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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]

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]

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]

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]

The analysis of nonlinear fractional-order circuits is a challenging problem. This is due to the lack of nonlinear circuit theorems and designs particularly in the presence of memristive elements. The response of a series connection of a simple resistor with fractional order capacitor and its analytical formulation in both charging and discharging phases is considered. The numerical simulation of fractional order HP memristor in series with a fractional order capacitor is also discussed. It is a demonstration of a simple nonlinear fractional-order memristive circuit in both charging and discharging cases. Furthermore, this paper introduces an approach to approximate nonlinear fractional-order memrisitve circuits by linear circuits using a minimax optimization technique. Hence, the new circuit can be analyzed using the conventional linear circuit theorems. The charging and discharging of a series fractional-order memristor with a fractional-order capacitor are discussed numerically. The effect of fractional-order parameters and memristor polarity are also investigated. Using a suitable optimization technique, an accurate approximation by a circuit that include a resistor and a fractional-capacitor is obtained for both charging and discharging cases. A great matching was observed between the frequency responses of the fractional-order nonlinear low pass filter based on fractional-order memristor and fractional-order capacitor and that of the optimized linear fractional order case. Similar matching is observed for the nonlinear and optimized cases when a periodic triangular waveform is applied using Fourier series expansion. [ABSTRACT FROM AUTHOR]

We propose two methods to find analytic periodic approximations intended for differential equations of Hill type. Here, we apply these methods on the simplest case of the Mathieu equation. The former has been inspired in the harmonic balance method and designed to find, making use on a given algebraic function, analytic approximations for the critical values and their corresponding periodic solutions of the Mathieu differential equation. What is new is that these solutions are valid for all values of the equation parameter q , no matter how large. The second one uses truncations of Fourier series and has connections with the least squares method. [ABSTRACT FROM AUTHOR]

The paper is primarily devoted to the problem of smooth interpolation of data in 1D. In addition to the exact interpolation of data at nodes, we are also concerned with the smoothness of the interpolating curve and its derivatives. The interpolating curve is defined as the solution of a variational problem with constraints. The system of functions exp ( - i kx ) , k being an integer, is taken for the basis of the space where we measure the smoothness of the result. It also generates the functions used for the interpolation itself. Choosing different norms when measuring the smoothness, we arrive at different interpolating functions. We also mention the problem of smooth curve fitting (data smoothing). We discuss the proper choice of different norms for this way of approximation and present the results of several 1D numerical examples that show the advantages and drawbacks of smooth interpolation. [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]

In this paper, a new method based on the Fourier series is introduced to obtain approximate solutions to the systems of the point kinetics equations. These systems are stiff involving equations with slowly and rapidly varying components. They are solved numerically using Fourier series expansion over a partition of the total time interval. Approximate solution requires determining the series coefficients over each time step in that partition. These coefficients are determined using the high order derivatives of the dependent variables at the beginning of the time step introducing a system of linear algebraic equations to be solved at each step. The obtained algebraic system is similar to the Vandermonde system. Evaluation of the inverse of the Vandermonde matrix is required to determine the coefficients of the Fourier series. Because the obtained Vandermonde matrix has a special structure, due to the properties of the sine and cosine functions, a new formula is introduced to determine its inverse using standard computations. The new method solves the general linear and non-linear kinetics problems with six groups of delayed neutrons. The validity of the algorithm is tested with five different types of reactivities including step reactivity insertion, ramp input, oscillatory reactivity changes, a reactivity as a function of the neutron density and finally temperature feedback reactivity. Comparisons are made with analytical and conventional numerical methods used to solve the point kinetics equations. The results confirm the theoretical analysis and indicate the validity of the method for all applications considered. [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]

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 the present paper, a new theorem on the degree of approximation of function conjugate to a 2π periodic function f belonging to the generalized weighted Lipschitz -class by dropping the monotonicity condition on the generating sequence {pn } has been established which in turn generalizes the results of Lal (2009) [12] on Fourier series. We also note some errors appearing in the paper of Lal (2009) [12] and rectify them in the view of observations of Rhoades et al. (2011) [22]. [Copyright &y& Elsevier]

Highlights: [•] Introduce a finite Fourier-series model for evaluating monthly movement of annual average solar insolation. [•] Present a forecast method for predicting its movement based on the extended Fourier-series model in the least-squares. [•] Shown its movement is well described by a low numbers of harmonics with approximately 6-term Fourier series. [•] Predict its movement most fitting with less than 6-term Fourier series. [ABSTRACT FROM AUTHOR]

Abstract: In order to compute an approximate solution of a weakly singular integral equation, we first regularize the kernel and then truncate the associated Fourier series. Applications to Green and Abel operators are given. [Copyright &y& Elsevier]

Abstract: In this paper, two new sub-classes of linear-in-the-parameters nonlinear discrete-time filters, derived from the truncation of multidimensional generalized Fourier series, are presented. The filters, called Fourier nonlinear filters and even mirror Fourier nonlinear filters, are universal approximators for causal, time-invariant, finite-memory, continuous nonlinear systems, according to the Stone–Weierstrass approximation theorem. Their properties and limitations are discussed in detail. In particular, we show, by means of appropriate simulation examples, that an orthogonality property they satisfy for white uniform input signals is useful for improving the identification of nonlinear systems. [Copyright &y& Elsevier]

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]

Abstract: We present an analysis of the convergence of recently developed Fourier continuation techniques that incorporates the required truncation of the Singular Value Decomposition (SVD). Through the analysis, the convergence of SVD-based continuations are related to the convergence of any Fourier approximation of similar form, demonstrating the efficiency and accuracy of the numerical method. The analysis determines that the Fourier continuation approximation error can be bounded by a key value that depends only on the parameters of the Fourier continuation and on the points over which it is applied. For any given distribution of points, a finite number of calculations can be performed to obtain this important value. Our numerical computations on evenly spaced points show that as the number of points increases, this quantity converges to a fixed value, allowing for broad conclusions on the convergence of Fourier continuations calculated with truncated SVDs. We conclude that Fourier continuations can obtain super-algebraic or even exponential convergence on evenly spaced points for non-periodic functions until the convergence is limited by a parameter normally chosen near the machine precision accuracy threshold. [Copyright &y& Elsevier]

Abstract: In this paper, a study of the approximation of functions from by Fourier–Laplace series is performed. It is proved that the maximal operator of the Riesz means of the Fourier–Laplace series is bounded, from to . The result provides a natural and intrinsic characterization of the approximation of the functions by Fourier–Laplace series. [Copyright &y& Elsevier]

Abstract: Recently, the micropositioner has become an important developing target for achieving the requirements of precision machinery. The piezo-actuating device plays a very important role in this application area. In this paper, a model-free adaptive sliding-mode controller is proposed for a 3D piezo-actuating system because of the system’s hysteresis nonlinearity and time-varying characteristics. This control strategy employs the functional approximation technique to establish the unknown function for releasing the model based requirements of the sliding-mode control. The update laws for the coefficients of the Fourier series function parameters are derived from a Lyapunov function to guarantee the control system stability. To verify the effectiveness of the proposed controller, drilling process control using the designed controller is investigated in this paper. [Copyright &y& Elsevier]

Abstract: The flow of a gravity-driven thin liquid film over a substrate containing topography, in the presence of a normal electric field, is investigated. The liquid is assumed to be a perfect conductor and the air above it a perfect dielectric. Of particular interest is the interplay between inertia, for finite values of the Reynolds number, Re, and electric field strength, expressed in terms of the Weber number, We, on the resultant free-surface disturbance away from planarity. The hydrodynamics of the film are modelled via a depth-averaged form of the Navier–Stokes equations which is coupled to a Fourier series separable solution of Laplace’s equation for the electric potential: detailed steady-state solutions of the coupled equation set are obtained numerically. The case of two-dimensional flow over different forms of discrete and periodically varying spanwise topography is explored. In the case of the familiar free-surface capillary peaks and depressions that arise for steep topography, and become more pronounced with increasing Re, greater electric field strength affects them differently. In particular, it is found that for topography heights commensurate with the long-wave approximation: (i) the capillary ridge associated with a step-down topography at first increases before decreasing, both monotonically, with increasing We and (ii) the free-surface hump which arises at a step-up topography continues to increase monotonically with increasing We, the increase achieved being smaller the larger the value of Re. A series of results for the more practically relevant problem of three-dimensional film flow over substrate containing a localised square trench topography is provided. These exhibit behaviour and features consistent with those observed for two-dimensional flow, in that as We is increased the primary free-surface capillary ridges and depressions are at first enhanced, with a corresponding narrowing, before becoming suppressed. In addition, it is found that, while the well-known horse-shoe shaped disturbance characteristic of such flows continues to persist with increasing Re in the absence of an electric field, when the latter is present and We increased in value the associated comet tail disappears as does the related downstream surge. The phenomenon is explained with reference to the competition between the corresponding capillary pressure and Maxwell stress distributions. [Copyright &y& Elsevier]

Abstract: The Variational Theory of Complex Rays (VTCR) is a wave-based computational approach dedicated to the resolution of medium-frequency problems. It uses a variational formulation of the problem which enables one to use any type of shape function within the substructures provided that it satisfies the governing equation. Thus, the solution can be approximated using plane waves, which is very interesting in the medium-frequency vibration domain and also leads to a strong convergence of the method. In the previous works, this was shown in the case of acoustic problems in which the amplitudes of the plane waves were calculated as wavebands. In this paper, we propose a new approximation of these amplitudes based on Fourier series. We show that this approach increases the robustness of the method, makes it more efficient numerically and extends its applicability to somewhat higher frequencies. [Copyright &y& Elsevier]

Abstract: First-principles calculations are used to predict the generalized-stacking-fault energy (GSFE) surfaces of AlRE intermetallics. The calculations employ the projector augmented-wave (PAW) method within the generalized gradient approximation (GGA) using the density functional theory (DFT). GSFE curves along {110} direction, {110} direction and {110} direction have been calculated. The fitted GSFE surfaces have been obtained from the Fourier series based on the translational symmetry. In order to illuminate the reasonable of our computational accuracy, we have compared our theoretical results of B2 intermetallics YCu with the previous calculated results. The unstable-stacking-fault energy () on the {110} plane has the laws of along the , and directions. For the antiphase boundary (APB) energy, that of AlSc is the lowest in the calculated AlRE intermetallics. So the superdislocation with the Burgers vector along direction of AlSc will easily split into two superpartials. [Copyright &y& Elsevier]

Abstract: In the conventional semi-analytical finite strip analysis of folded plates, the boundary conditions and the intermediate support conditions must be satisfied a priori. The admissible functions used as the longitudinal part of the displacement functions are sometimes difficult to find, and they are valid for specific conditions only. In this paper, a general finite strip is developed for the static and vibration analyses of folded plate structures. The geometric constraints of the folded plates, such as the conditions at the end and intermediate supports, are modelled by very stiff translational and rotational springs as appropriate. The complete Fourier series including the constant term are chosen as the longitudinal approximating functions for each of the displacements. As these displacement functions are more general in nature and independent of one another, they are capable of giving more accurate solutions. The potential problem of ill-conditioned matrices is investigated and the appropriate choice of the very stiff springs is also suggested. The formulation is done in such a way to obtain a unified approach, taking full advantage of the power of modern computers. A few numerical examples are presented for comparison with numerical results from published solutions or solutions obtained from the finite element method. The results show that this kind of strips is versatile, efficient and accurate for the static and vibration analyses of folded plates. [Copyright &y& Elsevier]

Abstract: The enthalpy method is exploited in tackling a heat transfer problem involving a change of state. The resulting governing equation is then solved with a hybrid finite element – boundary element technique known as the Green element method (GEM). Two methods of approximation are employed to handle the time derivative contained in the discrete element equation. The first involves a finite difference method, while the second utilizes a Galerkin finite element approach. The performance of both methods are assessed with a known closed form solution. The finite element based time discretization, despite its greater challenge, yields less reliable numerical results. In addition a numerical stability test of both methods based on a Fourier series analysis explain the dispersive characters of both techniques, and confirms that replication of correct results is largely attributed to their ability to handle the harmonics of small wavelengths which are usually dominant in the vicinity of a front. [Copyright &y& Elsevier]

Abstract: The paper presents a new meshless numerical technique for solving one-dimensional problems with moving boundaries including the Stefan problems. The technique presented is based on the use of the delta-shaped functions and the method of approximate fundamental solutions (MAFS) firstly suggested for solving elliptic problems and for heat equations in domains with fixed boundaries. The numerical examples are presented and the results are compared with analytical solutions. The comparison shows that the method presented provides a very high precision in determining the position of the moving boundary even for a region that initially has zero thickness. [Copyright &y& Elsevier]

Abstract: In a recent paper Lal and Yadov obtained a theorem on the degree of approximation for a function belonging to the Lipschitz class Lipα using the product of the Cesàro and Euler means of order one of its Fourier series. In this paper we extend this result to any regular Hausdorff matrix for the same class of functions. [Copyright &y& Elsevier]

Abstract: A detailed analysis of the impact of a hysteretic quantizer on a multibit, Sigma-Delta modulator has been carried out in this paper. Both discrete-time and continuous-time modulators have been considered. A qualitative modeling of the hysteretic quantizer based on a hysteretic block followed by an ideal quantizer was proposed. Due to the hysteresis effect, the quantizer output signal is delayed and distorted with respect to the quantizer input signal, where the delay causes a phase-shift independent on the signal frequency. Yet, the effect of the hysteresis depends on the input signal amplitude. This model was validated by using system-level simulations for a second order, 3-bit, discrete-time Sigma-Delta modulator. A linear model for hysteresis was derived by assuming a narrow hysteresis cycle. The quantizer input signal plays a fundamental role in the discussion. In order to include this signal into the linear analysis some approximations are proposed. The quantizer output signal is decomposed by the use of the Fourier series analysis only into the in-phase and quadrature components (with respect to the input signal) whose Fourier series coefficients can be analytically calculated. A quantitative analysis for both a second order, 3-bit, DT and CT Sigma-Delta modulators including a hysteretic quantizer was carried out. For the CT modulator, finite GBW in amplifiers, excess loop delay, and a hysteretic quantizer were considered separately and combined. A good agreement with both system-level simulations and experimental results is found, despite the approximations considered for the quantizer input signal. [ABSTRACT FROM AUTHOR]

Abstract: The interior and exterior Neumann problems for the Helmholtz equation in starlike planar domains are addressed by using a suitable Fourier-like technique. Attention is in particular focused on normal-polar domains whose boundaries are defined by the so called “superformula” introduced by Gielis. A dedicated numerical procedure based on a computer algebra system is developed in order to validate the proposed approach. In this way, highly accurate approximations of the solution, featuring properties similar to classical ones, are obtained. Computed results are found to be in good agreement with theoretical findings on Fourier series expansion presented by Carleson. [Copyright &y& Elsevier]

Abstract: A rapidly convergent series, based on Fourier expansion of the exponential multiplier, is presented for highly accurate approximation of the Voigt function (VF). The corresponding algorithm enables the rapid calculation, required for its implementation as a subprogram in an interpolation approach. The numerical analysis of this VF approximation suggests that it may be more accurate than 10−9 in the Humlíček regions 3 and 4. [Copyright &y& Elsevier]

Abstract: We study the Poisson problem with zero boundary datum in a (finite) polyhedral cylinder with a non-convex edge. Applying the Fourier sine series to the equation along the edge and by a corner singularity expansion for the Poisson problem with parameter, we define the edge flux coefficient and the regular part of the solution on the polyhedral cylinder. We present a numerical method for approximating the edge flux coefficient and the regular part and show the stability. We derive an error estimate and give some numerical experiments. [Copyright &y& Elsevier]

Abstract: This paper addresses the approximation problem of functions affected by unknown periodically time-varying disturbances. By combining Fourier series expansion into multilayer neural network or radial basis function neural network, we successfully construct two kinds of novel approximators, and prove that over a compact set, the new approximators can approximate a continuously and periodically disturbed function to arbitrary accuracy. Then, we apply the proposed approximators to disturbance rejection in the first-order nonlinear control systems with periodically time-varying disturbances, but it is straightforward to extend the proposed design methods to higher-order systems by using adaptive backstepping technique. A simulation example is provided to illustrate the effectiveness of control schemes designed in this paper. [Copyright &y& Elsevier]

Abstract: Pulsed photothermal techniques provide useful methods based on linear relations between measurable quantities to obtain the thermal diffusivity and thermal conductivity of homogeneous materials. In this work, the effective thermal parameters of two-layered films are defined starting from an homogeneous layer which at the surfaces, produces the same temperature fluctuations and the same photothermal signal that the composite heated by a fast pulse-laser. Our theoretical model predicts that the effective thermal parameters of the layered system can only be calculated in the limit when the laser pulse duration is smaller tan the characteristic time of each layer, respectively. The temperature distribution is calculated in each layer by using the Fourier integral and the time-dependent one-dimensional heat diffusion equation with appropriate boundary conditions according to the experimental conditions. Within this approximation, we found an analytical expression for both, the effective thermal diffusivity and thermal conductivity which depend significantly on the thickness and the thermal parameters of each film. [Copyright &y& Elsevier]

Abstract: In this paper, two new theorems on the degree of approximation of the function f ∈Lipα and f ∈ W(L r , ξ(t)) have been established. [Copyright &y& Elsevier]

Abstract: In this study, by obtaining some Korovkin type approximation results in statistical sense for certain positive linear operators constructed by means of the Chan–Chyan–Srivastava multivariable polynomials [W.-C.C. Chan, C.-J. Chyan, H.M. Srivastava, The Lagrange polynomials in several variables, Integral Transform. Spec. Funct. 12 (2001) 139–148], we show that our approximation method is stronger than the corresponding classical aspects in the approximation theory settings. Furthermore, we investigate their statistical rates by means of the modulus of continuity and the elements of the Lipschitz class. [Copyright &y& Elsevier]

The approximation of a function affected by noise in several dimensions suffers from the so-called “curse of dimensionality”. In this paper a Fourier series method based on regularization is developed both for uniform and random design when a restriction on the complexity of the curve such as additivity is considered in order to circumvent the problem. Optimal convergence theorems are stated and numerical experiments are shown on several test problems available in the literature together with comparisons with alternative methods. [Copyright &y& Elsevier]

Abstract: The adaptive learning approach proposed in and , which guarantees output tracking of sufficiently smooth periodic reference signals (of known period) for classes of time-invariant nonlinear systems with sufficiently smooth unstructured uncertainties, is considered. By using a slightly different analysis (and in particular less conservative bounds), we show how improved convergence properties can be established while maintaining the same learning control structures. The delayed first arithmetic mean of the Fourier series of the uncertain periodic time function (of known period) can be used instead of the partial sums of the Fourier series of to further improve the result: a trigonometric polynomial, whose approximation to is almost as good as the best approximation of , is obtained. [Copyright &y& Elsevier]

Recently, L.N. Mishra et al. obtained a new theorem on the degree of approximation of function f ̃ : conjugate to a 2 π periodic function f belonging to the generalized weighted Lipschitz W ( L r , ξ ( t ) ) ( r ⩾ 1 ) -class (Mishra et al., 2014). In this note, we point out that the conclusion of the theorem only holds for the constant functions, and thus the result is trivial. [ABSTRACT FROM AUTHOR]