 Results. The boundary value problem for the Laplace equation in a rectangular parallelepiped has been broken down into three problems. Partial solutions to these problems under given boundary conditions have been obtained. The Fourier variable separation constants are estimated.
 
 Findings. The solution to the Laplace’s equation for a parallelepiped is the sum of the solutions to three partial problems. The boundary functions of a parallelepiped are odd periodic over two variable functions whose periods are equal to the lengths of the corresponding sides of the parallelepiped. The Fourier constants of partial solutions to the problem are the coefficients of the expansion of the boundary periodic functions of two variables into a trigonometric Fourier series. In two-dimensional series of the solution to the Laplace’s equation for odd-numbered harmonics and for a set of simultaneously even and odd harmonics, the Fourier constants differ only in signs.

On Pringsheim Convergence of a Subsequence of Partial Sums of a Multiple Trigonometric Fourier Series

On Pringsheim Convergence of a Subsequence of Partial Sums of a Multiple Trigonometric Fourier Series

SOME APPROXIMATE PROPERTIES OF CESARO MEANS FOURIER SERIES

The paper discusses the new property of the negative order Cesaro mean of the Fourier trigonometric series

Poroelastic response of unsaturated soils to cyclic loads with arbitrary waveforms

AbstractWe develop an analytical solution to the problem of one‐dimensional consolidation of unsaturated soil subjected to cyclic loads with arbitrary waveforms. The solution predicts the excess pore water and pore air pressures and the accompanying vertical compression in a poroelastic, unsaturated soil material. Cyclic loading occurs in a variety of engineering applications and often generates higher excess pore fluid pressures and larger vertical compression than does a time‐invariant load. In the present study, the loading function is allowed to take on any arbitrary waveform represented by a Fourier trigonometric series. Analytical solution to the boundary‐value problem in one dimension is given in closed form describing the frequency‐independent and frequency‐dependent components of the poroelastic response. We verify the analytical solution through representative examples involving cyclic loads with square and triangular patterns. Apart from the shape of the forcing function, we also investigate the effects of initial water saturation, soil texture, and excitation frequency on the system response.

Almost everywhere divergence of Cesàro means of subsequences of partial sums of trigonometric Fourier series

In this paper, we investigate the relationship between pointwise convergence of the arithmetic means corresponding to the subsequence of partial Fourier sums (Skjf:j∈N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(S_{k_j}f: j\\in \\mathbb {N})$$\\end{document} (for f∈L1(T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\in L^1(\\mathbb {T})$$\\end{document}) and the structure of the chosen subsequence of the sequence of natural numbers (kj:j∈N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(k_j: j\\in \\mathbb {N})$$\\end{document}. More precisely, the problem we deal with is to provide necessary and sufficient conditions for a subsequence N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}$$\\end{document} of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {N}$$\\end{document} that has the following property: for any subsequence N′=(kj:j∈N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N^{\\prime }} = (k_j: j\\in \\mathbb {N})$$\\end{document} of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}$$\\end{document} and any f∈L1(T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\in L^1(\\mathbb {T})$$\\end{document} one has 1N∑j=1NSkjf(x)→f(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{N}\\sum _{j=1}^N S_{k_j}f(x) \\rightarrow f(x)$$\\end{document} for a.e. x∈T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x\\in \\mathbb {T}$$\\end{document}. A direct corollary of this paper’s main theorem is that there exists a subsequence (kj)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(k_j)$$\\end{document} of the sequence of natural numbers and an integrable function f such that the arithmetic means of Skjf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{k_j}f$$\\end{document} do not converge to f almost everywhere. This is a negative answer to a question that originated in an article by Zalcwasser in 1936 Zalcwasser (Stud. Math. 6, 82–88 (1936)) for some increasing sequences (kj)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(k_j)$$\\end{document} of natural numbers.

Averaging equation of three-dimensional tracer transport by a dual approach

The conventional method for calculating averaged tracer concentration in three-dimensional tracer transport equation relies on classical averaging, which averages tracer concentration data obtained from the tracer transport equation in turbulent flow. However, the averaging tracer concentration quantity calculated by this method is not precisely the same as the one obtained by the dual approach. To address this issue, the author proposes a more intricate dual approach for averaging the three-dimensional tracer transport equation in turbulent flow. This paper describes how the author performs two integrations: one from time t to t + r (where r < T, T being the repeated period of fluctuating concentration), and another from time t to time t + T. The fluctuating concentration quantity in turbulent flow is modeled using the trigonometric Fourier series. The resulting three-dimensional mathematical model of tracer transport obtained by this dual approach is more comprehensive than the one obtained by the classical method.

Averaging Navier–Stokes Equations by a Dual Approach

The current three-dimensional averaging mathematical model of flow, also known as the Reynolds averaged Navier–Stokes equations or Reynolds equations, was developed based on the idea of Reynolds in 1895. This model is given by the classical averaging of velocity and pressure parameters from the three-dimensional Navier–Stokes equations. However, by doing this, these averaging parameters obtained by this classical approach are not generalized in comparison to ones estimated by the dual approach. This paper proposes a dual approach to establishing the three-dimensional flow equation. The model setup is more complicated than the classical model in terms of integration because the procedure can be repeated several times. In this paper, the authors perform twice: (1) first, integration of the velocity and pressure parameters from time [Formula: see text] to [Formula: see text], with time [Formula: see text], where [Formula: see text] is the repeated period of parameters; (2) second, integration from time t to [Formula: see text]. Fluctuating quantities such as velocity and pressure in turbulent flow, over time, are simulated using trigonometric Fourier series. The three-dimensional flow model obtained from this dual approach could provide more accurate results than those given by the Reynolds averaged Navier–Stokes equations.

Performance, combustion and emission characteristics of a diesel engine fuelled with diesel fuel + corn oil + alcohol ternary blends.

A blend of diesel fuel and corn oil in the ratio of 80:20 (v/v) is prepared. 1-butanol and 1-pentanol are mixed separately with the binary blend in different ratios (4:96, 7:93, and 10:90 v/v) to prepare ternary blends. Pure diesel fuel and ternary blends are tested at various engine speeds (1000-2500rpm) and at full throttle position. A regression model and its trigonometric Fourier series are proposed to represent the variation of in-cylinder pressure vs. crank angle measured by the author. The regression model and its Fourier series are compared to the Gaussian function of second-order using the in-cylinder pressure data measured by the author and different authors. On average, the ternary blends have lower brake effective efficiency (0.7347 [Formula: see text]-4.0553 [Formula: see text]) and peak heat release rate (5.1113 [Formula: see text]-6.3083 [Formula: see text]), compared to diesel fuel. On average, the ternary blends have a shorter combustion duration (0.4045 [Formula: see text]-7.0236 [Formula: see text]) and longer ignition delay (8.3635 [Formula: see text]-13.9110 [Formula: see text]) relative to diesel fuel. The ternary blends produce lower CO (8.4769 [Formula: see text]-13.1598 [Formula: see text]), HC (30.0073 [Formula: see text]-36.2523 [Formula: see text]), and smoke (4.8566 [Formula: see text]-7.4181 [Formula: see text]) emissions while higher NOX (3.2691 [Formula: see text]-10.8795 [Formula: see text]) emission. The estimated values from the proposed regression model and its Fourier series coincide quite well with in-cylinder pressure data measured by the author and different authors.

A Summation Method for Trigonometric Fourier Series Based on Sinc-Approximations

A Summation Method for Trigonometric Fourier Series Based on Sinc-Approximations

Freya: An educational MATLAB GUI-based tool for generalized Fourier series

The Fourier analysis is a very powerful mathematical tool to decompose functions into their frequency components. Due to this, it has applications in a wide variety of fields inside the realm of science and engineering. As usual, this theory starts with a discussion about the trigonometric Fourier series, the expansion of a function in terms of sines and cosines, and then is generalized in the sense that other functions rather than the trigonometric ones can be used as an orthogonal basis, as the eigenfunctions of some specific Sturm-Liouville problems, such as Bessel functions and Legendre polynomials. In this direction, we present the so- called Freya, an educational graphical user interface (GUI) for the generalized Fourier series developed using the interactive MATLAB (MATrix Laboratory) App Designer environment. We aim to provide a user-friendly tool as a learning aid system for students to gain a comprehensive understanding of the subject as well as for teaching.

LONGITUDINAL TRANSIENT VIBRATIONS OF A FINITE MOMENT ELASTIC ROD

Non-stationary longitudinal vibrations of a moment elastic rod of finite length are investigated. To describe the motion of the rod, the system of equations of the general model of moment elastic thin bodies is used without additional hypotheses. The equations of this model take into account longitudinal movements, changes in the angle of independent microrotation, as well as transverse compression of the rod. The rod material is assumed to be homogeneous and isotropic. The system of equations of motion is supplemented by physical relations that describe the relationship of displacements, changes in angles and transverse compression with forces. In contrast to classical models, in addition to normal forces, additional force factors arise in a moment rod. They are: additional moments, moment cutting forces, moments of moment stresses. Accordingly, in addition to the elastic constants of the material, additional physical parameters of the medium are taken into account, which are necessary when taking into account moment effects in the material. The conditions of generalized hinged support are used as boundary conditions at the ends of the rod. The initial conditions are assumed to be zero. To construct the solution, expansions of the desired functions and the external load into trigonometric Fourier series are used. Substituting these expansions into the original relations leads to a system of equations for the coefficients of time-dependent series. To solve it, the integral Laplace transform a in time is used. As a result, expressions for the required coefficients of expansion series in the image space are found. Each of these expressions is the sum of three products. The factors in these products are the Laplace images of the coefficients of the Fourier expansions for the load and for the influence functions. Influence functions are fundamental solutions (Green's functions) of the problem under study. The original coefficients of the series for the influence functions are found analytically using residues. The final expressions for the coefficients of the expansion series of solutions have the form of convolutions in time. The cores of these integral representations are the original coefficients of the series for the influence functions. As an example, the response of a moment elastic rod to the action of a non-stationary axial load is considered. The results obtained are illustrated graphically. The practical convergence of expansion series is estimated.

Learning Orientation-Aware Distances for Oriented Object Detection

Oriented object detectors have suffered severely from the discontinuous boundary problem for a long time. In this work, we ingeniously avoid this problem by relating regression outputs to regression target orientations. The core idea of our method is to build a contour function which imports orientations and outputs the corresponding distance predictions. Inspired by Fourier transformations, we assume this function can be represented as a linear combination of trigonometric functions and Fourier series. We replace the final 4D layer in the regression branch of fully convolutional one-stage object detector (FCOS) with a Fourier Series Transformation (FST) module and term this new network FCOSF. By this unique design, the regression outputs in FCOSF can adaptively vary according to the regression target orientations. Thus, the discontinuous boundary has no impact on our FCOSF. More importantly, FCOSF avoids building complicated oriented box representations, which usually cause extra computations and ambiguities. With only flipping augmentation and single-scale training and testing, FCOSF with ResNet-50 achieves 73.64% mAP on the DOTA-v1.0 dataset with up to 23.6 FPS speed, surpassing all one-stage oriented object detectors. On the more challenging DOTA-v2.0 dataset, FCOSF also achieves the highest results of 51.75% mAP among one-stage detectors. More experiments on DIOR-R and HRSC2016 are also conducted to verify the robustness of FCOSF. Code and models will be available at https://github.com/DDGRCF/FCOSF.

On absolute summability of infinite series and Fourier series

Abstract In the present paper, a theorem on θ −|T; δ|k summability method of an infinite series is proved, and also by using this method, a result on summability of a trigonometric Fourier series is obtained.

Mean-square Approximation of Iterated Ito and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Integration of Ito Sdes and Semilinear Spdes (third Edition)

This is the third edition of the monograph (first edition 2020, second edition 2021) devoted to the problem of mean-square approximation of iterated Ito and Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. The mentioned problem is considered in the book as applied to the numerical integration of non-commutative Ito stochastic differential equations and semilinear stochastic partial differential equations with nonlinear non-commutative trace class noise. The book opens up a new direction in researching of iterated stochastic integrals. For the first time we use the generalized multiple Fourier series converging in the sense of norm in Hilbert space for the expansion of iterated Ito stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process (Chapter 1). Sections 1.11-1.13 (Chapter 1) are new and generalize the results of Chapter 1 obtained earlier by the author and are also closely related to the multiple Wiener stochastic integral introduced by Ito in 1951. The convergence with probability 1 as well as the convergence in the sense of n-th (n=2, 3,...) moment for the expansion of iterated Ito stochastic integrals have been proved (Chapter 1). Moreover, the rate of both types of convergence has been established. The main difference between the third and second editions of the book is that the third edition includes original material (Chapter 2, Sections 2.10-2.19) on a new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process. The above approach allowed us to generalize some of the author's earlier results and also to make significant progress in solving the problem of series expansion of iterated Stratonovich stochastic integrals. In particular, for iterated Stratonovich stochastic integrals of the fifth and sixth multiplicity, series expansions based on multiple Fourier-Legendre series and multiple trigonometric Fourier series are obtained. In addition, expansions of iterated Stratonovich stochastic integrals of multiplicities 2 to 4 were generalized. These results (Chapter 2) adapt the results of Chapter 1 for iterated Stratonovich stochastic integrals. Two theorems on expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k based on generalized iterated Fourier series with pointwise convergence are formulated and proved (Chapter 2). The results of Chapters 1 and 2 can be considered from the point of view of the Wong-Zakai approximation for the case of a multidimensional Wiener process and the Wiener process approximation based on its series expansion using Legendre polynomials and trigonometric functions. The integration order replacement technique for iterated Ito stochastic integrals has been introduced (Chapter 3). Exact expressions are obtained for the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity k (Chapter 1) and iterated Stratonovich stochastic integrals of multiplicities 1 to 4 (Chapter 5). Furthermore, we provided a significant practical material (Chapter 5) devoted to the expansions and mean-square approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the unified Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4). These approximations were obtained using Legendre polynomials and trigonometric functions. The methods constructed in the book have been compared with some existing methods (Chapter 6). The results of Chapter 1 were applied (Chapter 7) to the approximation of iterated stochastic integrals with respect to the finite-dimensional approximation of the Q-Wiener process (for integrals of multiplicity k) and with respect to the infinite-dimensional Q-Wiener process (for integrals of multiplicities 1 to 3).

On Degree of Approximation of Signals in the Generalized Zygmund Class by Using (E, r)(N,q_n) Mean

In the present article, we have established a result on degree of approximation of function (or signal) in the generalized Zygmund class Zl(m),(l ≥ 1) by using (E,r)(N,qn)- mean of Trigonometric Fourier series.