Trigonometric Fourier Series Research Articles

One-dimensional Fourier series of a function of many variables

It is well known that to each summable in the n-dimensional cube [−π,π]n function f of variables x1,…,xn there corresponds one n-multiple trigonometric Fourier series S[f] with constant coefficients.In the present paper, with the function f we associate n one-dimensional Fourier series S[f]1,…,S[f]n, with respect to variables x1,…,xn, respectively, with nonconstant coefficients and announce the preliminary results. In particular, if a continuous function f is differentiable at some point x=(x1,…,xn), then all one-dimensional Fourier series S[f]1,…,S[f]n converge at x to the value f(x).For illustration we consider the well known example of Ch. Fefferman’s function F(x,y) whose double trigonometric Fourier series S[F] diverges everywhere in the sense of Prinsheim. Namely, we establish the simultaneous convergence of the one-dimensional Fourier series S[F]1 and S[F]2 at almost all points (x,y)∈[−π,π]2 to the values F(x,y).

Investigation of three-layer rectangular panel bending by variation method

The task of transverse bending of a rectangular three-layer panel with the lightweight filler is considered. The differential equation system of transverse bending of the panel is generated within the scope of Lagrange variational principle. An explicit solution of the differential equation system in double trigonometric Fourier series is obtained. It is shown that the maximum deflection of the three-layer panel can be compared with the three-layer beam deflection, with a multiple panel width excess over its length.

Полевая аналитическая модель магнитоэлектрического вентильного двигателя

Phenomena occurring in a switched magnetoelectric motor can be considered in its non-magnetic gap, which should also embrace the layer of a high-energy magnet. Complex periodic potential functions serve as a mathematical basis for analytically solving the Dirichlet problem in a non-magnetic gap having the shape of an infinite horizontal strip with boundaries confined by two parallel straight lines. The imaginary components of the complex potential functions at the boundaries of the above-mentioned strip represented by trigonometric Fourier series are the known scalar magnetic potentials of the stator winding and permanent magnets on the rotor, which are the magnetic field sources.A comparatively large width of the considered strip results in that the magnetic field in it is two-dimensional in nature due to the presence of permanent magnets. The boundary sources of magnetic field can be used for generating additional magnetic fields corresponding to the stator’s virtual teeth. The geometrical structure of the virtual teeth, which determines the amplitudes of tooth magnetic inductions, can be accurately taken into account using conformal transformation of a non-uniform air gap into an infinite strip or approximately based on the specific permeance method.

Absolute weighted arithmetic mean summability factors of infinite series and trigonometric fourier series

In this paper, we generalized a known theorem dealing with absolute weighted arithmetic mean summability of infinite series by using a quasi-f-power increasing sequence instead of a quasi-?-power increasing sequence. And we applied it to the trigonometric Fourier series

Аналитическая модель магнитного поля в ярме магнитоэлектрического вентильного двигателя

The processes occurring in a switched permanent magnet motor can be considered in the nonmagnetic gap. The high-energy magnet layer should also be related to this gap. Complex periodic potential functions serve as the mathematical basis for analytically solving the Dirichlet problem both in the nonmagnetic gap represented by an infinite horizontal strip with the boundaries delineated by two parallel straight lines and in similar strips with homogeneous magnetic permeability of the stator and rotor yokes. The imaginary components of the complex potential functions at the boundaries of the above-mentioned strips represented by trigonometric Fourier series are the known scalar magnetic potentials of magnetic field sources, namely, the stator winding and permanent magnets on the rotor. To fulfill the magnetic field boundary conditions in the yoke belts, a procedure for reducing these fields to the magnetic field in the air gap is implemented by means of special coefficients for each of the harmonic components in the magnetic induction radial and tangential components. The value of magnetic induction uniformly distributed in the yoke body is selected from the maximal magnetic induction inside the yoke without taking into account the induction impulse values at the boundaries with the air gap. Graphs showing the distribution of magnetic fields in the yokes corresponding to the obtained analytical correlations are given.

ON Λ-CONVERGENCE ALMOST EVERYWHERE OF MULTIPLE TRIGONOMETRIC FOURIER SERIES

We consider one type of convergence of multiple trigonometric Fourier series intermediate between the convergence over cubes and the \(\lambda \)-convergence for \(\lambda >1\). The well-known result on the almost everywhere convergence over cubes of Fourier series of functions from the class \( L (\ln ^ + L) ^ d \ln ^ + \ln ^ + \ln ^ + L ([0,2 \pi)^d ) \) has been generalized to the case of the \( \Lambda \)-convergence for some sequences \(\Lambda\).

A Perturbative-Based Generalized Series Expansion in Terms of Non-Orthogonal Component Functions

In this paper we present a generalized perturbative approximate series expansion in terms of non-orthogonal component functions. The expansion is based on a perturbative formulation where, in the non-orthogonal case, the contribution of a given component function, at each point, in the time domain or frequency in the Fourier domain, is assumed to be perturbed by contributions from the other component functions in the set. In the case of orthogonal basis functions, the formulation reduces to the non-perturbative case approximate series expansion. Application of the series expansion is demonstrated in the context of two non-orthogonal component function sets. The technique is applied to a series of non-orthogonalized Bessel functions of the first kind that are used to construct a compound function for which the coefficients are determined utilizing the proposed approach. In a second application, the technique is applied to an example associated with the inverse problem in electrophysiology and is demonstrated through decomposition of a compound evoked potential from a peripheral nerve trunk in terms of contributing evoked potentials from individual nerve fibers of varying diameter. An additional application of the perturbative approximation is illustrated in the context of a trigonometric Fourier series representation of a continuous time signal where the technique is used to compute an approximation of the Fourier series coefficients. From these examples, it will be demonstrated that in the case of non-orthogonal component functions, the technique performs significantly better than the generalized Fourier series which can yield nonsensical results.

The multipliers of multiple trigonometric Fourier series

Abstract We study the multipliers of multiple Fourier series for a regular system on anisotropic Lorentz spaces. In particular, the sufficient conditions for a sequence of complex numbers {λk}k∈Zn in order to make it a multiplier of multiple trigonometric Fourier series from Lp[0; 1]n to Lq[0; 1]n , p > q. These conditions include conditions Lizorkin theorem on multipliers.

On One Two-dimensional Binary Mixture’s Motion in a Flat Layer

In this paper is estimated a special solution for solving thermal diffusion equations, that describe motion of binary mixture in a flat layer. When Reynolds number (Re → 0) is small, it is possible to simplify these equations to some easier problems. In solving process to find pressure it is necessary to solve an inverse problem. Answers for non-stationary regime are presented in trigonometric Fourier series.

Determination of Jumps in Terms of Linear Operators

A theorem of Lukacs [J. Reine Angew. Math., 150, 107–112 (1920)] states that the partial sums of conjugate Fourier series of a periodic Lebesgue integrable function f diverge with a logarithmic rate at the points of discontinuity of f of the first kind. Moricz [Acta Math. Hung., 98, 259–262 (2003)] proved a similar theorem for the rectangular partial sums of double conjugate trigonometric Fourier series. We consider analogs of the Moricz theorem for the generalized Cesaro means and positive linear means. In the present paper we prove a similar theorem in terms of linear operators satisfying certain conditions.

A simplified nonlinear model of the Marangoni instability in gas absorption

The process of gas absorption into initially motionless liquid layer is investigated. The convective instability caused by the temperature dependence of the surface tension. The critical time of transition of the process to unstable convective regime, as well as the intensity of mass transfer in a surface convection are estimated numerically. The mathematical model includes the equations of convective diffusion, thermal conduction and fluid motion. The problem was solved numerically in the two-dimensional formulation. In the coordinate along the interface the concentration of the absorbed substance is represented by three terms of the trigonometric Fourier series. A difference approximation of equations with an exponentially changing grid in the direction normal to the interface is used. The simulations results agree with the well-known experimental data on the absorption of carbon dioxide in water.

Degree of approximation of functions belonging to $$Lip(\omega (t),p)$$ L i p ( ω ( t ) , p ) -class by linear operators based on Fourier series

Fourier series and operators based on it are of great importance in both theoretical and applied mathematics because they can be considered as representation of a function or signal. In this paper, we estimate the degree of approximation of \(f \in Lip(\omega (t),p)\)-class and its conjugate \(\tilde{f},\) by matrix means of their trigonometric Fourier series by relaxing the conditions on \(\omega (t)\) imposed by the earlier researchers. We also discuss some results which are analogous to our results.

Contact problems of the sliding of a punch with a periodic relief on a viscoelastic half-plane

A method is proposed for the analytical solution of problems of the sliding of a punch with a periodic relief on a viscoelastic half-plane under conditions when they are in complete contact as well as problems on the loading of a viscoelastic half-plane with a piecewise-constant pressure that moves at a constant velocity on its surface. The problems are considered in a quasistatic formulation. The method is based on the expansion of the sought functions in trigonometric Fourier series and finding the relation between the displacements of the boundary and the contact pressures. Examples of different shapes of punches are considered and the dependencies of the contact characteristics and the mechaical component of the friction force on the sliding velocity, the viscosity parameters of the base and the geometric characteristics of the surface are studied.

Analytical solution for simply supported laminated composite plates based on partial Layerwise theory

Anisotropic composition of layered composite plates requires more accurate mathematical and cal- culation models. By applying Reddy's layerwise theory we can cover a wide range of problems of layered composite plates with arbitrary arrangement of layers through the plate thickness. The analytical method of solving bending equations in Layerwise Theory is based on the assumed displacement field in the form of the double trigonometric Fourier's series. The analytical solution can be used as a test solution for solutions obtained by using numerical methods, including finite element method. For Partial Layerwise Theory, the paper presents the equations of bending for laminated composite plates and the algorithm for calculation of deflections and stresses in an arbitrary cross section of a simply supported rectangular plate loaded with distributed load. On the basis of the algorithm presented in this paper, author has prepared Fortran program called ANSLACOP (ANalytical SOlution of LAminated COmposite Plates), whose structure will be presented in the paper. It is presented that the solution obtained by using this program very quickly converges depending on the adopted number of members of double trigonometric series.

Sufficient Conditions for Convergence Almost Everywhere of Multiple Trigonometric Fourier Series with Lacunary Sequence of Partial Sums

Sufficient conditions for the convergence (almost everywhere) of multiple trigonometric Fourier series of functions $f$ in the classes $L_p$, $p>1$, are obtained in the case where rectangular partial sums $S_n(x;f)$ of this series have numbers $n=(n_1,\dots,n_N)\in \mathbb Z^N$, $N\geq 3$, such that of $N$ components only $k$ ($1\leq k\leq N-2$) are elements of some lacunary sequences. Earlier, in the case where $N-1$ components of the number $n$ are elements of lacunary sequences, convergence almost everywhere for multiple Fourier series was obtained for functions in the classes $L_p$, $p>1$, by M. Kojima (1977), and for functions in Orlizc classes by D. K. Sanadze, Sh. V. Kheladze (1977) and N. Yu. Antonov (2014). Note that presence of two or more “free” components in the number $n$, as follows from the results by Ch. Fefferman (1971) and M. Kojima (1977), does not guarantee the convergence almost everywhere of $S_n(x;f)$ for $N\geq 3$ even in the class of continuous functions.

On β-absolute convergence of Vilenkin-Fourier series with small gaps

The study of absolute convergence of Fourier series is one of the most important problems of Fourier Analysis and the problem has been studied intensively by many researchers in the setting of circle group in particular and classical groups in general. Recently, in [Math. Inequal. Appl. 17 (2) (2014), 749-760], we have studied the β-absolute convergence (0 < β ≤ 2) of Vilenkin-Fourier series for the functions of various classes of functions of generalized bounded fluctuation and given sufficient conditions in terms of modulus of continuity. In this paper, we prove that this is a matter only of local fluctuation for functions with the Vilenkin-Fourier series lacunary with small gaps. Our results, as in the case of trigonometric Fourier series, illustrate the interconnection between 'localness' of the hypothesis and type of lacunarity and allow us to interpolate the results.

Lp(T2)-수렴성과 모리츠에 관하여

This paper is concerned with the convergence of double trigonometric series and Fourier series. Since the beginning of the 20th century, many authors have studied on those series. Also, Ferenc <TEX>$M{\acute{o}}ricz$</TEX> has studied the convergence of double trigonometric series and double Fourier series so far. We consider <TEX>$L^p(T^2)$</TEX>-convergence results focused on the Ferenc <TEX>$M{\acute{o}}ricz^{\prime}s$</TEX> studies from the second half of the 20th century up to now. In section 2, we reintroduce some of Ferenc <TEX>$M{\acute{o}}ricz^{\prime}s$</TEX> remarkable theorems. Also we investigate his several important results. In conclusion, we investigate his research trends and the simple minor genealogy from J. B. Joseph Fourier to Ferenc <TEX>$M{\acute{o}}ricz$</TEX>. In addition, we present the research minor lineage of his study on <TEX>$L^p(T^2)$</TEX>-convergence.

Differential quadrature procedure for in-plane vibration analysis of variable thickness circular arches traversed by a moving point load

Point discretization methods such as the differential quadrature method (DQM) are well known to have difficulties in solving partial differential equations that involve the Dirac-delta function because the Dirac-delta function is a generalized singularity function and it cannot be discretized directly using the DQM. To overcome this difficulty, a simple differential quadrature methodology is proposed in this study, where the Dirac-delta function is expanded into a Fourier trigonometric series. By expanding the Dirac-delta function into a Fourier trigonometric series, this singular function is treated as non-singular functions, which can be discretized easily and directly using the DQM. The applicability of the proposed method is demonstrated by the in-plane vibration analysis of variable thickness circular arches traversed by a moving point load. The numerical results show that the proposed method is highly accurate and reliable.

Two-dimensional Motion of Binary Mixture such as Hiemenz in a Flat Layer

This paper considers solution of thermal diffusion equations in a special type, which describes twodimensional motion of binary mixture in a flat channel. Substituting this solution to equations of motion and heat and mass transfer equations results initial-boundary problems for unknown functions as velocity, pressure, temperature and concentration. If assume that Reynolds number is small (creeping motion),these problems become linear. In addition, they are inverse since unsteady pressure gradient is also desired. Solution of the problem is obtained by using trigonometric Fourier series, which are rapidly convergent for any time value. Exact solution of the stationary and non-stationary problems is presented.

Summability of trigonometric Fourier series at -points and a generalization of the Abel–Poisson method

We study the convergence of linear means of the Fourier series of a function to as at all points at which the derivative exists (i. e. at the -points). Sufficient conditions for the convergence are stated in terms of the factors and, in the case of , in terms of the condition that the functions and belong to the Wiener algebra . We also study a new problem concerning the convergence of means of the Abel– Poisson type, , as depending on the growth of the function on the semi-axis. It turns out that cannot differ substantially from a power-law function.