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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]

In approximation theory, the "direct theorem" is a fundamental concept that relates to how well more straightforward functions or signals can approximate a function or signal from a chosen approximation space. The direct theorem typically provides bounds on the error or the quality of the approximation in terms of various parameters, such as the degree of approximation, the dimension of the approximation space, and the properties of the approximated functions. The specific form of the direct theorem can vary depending on the context and the type of approximation being considered (e.g., polynomial approximation, Fourier series approximation, spline approximation, etc.). The direct theorem aims to establish the mathematical framework for comprehending how well elements from a given space can approximate a function, which is crucial in various fields like numerical analysis, signal processing, and scientific computing. Finally, the direct theorem is a crucial result in approximation theory. It describes how elements from a given approximation space get closer and better at approximating a given function as the number of terms in the approximation sequence grows. It provides essential insights into the behavior of approximations and plays a central role in various mathematical and computational disciplines. In this article, we aim to prove the direct theorem by defining a new appropriate operator for functions Lp ([[0, 2π]d) that uses our new k-mixed modulus of smoothness to estimate the best approximation degree and the new k-mixed difference. Also, to prove the direct theorem, we introduce the angular approximation in Lp ([0, 2π]d to reach the most accurate estimate. [ABSTRACT FROM AUTHOR]

We consider nonharmonic Fourier series defined in terms of arbitrarily close exponentials. Our aim is to use the positivity of quadratic polynomials in order to get new Ingham-type weighted inequalities. The proof relies on an Ingham proof technique inspired by Jaffard et al. (J Fourier Anal Appl 3:577–582, 1997). As applications, we consider families of frequencies with relevance in control approximation theory, for which we can prove the uniform (with respect to the mesh-size) controllability property of the semi-discrete model, when the spurious frequencies (the gap between them tends to zero when the mesh size goes to zero) are eliminated. [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]

Taylor series play a ubiquitous role in calculus courses, and their applications as approximants to functions are widely taught and used everywhere. However, it is not common to present the students with other types of approximations besides Taylor polynomials. These notes show that polynomials construed to satisfy certain boundary conditions at an interval of definition can be represented by rapidly converging Fourier series. These polynomials are shifted and re-scaled versions of the Bernoulli polynomials. From this construction, an argument to 'invert' the Fourier series to obtain an approximation to sines and cosines in terms of the Bernoulli polynomials is presented. We then show that approximating sines and cosines by the Bernoulli polynomials might be much better than using the truncated Taylor series, especially in problems where global proprieties are desired. These contents can be taught in advanced Calculus classes approaching series of functions and their applications. [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]

In the present work we investigate the approximation of the functions by Cesaro, Zygmund and Abel-Poisson means of Fourier series in Morrey spaces Lp,λ (T), 0 < λ = 2,1 < p < ∞ in the terms of the modulus of continuity. These results applied to the estimates of approximation of Cesaro, Zygmund and Abel sums of Faber series in the Morrey-Smirnov classes Ep,λ(G), 0 < λ = 2 and 1 < p < ∞ defined in the domains with Dini-smooth boundary of the complex plane. [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 compute the upper bounds for the H2-norms of partial sums of the Fourier series in the Laguerre basis on the unit ball in the space of bounded holomorphic functions on unit disk. We present an application of the main result to the solution of some extreme problems in the theory of approximation of holomorphic functions. [ABSTRACT FROM AUTHOR]

In this paper, we have proved some results on the degree of approximation in sup-norm of continuous functions by generalized deferred Voronoi-Norlund means of partial sums of their Fourier series. [ABSTRACT FROM AUTHOR]

The pseudo-hyperbolic equation with cubic nonlinearity and additive space-time noise is discussed. The space-time noise is assumed to be Gaussian in time and possesses a Fourier series expansion in space. First, we prove the existence and uniqueness of the approximate strong solutions of the equation and show that the truncated Fourier solution which can be approximated by the truncated finite-dimensional system, is an approximate solution. Second, a new transformation is used to convert pseudo-hyperbolic equation into a system of equations, which can construct an infinitesimal generator with good properties. After analyzing the related total energy evolution, we obtain that the energy growth will not blow-up in the limited time. Finally, we present a Fourier scheme of a procedure for its numerical approximation and give the stability and convergence analysis of the scheme. [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 this paper, for any smooth function on [0, 1]2, we give an asymptotic representation of hyperbolic cross approximations of its Fourier series whose principal part is determined by the values of the function at vertexes of [0, 1]2 and present a novel approach to estimates of the upper bounds of approximation errors. At the same time, we also give an asymptotic formula of partial sum approximations whose principal part is determined by not only partial derivatives at vertexes of [0, 1]2, but also mean values on each side. Comparing asymptotic representations of these two kinds of approximation, we find that although in general the hyperbolic cross approximation is better than the partial sum approximation, the partial sum approximation possibly work better under some cases, and we also give the corresponding necessary and sufficient condition to characterize these cases. [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]

We consider processes of the form μ(t) = μ((0, t]), where μ is a σ-additive in probability stochastic set function. Convergence of a random Fourier series to μ(t) is proved, and the approximation of integrals with respect to μ using Fejer sums is obtained. For this approximation, we prove the convergence of solutions of the heat equation driven by μ. [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]

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. [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 problem of obtaining degree of approximation for the 2π-periodic functions in the weighted Lipschitz class W(Lp; ξ(t)) (p ≥ 1) by Riesz means of the Fourier series have been studied by various investigators under Lp-norm. Recently, Deepmala and Piscoran [Approximation of signals(functions) belonging to certain Lipschitz classes by almost Riesz means of its Fourier series, J. Inequal. Appl., (2016), 2016:163. DOI 10.1186/s13660-016-1101-5] obtained a result on degree of approximation for weighted Lipschitz class by Riesz means. In this note, we extend this study to the weighted Lp-norm which in turn improves some of the previous results. We also derive some corollaries from our result. [ABSTRACT FROM AUTHOR]

The generalized Lorenz model for studying the Rayleigh-Bénard convection in water-multiwalled carbon nanotubes and water-alumina in the presence of heat source/sink is derived using a tri-modal, Fourier series representation under the assumption of the Boussinesq approximation and small-scale convective motion. On the premise that the multiwalled carbon nanotubes/alumina nanoparticles undergo all motions which water particles undergo the single-phase description of the Khanafer-Vafai-Lightstone model can be adopted. The thermophysical properties of the two nanoliquids are calculated using the phenomenological laws and traditional mixture theory. The transition of the dynamical Lorenz system from stability to chaos followed by periodic motion in a window of periodicity and then a sequence of such motions are explained in detail by calculating the maximum Lyapunov exponent. Specifically, the individual effect of multiwalled carbon nanotubes and alumina nanoparticles on the stability of the dynamical system is investigated from a new perspective of time series solution and attractors of the Lorenz model. We then study the individual influence of multiwalled carbon nanotubes and alumina nanoparticles on the onset of convection and heat transport. The findings of a feasibility study indicate that less expensive spherical alumina nanoparticles are preferable in heat transfer applications over the more expensive multiwalled carbon nanotubes, in spite of the latter having a larger surface area and higher thermal conductivity. It is found that exactly the same amount of heat transfer obtained by using multiwalled carbon nanotubes can be achieved by replacing them with a slightly higher volume fraction of the much cheaper alumina nanoparticles. [ABSTRACT FROM AUTHOR]

Fourier extension is an approximation scheme in which a function on an arbitrary bounded domain is approximated using classical Fourier series on a bounding box. On the domain of interest the Fourier series exhibits redundancy, and it has the mathematical structure of a frame rather than a basis. It is not trivial to construct approximations in this frame using function evaluations in points that belong to the domain only, but one way to do so is through a discrete least squares approximation. The corresponding system is extremely ill-conditioned, due to the redundancy in the frame, yet its solution via a regularized singular value decomposition is known to be accurate to very high (and nearly spectral) precision. Still, this computation requires Q (N3) operations. In this paper we describe an algorithm to compute such Fourier extension frame approximations in only Q (N2 log2 N) operations for general 2D domains. The cost improves to Q (N log2 N) operations for simpler tensor-product domains. The algorithm exploits a phenomenon called the plunge region in the analysis of time-frequency localization operators, which manifests itself here as a sudden drop in the singular values of the least squares matrix. It is known that the size of the plunge region scales like Q (logN) in 1D problems. In this paper we show that for most 2D domains in the fully discrete case the plunge region scales like Q (N logN), proving a discrete equivalent of a result that was conjectured by Widom for a related continuous problem. The complexity estimate depends on the Minkowski or box-counting dimension of the domain boundary, and as such it is larger than Q (N logN) for domains with fractal shape. [ABSTRACT FROM AUTHOR]

We find the exact order of the entropy numbers and the Kolmogorov widths of a class of functions defined by a condition on the uniform norm of blocks of the Fourier series taken over a lacunar sequence of indices. This result generalizes a result by B. S. Kashin and V. N. Temlyakov. [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]

We obtain some order-sharp estimates for the Fourier widths of Nikol'skii-Besov and Lizorkin-Triebel function classes with given majorant of the mixed modulus of smoothness in the Lebesgue space for a few relations between the parameters of the class and the space. The upper bounds follow from estimates of the approximation of functions of these classes by special partial sums of their Fourier series with respect to the multiple system of periodized Meyer wavelets. [ABSTRACT FROM AUTHOR]

The spline theory can be derived from two sources: the algebraic one (where splines are understood as piecewise smooth functions satisfying some continuity conditions) and the variational one (where splines are obtained via minimization of some quadratic functionals with constraints). We show that the general variational approach called smooth interpolation introduced by Talmi and Gilat covers not only the cubic spline but also the tension spline (called also spline in tension or spline with tension) in one or more dimensions. We show the results of a 1D numerical example that present the advantages and drawbacks of the tension spline. [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]

In the three-dimensional statement, by the method of approximation of functions by discrete Fourier series, we perform the analysis of the dependence of the stress state of a hollow cylinder whose cross section has the form of joined concave semicorrugations subjected to the action of internal pressure under certain conditions imposed on the end faces on the thicknesses and curvatures of the cross sections of the cylinders caused by the changes in the number of semicorrugations. The results of the solution of the problem are presented in the form of the plots of distributions of the fields of displacements and stresses. [ABSTRACT FROM AUTHOR]

The problem of approximation of electrocardiograms has a great practical importance and it is constantly in the field of research. Solution of this problem allows us to automate and computerize the process of diagnosis and to detect timely deviations from normal characteristics of cardiograms of heartbeat in the early stages of appearance and development of a disease. There are different methods for approximation of cardiograms. All these methods have their drawbacks. The article deals with new methods of approximation of cardiograms, which do not have any disadvantages of the known methods. The developed mathematical models are based on new methods of approximation of step functions, and these methods can significantly reduce errors of the approximation of cardiograms. In this paper new methods of approximation of step functions with estimation of errors of the approximation are supposed. The proposed methods do not have any drawbacks of traditional expansions of step functions in Fourier series and can be used in problems of mathematical modeling of a wide class of processes and systems, in particular, for solution of medical and biological problems. [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]

In this work the approximation of functions by linear means of Fourier series in reflexive weighted Orlicz spaces with Muckenhoupt weights is studied. This result is applied to the approximation of functions by linear means of Faber series in weighted Smirnov-Orlicz classes defined on simply connected domain of the complex plane. [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]

The problem of approximation of a differentiable function of two variables by partial sums of a double Fourier-Bessel series is considered. Sharp estimates of the rate of convergence of the double Fourier-Bessel series on the class of differentiable functions of two variables characterized by a generalized modulus of continuity are obtained. The proofs of four theorems on this issue, which can be directly applied to solving particular problems of mathematical physics, approximation theory, etc., are presented. [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]

We study the asymptotic behavior of the least upper bounds of the approximations of functions from the classes $$ {W}_{\beta}^r{H}^{\alpha } $$ by Weierstrass integrals in the uniform metric. [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 paper, we determine the degree of trigonometric approximation of -periodic functions and their conjugates, in terms of the moduli of continuity associated with them, by matrix means of corresponding Fourier series. We also discuss some analogous results with remarks and corollaries. [ABSTRACT FROM AUTHOR]