Search

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]

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]

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 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]

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]

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]

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 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 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]

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]

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 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]

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]

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]

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]

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]

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]

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]

Abstract: In this paper we study the problem of computing wavelet coefficients of compactly supported functions from their Fourier samples. For this, we use the recently introduced framework of generalized sampling. Our first result demonstrates that using generalized sampling one obtains a stable and accurate reconstruction, provided the number of Fourier samples grows linearly in the number of wavelet coefficients recovered. For the class of Daubechies wavelets we derive the exact constant of proportionality. Our second result concerns the optimality of generalized sampling for this problem. Under some mild assumptions we show that generalized sampling cannot be outperformed in terms of approximation quality by more than a constant factor. Moreover, for the class of so-called perfect methods, any attempt to lower the sampling ratio below a certain critical threshold necessarily results in exponential ill-conditioning. Thus generalized sampling provides a nearly-optimal solution to this problem. [Copyright &y& Elsevier]

In the present study, a new theorem on the degree of approximation of function f, conjugate to a periodic function f belonging to weighted W(Lr, ξ(t))-class using semimonotonicity on the generating sequence {pn} has been established. [ABSTRACT FROM AUTHOR]

A new estimate for the degree of approximation of a function ... ∊ Hω class by (Np · E¹) means of its Fourier series has been determined. Here, we extend the results of Singh and Mahajan (2008) which in turn generalize the result of Lal and Yadav (2001). Some corollaries have also been deduced from our main theorem. [ABSTRACT FROM AUTHOR]

The technique of approximate partition of unity, the way of Fourier series, and inequality technique are used to construct a neural network with two weights and with sigmoidal functions. Furthermore by using inequality technique, we prove that the neural network with two weights can more precisely approximate any nonlinear continuous function than BP neural network constructed in (Chen et al., 2012). [ABSTRACT FROM AUTHOR]

The end problem referring to anti-plane shear deformation of a nonhomogeneous semi-infinite strip is investigated here, by using the analogous methodology proposed by Papkovich and Fadle in plane problem. Two types of nonhomogeneity are considered: (i) the shear modulus varies with the thickness coordinate x exponentially; (ii) it varies with the length coordinate y exponentially. The closed elastic solutions in trigonometric series form are derived by the eigenfunctions expansion, and the completeness of the solutions is also proved. Therefore, the elastic field caused by a self-equilibrating traction on the end could be solved in an arbitrary accuracy by taking a necessary number of terms in the series, approximatively, which is usually neglected by invoking Saint-Venant principle. By considering the biggest negative eigenvalue, the Saint-Venant Decay rates of the problem is also estimated in the last section. [ABSTRACT FROM AUTHOR]

We study the Walsh series expansion of multivariate functions in Lp (1 ≤ p ≤ ∞) and, in particular, in Lip(α, p). The rate of uniform approximation by T-transformation of rectangular partial sums of double Walsh to these functions is investigated. By extending the concepts of rest (head) bounded variation series, which was introduced by Leindler (2004), we generalize the related results of Móricz and Rhoades (1996), Nagy (2012). Our results can be applied to many summability methods, including the Norlund summability and weighted summability. [ABSTRACT FROM AUTHOR]

The current paper offers a comprehensive dynamical analysis and Fourier series approximations of J2-bounded equatorial orbits. The initial conditions of heterogeneous families of J2-perturbed equatorial orbits are determined first. Then the characteristics of two types of J2-bounded orbits, namely, pseudo-elliptic orbit and critical circular orbit, are studied. Due to the ambiguity of the closed-form solutions which comprise the elliptic integrals and Jacobian elliptic functions, showing little physical insight into the problem, a new scheme, termed Fourier series expansion, is adopted for approximation herein. Based on least-squares fitting to the coefficients, the solutions are expressed with arbitrary high-order Fourier series, since the radius and the flight time vary periodically as a function of the polar angle. As a consequence, the solutions can be written in terms of elementary functions such as cosines, rather than complex mathematical functions. Simulations enhance the proposed approximation method, showing bounded and negligible deviations. The approximation results show a promising prospect in preliminary orbits design, determination, and transfers for low-altitude spacecrafts. [ABSTRACT FROM AUTHOR]

We study the Walsh series expansion of multivariate functions in Lp (1 = p = ∞) and, in particular, in Lip(α, p). The rate of uniform approximation by T-transformation of rectangular partial sums of double Walsh to these functions is investigated. By extending the concepts of rest (head) bounded variation series, which was introduced by Leindler (2004), we generalize the related results of Móricz and Rhoades (1996),Nagy (2012).Our results can be applied tomany summability methods, including the Nörlund summability and weighted summability. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the approximation properties of the matrix means of trigonometric Fourier series of f belonging to weighted Lipschitz class Lip(α, p,w) with Muckenhoupt weights generated by T ≡ (an;k) under relaxed conditions. Our theorem extends some of the previous results pertaining to the degree of approximation of functions in weighted Lipschitz class Lip(α,p,w) and the ordinary Lipschitz class Lip(α,p). [ABSTRACT FROM AUTHOR]

In the present paper, a new theorem on the degree of approximation of a function [InlineEquation not available: see fulltext.], conjugate to a 2 π periodic function f belonging to the Lip α ([InlineEquation not available: see fulltext.]) class without the monotonicity condition on the generating sequence [InlineEquation not available: see fulltext.] has been established, which in turn generalizes the results of Lal (Appl. Math. Comput. 209: 346-350, 2009) on a Fourier series. MSC: 40G05, 41A10, 42B05, 42B08. [ABSTRACT FROM AUTHOR]

Sharp Jackson-Stechkin type inequalities in which the modulus of continuity of mth order of functions is defined via the Steklov function are obtained. For the classes of functions defined by these moduli of continuity, exact values of various n-widths are derived. [ABSTRACT FROM AUTHOR]

Very recently, Sonker and Singh [21] determined the degree of approximation of the conjugate of 2π-periodic signals (functions) belonging to Lip(α, r)(0 < α = 1, r = 1)-class by using Cesàro-Euler (C,1) (E,q) means of their conjugate trigonometric Fourier series. In the present paper, we generalize the result of Sonker and Singh [21] on the generalized weighted Lipschitz W(Lr, ξ(t)), (r = 1)- class of signals (functions) by product summability (C,1) (E,q) transform of conjugate trigonometric Fourier series. Our result also generalizes the result of Lal and Singh [6]. Few applications and example of approximation of functions will also be highlighted. [ABSTRACT FROM AUTHOR]

In this paper, new best approximations of the function f E Zr(w), (r = 1) class by Matrix-Euler means (Δ.E1) of its Fourier Series have been determined. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we present low-residual approximate solutions for nonlocal 1D and 2D elasticity problems defined according to Eringen’s integral model. The benchmarks in the 1D cases are defined by prescribing the stress field while the unknown fields are the strains or the displacements. For the 2D cases we define problems with equilibrated tractions and evaluate the approximate displacement field. Meanwhile a Fourier series as well as a set of Chebyshev polynomials are used as the basis functions for the main unknown fields. We increase the number of the approximation functions to decrease the norm of the residuals and repeat the procedure until reasonable accuracy is obtained for the final solution. Since the procedure is very time consuming, in some benchmark problems we present the calculated coefficients and in some other we give some point-wise values for further use. The results presented in this paper are particularly useful for the validation and convergence studies when numerical methods are to be used for the solution of the nonlocal elasticity problems. [Copyright &y& Elsevier]

Recently, Lal [9] has determined the degree of approximation of function belonging to Lip α and weighted classes using Product C¹. Np summability with non-increasing weights{pn}. In this paper, we determine the degree of approximation of function f, conjugate to a 2π-periodic function f belonging to Lip (ξ (t), r)-class using semi-monotonicity on the generating sequence {pn} with proper set of conditions. Few applications of approximation of functions will also be highlighted. [ABSTRACT FROM AUTHOR]