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The article is devoted to the problem of approximation of classes of periodic functions by rectangular linear means of Fourier series. Asymptotic equalities are found for upper bounds of deviations in the uniform metric of rectangular Fejér means on classes of periodic functions of several variables generated by sequences that tend to zero at the rate of geometric progression. In one-dimensional cases, these classes consist of Poisson integrals, namely functions that can be regularly extended in the fixed strip of a complex plane. [ABSTRACT FROM AUTHOR]

This paper is devoted to solving one of the extremal problems in the theory of approximation of functional classes by linear methods of summation of the Fourier series in the integral metric, namely, approximation of classes Lβ,1ψ by biharmonic Poisson integrals. As a result of the research, we have found the asymptotic equalities for the approximation values of classes of (ψ, β)-differentiable functions by biharmonic Poisson integrals, that is, have found solutions of the Kolmogorov-Nikol’skii problem for biharmonic Poisson integrals on classes Lβ,1ψ in the integral metric. [ABSTRACT FROM AUTHOR]

The paper is devoted to the investigation of problem of approximation of continuous periodic functions by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. The simplest example of a linear approximation of periodic functions is the approximation of functions by partial sums of their Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. Therefore, many studies devoted to the research of the approximative properties of approximation methods, which are generated by transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for the whole class of continuous functions. Particularly, Fej'er sums have been widely studied recently. One of the important problems in this area is the study of asymptotic behavior of the sharp upper bounds over a given class of functions of deviations of the trigonometric polynomials. In the paper, we study upper asymptotic estimates for deviations between a function and the Fej'er means for the Fourier series of the function. The asymptotic behavior is considered for the functions represented by the Poisson integrals of periodic functions of a real variable. The mentioned classes consist of analytic functions of a real variable. These functions can be regularly extended into the corresponding strip of the complex plane. An asymptotic equality for the upper bounds of Fej'er means deviations on classes of Poisson integrals was obtained. [ABSTRACT FROM AUTHOR]

We essentially extend and improve the classical result of G. H. Hardy and J. E. Littlewood on strong summability of Fourier series. We will present an estimation of the generalized strong mean (H, Φ) as an approximation version of the Totik type generalization of the result of G. H. Hardy, J. E. Littlewood, in case of integrable functions from LΨ. As a measure of such approximation we will use the function constructed by function Ψ complementary to Φ on the base of definition of the LΨ points. Some corollary and remarks will also be given. [ABSTRACT FROM AUTHOR]

We study an inverse problem of recovering initial datum in a one-dimensional linear evolution equation with the Dirichlet boundary conditions when the solution to the equation is known only at a suitably fixed space location and suitably chosen finitely many later time instances. To be more explicit, we consider a one-dimensional linear evolutionary equation involving a Dirichlet fractional Laplacian and an unknown initial datum f which is assumed to be in a suitable subset of a Sobolev space, and then construct n future times so that from the known values of the solution at a suitably fixed space location and at these n future times, we recover f with a desired accuracy. [ABSTRACT FROM AUTHOR]

In the present paper, we establish a theorem on best approximation of a function g 2 Bλ q (Lr) of its Fourier series. Our main theorem generalizes some known results of this direction of work. Thus, the results of [10], [26] and [27] become the particular case of our main Theorem 3.1. [ABSTRACT FROM AUTHOR]

We show that the problems of approximating the shortest and closest vector in a lattice to within a factor of √n lie in NP intersect coNP. The result (almost) subsumes the three mutually-incomparable previous results regarding these lattice problems: Banaszczyk [1993], Goldreich and Goldwasser [2000], and Aharonov and Regev [2003]. Our technique is based on a simple fact regarding succinct approximation of functions using their Fourier series over the lattice. This technique might be useful elsewhere -- we demonstrate this by giving a simple and efficient algorithm for one other lattice problem (CVPP) improving on a previous result of Regev [2003]. An interesting fact is that our result emerged from a "dequantization" of our previous quantum result in Aharonov and Regev [2003]. This route to proving purely classical results might be beneficial elsewhere. [ABSTRACT FROM AUTHOR]

In this paper, a generalized switching modification method is proposed to reduce the root-mean-square (rms) current flowing into a dc-link capacitor in a dc–dc–ac structure consisting of a boost converter and a three-phase inverter. To analyze the current flowing into the capacitor, we analyze the currents at the respective converter and inverter sides. Based on this analysis, we derive an analytical method to find the modification time between the converter and inverter pulsewidth modulation (PWM) carriers through a Fourier series and approximation to facilitate real-time implementation. Real-time implementation of the modification time calculated by the derived method is described. To verify the validity of the proposed method, experimental results are presented based on a 1-kVA dc–dc–ac prototype. The proposed method has a better capacitor rms current reduction performance than the conventional method in various inverter power factors and PWM strategies. Especially, it has an excellent performance in mid-power factor level (0.95–0.1). [ABSTRACT FROM AUTHOR]

In the paper, we solve one extremal problem of the theory of approximation of functional classes by linear methods. Namely, questions are investigated concerning the approximation of classes of differentiable functions by l-methods of summation for their Fourier series, that are defined by the set ʌ = {lδ(·)} of continuous on [0,∞) functions depending on a real parameter δ. The Kolmogorov-Nikol’skii problem is considered, that is one of the special problems among the extremal problems of the theory of approximation. That is, the problem of finding of asymptotic equalities for the quantity ξ(N;Uδ)X = sup ∣∣f(·) − Uδ (f;·;L)∣∣X, where X is a normalized space, N⊂X is a given function class, Uδ( f ; x;ʌ) is a specific method of summation of the Fourier series. In particular, in the paper we investigate approximative properties of the three-harmonic Poisson integrals on the Weyl-Nagy classes. The asymptotic formulas are obtained for the upper bounds of deviations of the three-harmonic Poisson integrals from functions from the classes Wrb,∞. These formulas provide a solution of the corresponding Kolmogorov-Nikol’skii problem. Methods of investigation for such extremal problems of the theory of approximation arised and got their development owing to the papers of A.N. Kolmogorov, S.M. Nikol’skii, S.B. Stechkin, N.P. Korneichuk, V.K. Dzyadyk, A.I. Stepanets and others. But these methods are used for the approximations by linear methods defined by triangular matrices. In this paper we modified the mentioned above methods in order to use them while dealing with the summation methods defined by a set of functions of a natural argument. [ABSTRACT FROM AUTHOR]

In this paper we find a class of functions for which the Lebesgue estimate can be improved. [ABSTRACT FROM AUTHOR]

A one-dimensional stochastic wave equation driven by a general stochastic measure is studied in this paper. The Fourier series expansion of stochastic measures is considered. It is proved that changing the integrator by the corresponding partial sums or by Fejèr sums we obtain the approximations of mild solution of the equation. [ABSTRACT FROM AUTHOR]

The Fourier series method for implementing the Rayleigh hypothesis [Rayleigh-Fourier method (RFM)] is used as a reference solution to assess the Kirchhoff approximation of the Helmholtz integral [Helmholtz-Kirchhoff approximation (HKA)] for modeling broadband scatter from sinusoidal surfaces at low grazing angles. The HKA is a valuable solution because it has an eigen-ray interpretation without unbounded caustic amplitudes and discontinuous shadow zones. Plane wave studies of the HKA, however, show it becomes inaccurate at low grazing angles. This study quantifies how this limitation manifests with increasing transmission distance for time domain scattering simulations. Scattering results are compared over a complete surface wave cycle with parameters modeling sea surface-swell. The HKA agrees reasonably well with the RFM in point source calculations for limited extensions of transmission distances beyond where plane wave comparisons begin to diverge. Past these distances, HKA solutions begin to show significant over-prediction of the acoustic amplitude around late arrivals. This over-prediction is frequency dependent and eigen-ray interference offers an explanation of this behavior. Further extending the transmission range leads to a significant HKA error, and a range is found at which flat surface reflections have less error. [ABSTRACT FROM AUTHOR]

This work is concerned with the asymptotic analysis of linearly elastic plates with periodically rapidly varying heterogeneities. For the sake of simplicity we assume that the structure of heterogeneity is homogeneous in the direction perpendicular to the mid-surface of the plate. We want to derive a homogenized two-dimensional model which is independent of the magnitude of the applied load. Consequently we have to proceed in the following manner. Firstly, we consider a two-dimensional model of the plate obtained by expanding the displacement field by Fourier-Series expansion in thickness direction of the plate with respect to a basis of scaled Legendre polynomials. We consider a second order approximation of the displacement field which gives a good compromise between the accuracy of the approximate solution and the complexity of the approximate problem. This approximation result from an approximation of the Fourier series expansion of the displacement field up to order h6 (h denotes the thickness of the plate). By considering standard argument for this type of problem, we can rigorously formulate a two-dimensional homogenized boundary value problem for the plate. [ABSTRACT FROM AUTHOR]

The deferred Cesàro transformations which have useful properties not possessed by the Cesàro transformation was considered by R.P. Agnew in [1]. In [9], Değer and Küçükaslan introduced a generalization of deferred Cesaro transformations by taking account of some well known transformations such as Woronoi-Nörlund and Riesz, and considered the degree of approximation by the generalized deferred Cesaro means in the space H(α, p), p ≥ 1, 0 < α ≤ 1 by concerning with some sequence classes. In 2014, Nayak et al. studied the rate of convergence problem of Fourier series by deferred Cesaro mean in the space Hp(w) introduced by Das et al. in [7]. In this study we shall give the degree of approximation by the generalized deferred Cesàro means in the space Hp(w). Therefore the results given in [ ] are generalized according to the summability method. [ABSTRACT FROM AUTHOR]

We consider the problem of the mean-square approximation of complex functions regular in a domain D ⊂ C by Fourier serieswith respect to an orthogonal (inD) systemof functions {ϕk(z)}, k = 0, 1, 2,.... For the case inwhich D = {z ∈ C: |z| < 1}, we obtain sharp estimates for the rate of convergence of the Fourier series in the orthogonal system {zk}, k = 0, 1, 2,..., for classes of functions defined by a special mth-order modulus of continuity. Exact values of the series of n-widths for these classes of functions are calculated. [ABSTRACT FROM AUTHOR]

In this study, a comprehensive approach for model order reduction based on a Fourier series of a discrete system representation is proposed. The developed method represents an alternative to model reduction of large‐scale dynamical systems and can be used for the analysis of unstable dynamic systems and the study of interactions among power system controls. Drawing on the principle of conformal mapping, the linearised system model is first transformed into its discrete equivalent. Then, the Fourier series of the discrete‐time transfer function is used to obtain a reduced‐order model (ROM). Using this model, approximate Hankel‐based interaction measures are proposed to efficiently analyse large, unstable linear system representations and determine pairs of input outputs to design controller. The high degree of applicability and accuracy offered by the method, and its ability to extract ROMs from stable and unstable systems is demonstrated on three test systems. [ABSTRACT FROM AUTHOR]

Shape-based methods are becoming popular in low-thrust trajectory optimization due to their fast computation speeds. In existing shape-based methods constraints are treated at the acceleration level but not at the thrust level. These two constraint types are not equivalent since spacecraft mass decreases over time as fuel is expended. This paper develops a shape-based method based on a Fourier series approximation that is capable of representing trajectories defined in spherical coordinates and that enforces thrust constraints. An objective function can be incorporated to minimize overall mission cost, i.e., achieve minimum Δ V . A representative mission from Earth to Mars is studied. The proposed Fourier series technique is demonstrated capable of generating feasible and near-optimal trajectories. These attributes can facilitate future low-thrust mission designs where different trajectory alternatives must be rapidly constructed and evaluated. [ABSTRACT FROM AUTHOR]

In this paper, we investigate convergence and divergence of partial sums with respect to the 2-dimensional Walsh system on the martingale Hardy spaces. In particular, we find some conditions for the modulus of continuity which provide convergence of partial sums of Walsh-Fourier series. We also show that these conditions are in a sense necessary and sufficient. [ABSTRACT FROM AUTHOR]

We demonstrate that the Fourier partial sums of a Lipschitz continuous function on the two-dimensional special unitary group converge uniformly to the function. [ABSTRACT FROM AUTHOR]

Let the function f belong to the Hölder class Hα(Ω), 0 < α ≤ 1, where Ω is the spectral set of the hexagonal lattice in the Euclidean plane: Also, let p = (pn) and q = (qn) be two sequences of non-negative real numbers such that pn < qn and qn → ∞ as n → ∞: The order of approximation of f by deferred Cesàro means Dn (p; q; f) of its hexagonal Fourier series is estimated in the uniform and Hölder norms. [ABSTRACT FROM AUTHOR]

We extend the results presented by Xh. Z. Krasniqi (Slight extensions of some theorems on the rate of pointwise approximation of functions from some subclasses of Lp. Acta Comment. Univ. Tartu. Math., 2013, 17, 89-101) and W. Lenski and B. Szal (Approximation of functions belonging to the class Lp(w) by linear operators. Acta Comment. Univ. Tartu. Math., 2009, 13, 11-24) to the case when a conjugate function depends on r and where in the measures of estimations r-differences of the entries are used. [ABSTRACT FROM AUTHOR]

We introduce a generalized sigmoidal transformation wm(r;x) on a given interval [a,b] with a threshold at x=r∈(a,b). Using wm(r;x), we develop a weighted averaging method in order to improve Fourier partial sum approximation for a function having a jump-discontinuity. The method is based on the decomposition of the target function into the left-hand and the right-hand part extensions. The resultant approximate function is composed of the Fourier partial sums of each part extension. The pointwise convergence of the presented method and its availability for resolving Gibbs phenomenon are proved. The efficiency of the method is shown by some numerical examples. [ABSTRACT FROM AUTHOR]

The study of infinite matrices is important in the theory of summability and in approximation. In particular, Toeplitz matrices or regular matrices and almost regular matrices have been very useful in this context. In this paper, we propose to use a more general matrix method to obtain necessary and sufficient conditions to sum the conjugate derived Fourier series. [ABSTRACT FROM AUTHOR]

Here we develop an option pricing method based on Legendre series expansion of the density function. The key insight, relying on the close relation of the characteristic function with the series coefficients, allows to recover the density function rapidly and accurately. Based on this representation for the density function, approximations formulas for pricing European type options are derived. To obtain highly accurate result for European call option, the implementation involves integrating high degree Legendre polynomials against exponential function. Some numerical instabilities arise because of serious subtractive cancellations in its formulation (96) in Proposition A.1. To overcome this difficulty, we rewrite this quantity as solution of a second-order linear difference equation and solve it using a robust and stable algorithm from Olver. Derivation of the pricing method has been accompanied by an error analysis. Errors bounds have been derived and the study relies more on smoothness properties which are not provided by the payoff functions, but rather by the density function of the underlying stochastic models. This is particularly relevant for options pricing where the payoffs of the contract are generally not smooth functions. The numerical experiments on a class of models widely used in quantitative finance show exponential convergence. [ABSTRACT FROM AUTHOR]

A new accurate iterative and asymptotic method is introduced to construct analytical approximate solutions to strongly nonlinear conservative symmetric oscillators. The method is based on applying a second-order expansion with the harmonic balance method and it excludes the requirement of solving a set of coupled nonlinear algebraic equations. Newton's iteration or the linearized model may be readily deduced by considering only the first-order terms in the model. According to this iterative approach, only Fourier series expansions of restoring force function, its first- and second-order derivatives for each iteration are required. It is concluded here that by only one single iteration, very brief and yet accurate analytical approximate solutions can be attained. Three physical examples are solved and accurate solutions are presented to illustrate the physics of the system and the effectiveness of the proposed asymptotic method. [ABSTRACT FROM AUTHOR]

In the paper a new theorem on the degree of approximation of Lipschitz function on by (c,1) means is established. [ABSTRACT FROM AUTHOR]

In this work the approximation of the functions by trigonometric polynomials Nλn(f,x) of degree n in weighted Lebesgue spaces with Muckenhoupt weights are studied. [ABSTRACT FROM AUTHOR]

In a thermal tomography measurement setup, a physical body is sequentially heated at different source locations and temperature evolutions are measured at several measurement locations on the surface of the body. Based on these transient measurements, the thermal conductivity, the volumetric heat capacity and the surface heat transfer coefficient of the body are estimated as spatially distributed parameters, typically by minimizing a modified data misfit functional between the measured data and the data computed with the estimated thermal parameters. In thermal tomography, heat transfer is modeled with the time-dependent heat diffusion equation for which direct time domain solving is computationally expensive. In this paper, the computations of thermal tomography are sped up by utilizing a truncated Fourier series approximation approach. In this approach, a frequency domain equivalent of the time domain heat diffusion equation is solved at multiple frequencies and the solutions are used to obtain a truncated Fourier series approximation for the solution and the Jacobian of the time domain heat transfer problem. The feasibility of the approximation is tested with simulated and experimental measurement data. When compared to a previously used time domain approach, it is shown to lead to a significant reduction of computation time in image reconstruction with no significant loss of reconstruction accuracy. [ABSTRACT FROM AUTHOR]

We study Fourier series of Jacobi polynomials P ( x), k = r, r +1,..., orthogonal with respect to the Sobolev-type inner product of the following form: $$\left\langle {f,g} \right\rangle = \sum\limits_{v = 0}^{r - 1} {{f^{\left( v \right)}}} \left( { - 1} \right){g^{\left( v \right)}}\left( { - 1} \right) + \int_{ - 1}^1 {{f^{\left( r \right)}}} {g^{\left( r \right)}}\left( t \right){\left( {1 - t} \right)^\alpha }dt$$ . It is shown that such series are a particular case of mixed series of Jacobi polynomials P ( x), k = 0, 1,..., considered earlier by the author. We study the convergence of mixed series of general Jacobi polynomials and their approximation properties. The results obtained are applied to the study of the approximation properties of Fourier series of Sobolev orthogonal Jacobi polynomials P ( x). [ABSTRACT FROM AUTHOR]

Degree of approximation of functions of different classes has been studied by several researchers by different summability methods. In the proposed paper, we have established a new theorem for the approximation of a signal (function) belonging to the W(Lr, ξ(t))-class by (..., pn, qn)(E, s)-product summability means of a Fourier series. The result obtained here, generalizes several known theorems. [ABSTRACT FROM AUTHOR]

In this paper, we have proved the degree of approximation of functions belonging to L[0, ∞) by Harmonic-Euler means of its Fourier-Laguerre series at x = 0. The aim of this paper is to concentrate on the approximation properties of the functions in L[0, ∞) by Harmonic-Euler means of its Fourier-Laguerre series associated with the function f. [ABSTRACT FROM AUTHOR]

In this study, an optimally scaled Fourier expansion method with half-order technique, referred to as OSFEM(H), is developed based on the combination of the optimal scaling and the half-order techniques. The optimal scaling technique is employed to reduce the ill-posedness which may arise from using high order Fourier expansion series to approximate periodic solutions. The half-order technique is a powerful tool with which one can use an m -order Fourier expansion to interpolate as many as up to 4 m + 1 points. In this paper, the best scales in the multi-scale Fourier expansion interpolation are derived based on the idea of equating the norm of each column of the interpolated matrix, such that the condition number of the preconditioned matrix is minimized. Then, the present half-order OSFEM, i.e. OSFEM(H), is used to solve the Duffing equation. It is shown that excellent results are achieved by comparing with the differential transformation method and the harmonic balance method. Numerical simulations verify the accuracy and effectiveness of the presently proposed method. [ABSTRACT FROM AUTHOR]

Filtering for multiple, possibly dependent angular variates on higher-dimensional manifolds such as the hypertorus is challenging as solutions from the circular case cannot easily be extended. In this paper, we present an approach to recursive multivariate angular estimation based on Fourier series. Since only truncated Fourier series can be used in practice, implications of the approximation errors need to be addressed. While approximating the density directly can lead to negative function values in the approximation, this problem can be solved by approximating the square root of the density. As this comes at the cost of additional complexity in the algorithm, we present both a filter based on approximating the density and a filter based on approximating its square root and closely regard the trade-offs. While the computational effort required for the filters grows exponentially with increasing number of dimensions, our approach is more accurate than a sampling importance resampling particle filter when comparing configurations of equal run time. [ABSTRACT FROM AUTHOR]

An approximation of functions in Müntz spaces M Λ , L 1 of L 1 functions is studied with the help of Fourier series. Müntz spaces are considered with the Müntz condition and the gap condition imposed. It is proved that up to an isomorphism such spaces are contained in Weil–Nagy's class. Moreover, existence of Schauder bases in Müntz spaces M Λ , L 1 is investigated. [ABSTRACT FROM AUTHOR]

For given p ∈ [ 1 , ∞ ] and g ∈ L p ( R ) , we establish the existence and uniqueness of solutions f ∈ L p ( R ) , to the equation f ( x ) − a f ( b x ) = g ( x ) , where a ∈ R , b ∈ R ∖ { 0 } , and | a | ≠ | b | 1 / p . Solutions include well-known nowhere differentiable functions such as those of Bolzano, Weierstrass, Hardy, and many others. Connections and consequences in the theory of fractal interpolation, approximation theory, and Fourier analysis are established. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to analyze the degree of approximation of a function f that is a conjugate of a function f belonging to the Lipschitz class by Hausdorff means of a conjugate series of the Fourier series. [ABSTRACT FROM AUTHOR]

A new theorem on the approximation of almost Lipa function by Kλ summabilitv method of Fourier series. [ABSTRACT FROM AUTHOR]

We estimate the pointwise approximation of periodic func- tions belonging to Lp(ω)β-class, where ω is an integral modulus of conti- nuity type function associated with f, using product means of the Fourier series of f generated by the product of two general linear operators. We also discuss the case p = 1 separately. This case has not been mentioned in the earlier results given by various authors. The deviations obtained in our theorems are free from p and more sharper than the earlier results. [ABSTRACT FROM AUTHOR]

In this paper, we prove direct and inverse theorems on the approximation of functions by Fourier-Laplace sums in the spaces S(σ), m ≥ 3, in terms of best approximations and moduli of continuity and consider the constructive characteristics of function classes defined by the moduli of continuity of their elements. The given statements generalize the results of the author's work carried out in 2007. [ABSTRACT FROM AUTHOR]

Various investigators have studied the degree of approximation of a function using different summability (Cesáro means of order α: Cα, Euler Eq, and Nörlund Np) means of its Fourier-Laguerre series at the point x=0 after replacing the continuity condition in Szegö theorem by much lighter conditions. The product summability methods are more powerful than the individual summability methods and thus give an approximation for wider class of functions than the individual methods. This has motivated us to investigate the error estimation of a function by (T·Eq)-transform of its Fourier-Laguerre series at frontier point x=0, where T is a general lower triangular regular matrix. A particular case, when T is a Cesáro matrix of order 1, that is, C1, has also been discussed as a corollary of main result. [ABSTRACT FROM AUTHOR]

Differential equations play a noticeable role in engineering, physics, economics, and other disciplines. Approximate approaches have been utilized when obtaining analytical (exact) solutions requires substantial computational effort and often is not an attainable task. Hence, the importance of approximation methods, particularly, metaheuristic algorithms are understood. In this paper, a novel approach is suggested for solving engineering ordinary differential equations (ODEs). With the aid of certain fundamental concepts of mathematics, Fourier series expansion, and metaheuristic methods, ODEs can be represented as an optimization problem. The target is to minimize the weighted residual function (error function) of the ODEs. The boundary and initial values of ODEs are considered as constraints for the optimization model. Generational distance and inverted generational distance metrics are used for evaluation and assessment of the approximate solutions versus the exact (numerical) solutions. Longitudinal fins having rectangular, trapezoidal, and concave parabolic profiles are considered as studied ODEs. The optimization task is carried out using three different optimizers, including the genetic algorithm, the particle swarm optimization, and the harmony search. The approximate solutions obtained are compared with the differential transformation method (DTM) and exact (numerical) solutions. The optimization results obtained show that the suggested approach can be successfully applied for approximate solving of engineering ODEs. Providing acceptable accuracy of the proposed technique is considered as its important advantage against other approximate methods and may be an alternative approach for approximate solving of ODEs. [ABSTRACT FROM AUTHOR]

In the circular restricted three-body problem (CR3BP), transit orbit is a class of orbit which can pass through the bottleneck region of the zero velocity curve and escapes from the vicinity of the primary or the secondary. This kind of orbit plays a very important role in the design of space exploration missions. A kind of low-energy interplanetary transfer, which is called Interplanetary Superhighway (IPS), can be realized by utilizing transit orbits. To use the transit orbit in actual mission design, a key issue is to find an algorithm which can separate the states corresponding to transit orbits from the states corresponding to other types of orbits rapidly. In fact, the distribution of transit orbit in the phase space has been investigated by numerical method, and a Fourier series approximation method has been introduced to describe the boundary of transit orbits. However, the Fourier series approximation method needs several hundred sets of Fourier series. The coefficients of these Fourier series are neither easy to be computed nor convenient to be stored, which makes the method can hardly be used in actual mission design. In this paper, the support vector machine (SVM) is used to classify the trajectories in the CR3BP. Using the Gaussian kernel, the 6-dimensional states in the CR3BP are mapped into an infinite-dimensional space, and the bound of the transit orbits is described by a hyperplane. A training data generation method is introduced, which reduces the size of training data by generating the states near the hyperplane. The numerical results show that the proposed algorithm gives the good correct rate of classification, and its computing speed is much faster than that of the Fourier series approximation method. [ABSTRACT FROM AUTHOR]

The equivalence of the error of approximation by Fourier means and of general moduli of smoothness, provided that their generators are equivalent, is established. [ABSTRACT FROM AUTHOR]

A model of the optimal representation of spatial data for the task of complex correlation-extreme navigation is developed based on the criterion of minimum deviation of the correlation functions of the original and the resulting fields. Calculations are presented for one-dimensional case using the approximation of the correlation function by Fourier series. It is shown that in the presence of different geophysical map data fields their representation is possible by single template with optimal sampling without distorting the form of the correlation functions. [ABSTRACT FROM AUTHOR]

In this paper, a new estimate for the degree of approximation of a function f(x; y) belonging to generalized Lipschitz class Lip(ξ1, ξ2) by (N, pm, qn) summability method of its double Fourier series has been determined. [ABSTRACT FROM AUTHOR]

Recently, a simple time domain collocation (TDC) method was proposed by researchers including the first author of the present paper, and has been successfully applied to obtain the harmonic, subharmonic, and superharmonic responses of the nonlinear Duffing oscillator. The TDC method is based on the point-collocation method performing within an appropriate period of the periodic solution, wherein the approximate solution is assumed as a Fourier series. Upon using the TDC method, the ordinary differential equation is transformed into a system of non-linear algebraic equations (NAEs), which can be readily solved by an NAE solver. In this study, using the Duffing oscillator as the prototype, we develop a multiple scale time domain collocation (MSTDC) method, by introducing a series of optimal multiple scales to the Fourier series of the approximate solution, to alleviate the ill-posedness of the system of collocation-resulting NAEs due to the inclusion of very high order harmonics in the approximate solution. Besides the MSTDC method, a multiple scale differential transformation (MSDT) method is proposed by introducing the multiple scales to the classical differential transformation method. Finally, numerical experiments are carried out to verify the accuracy and efficiency of the MSTDC and the MSDT methods. [ABSTRACT FROM AUTHOR]

Questions dealing with the approximation of functions from the classes C H by Poisson integrals are studied. The Kolmogorov-Nikol'skii problem for Poisson integrals for the classes C H is solved in the uniform metric. [ABSTRACT FROM AUTHOR]

Abstract: Analysis of signals or time functions are of great importance, because it convey information or attributes of some phenomenon. In this paper, we consider the functions/signals defined in terms of their integral modulus of continuity and approximate them by matrix means of their trigonometric Fourier series. The error of approximation obtained in this paper depends on the integral modulus of continuity of the signal. We also discuss an example to show the application of the result in the situation when the Fourier series of the signal has Gibbs phenomenon. We also discuss some corollaries of our theorems. [Copyright &y& Elsevier]

This paper deals with the de la Vallée Poussin means of the triangular partial sums of multivariate Fourier series. We determine the exact order of the corresponding operator norms. The lower estimation of these norms will be extended to a class of projection operators having similar projection properties as the de la Vallée Poussin mean. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider fractional sub-diffusion equation on two-dimensional unbounded domains and obtain an exact and some approximating artificial boundary conditions for it by a joint application of the Laplace transform and the Fourier series expansion. In order to test the derived artificial boundary condition, after reducing the main problem to an initial–boundary value problem on bounded domain by imposing the obtained boundary condition, we just use classical Crank–Nicolson method for space variables and L1 approximation for the fractional time derivative. Some numerical examples are given which confirm the effectiveness of the proposed artificial boundary conditions. [Copyright &y& Elsevier]