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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]

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]

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 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]

A Brownian dynamics simulation technique is presented where a Fourier-based N log N approach is used to calculate hydrodynamic interactions in confined flowing polymer systems between two parallel walls. A self-consistent coarse-grained Langevin description of the polymer dynamics is adopted in which the polymer beads are treated as point forces. Hydrodynamic interactions are therefore included in the diffusion tensor through a Green’s function formalism. The calculation of Green’s function is based on a generalization of a method developed for sedimenting particles by Mucha et al. [J. Fluid Mech. 501, 71 (2004)]. A Fourier series representation of the Stokeslet that satisfies no-slip boundary conditions at the walls is adopted; this representation is arranged in such a way that the total O(N2) contribution of bead-bead interactions is calculated in an O(N log N) algorithm. Brownian terms are calculated using the Chebyshev polynomial approximation proposed by Fixman [Macromolecules 19, 1195 (1986); 19, 1204 (1986)] for the square root of the diffusion tensor. The proposed Brownian dynamics simulation methodology scales as O(N1.25 log N). Results for infinitely dilute systems of dumbbells are presented to verify past predictions and to examine the performance and numerical consistency of the proposed method. [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]

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]

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 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]

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]

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]

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]

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]

The order estimates of the best M-term trigonometric approximations of functions $$ {D}_{\beta}^{\psi } $$ and the classes of ( ψ , β)-differentiable periodic multivariate functions in the space L, 2 ≤ q < ∞ are obtained. It is shown that, under certain conditions imposed on the parameter q; the best M-term trigonometric approximations $$ {e}_M{\left({L}_{\beta, 1}^{\psi}\right)}_q $$ have better order than the best orthogonal trigonometric approximations $$ e\frac{1}{M}{\left({L}_{\beta, 1}^{\psi}\right)}_q $$. [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]

This paper presents an analytical method for designing a spatial filter that processes the data from an array of two-dimensional guided wave transducers. An inverse problem is defined where the spatial filter coefficients are determined in such a way that a prescribed beam shape, i.e., a desired array output is best approximated in the least-squares sense. Taking advantage of the 2 π -periodicity of the generated wave field, Fourier-series representation is used to derive closed-form expressions for the constituting matrix elements. Special cases in which the desired array output is an ideal delta function and a gate function are considered in a more explicit way. Numerical simulations are performed to examine the performance of the filters designed by the proposed method. It is shown that the proposed filters can significantly improve the beam quality in general. Most notable is that the proposed method does not compromise between the main lobe width and the sidelobe levels; i.e. a narrow main lobe and low sidelobes are simultaneously achieved. It is also shown that the proposed filter can compensate the effects of nonuniform directivity and sensitivity of array elements by explicitly taking these into account in the formulation. From an example of detecting two separate targets, how much the angular resolution can be improved as compared to the conventional delay-and-sum filter is quantitatively illustrated. Lamb wave based imaging of localized defects in an elastic plate using a circular array is also presented as an example of practical applications. [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]

In this paper, we reveal a new connection between approximation numbers of periodic Sobolev type spaces, where the smoothness weights on the Fourier coefficients are induced by a (quasi-)norm ∥⋅∥ on Rd, and entropy numbers of the embedding id : ℓ∥⋅∥d → ℓ∞d. This connection yields preasymptotic error bounds for approximation numbers of isotropic Sobolev spaces, spaces of analytic functions, and spaces of Gevrey type in L2 and H¹, which find application in the context of Galerkin methods. Moreover, we observe that approximation numbers of certain Gevrey type spaces behave preasymptotically almost identical to approximation numbers of spaces of dominating mixed smoothness. This observation can be exploited, for instance, for Galerkin schemes for the electronic Schrödinger equation, where mixed regularity is present. [ABSTRACT FROM AUTHOR]

The results corresponding to some theorems of S. Lal [Appl. Math. Comput. 209 (2009), 346-350] and the results of the authors [In: Banach Center Publ. 95, Polish Acad. Sci. Inst. Math., Warsaw, 2011, pp. 339-351] are shown. The same degrees of pointwise approximation as in mentioned papers by weaker assumptions on considered functions and examined summability methods are obtained. From presented pointwise results the estimation on norm approximation are derived. Some special cases as corollaries are also formulated. [ABSTRACT FROM AUTHOR]

We build a numerical approximation method, for Fredholm integral equation solution of the second type. This method is based on the regularization by convolution and Fourier series expansion. It provides a better convergence order. [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]

The aim of the paper is to determine the degree of approximation of functions by matrix means of their Fourier series in a new space of functions introduced by Das, Nath, and Ray. In particular, we extend some results of Leindler and some other results by weakening the monotonicity conditions in the results obtained by Singh and Sonker for some classes of numerical sequences introduced by Mohapatra and Szal and present new results by using matrix means. [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]

A new pseudo-spectral operator is developed for time-spectral harmonic balance solutions of periodic unsteady flows. The method utilises a mechanism similar to sigma-approximation technique with Lanczos filtering function that alters the inverse of the discrete Fourier transformation matrix, leading to a modified pseudo-spectral operator. The modified operator is then used instead of the original operator that mimics the time-derivative term of the unsteady governing equations. The modified operator is capable of damping high-frequency nonlinearities in the harmonic balance solution, thus alleviating the effects of high-frequency oscillations that result in Gibbs-type phenomena. The effectiveness and robustness of the technique are demonstrated through various test cases. [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]