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We revisit methods for computing flow over single and multi-element airfoils using more recently developed numerical methods for conformal mapping for simply and multiply connected domains. The corners at the trailing edges of the airfoils are successively removed by Karman–Trefftz maps. The map from the domain exterior to disks to the domain exterior to the smooth images of the airfoils is computed using extensions of a Fourier series method for the disk due to Fornberg. The velocity potential for flow in the circle domain, with circulation calculated to satisfy the Kutta condition in the airfoil domain, is computed by a reflection method based on the Milne-Thomson Circle Theorem. Convergence of the reflection method is proven if the domains are sufficiently well-separated. [ABSTRACT FROM AUTHOR]

This paper presents a method to obtain semi-analytical solutions to second order semilinear IBVPs where the standard Adomian decomposition may fail to yield nontrivial solutions or fail to produce the correct partial solutions. In contrast to the standard Adomian decomposition method, the proposed solution method is distinguished by simultaneous inversion of the linear differential operators using eigenfunctions expansion representations. The proposed method is applied to several examples of initial boundary value problems — a linear Advection–Diffusion problem, a Burgers equation and the deterministic Kardar–Parisi–Zhang (KPZ) equation. It is shown that the dependence of the series solution on the truncation order ( N ), the number of eigenfunctions ( M ) and the diffusion coefficient is rather complex. [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]

This paper is concerned with a variational approach solving the two-dimensional acoustic transmission problems. The original problem is reduced to an equivalent nonlocal boundary value problem by introducing an exact Dirichlet-to-Neumann (DtN) mapping in terms of Fourier expansion series on an artificial boundary. Uniqueness and existence of solutions in appropriate Sobolev spaces are established for the corresponding variational problem and its modification due to the truncation of DtN mapping. A priori error estimates containing the effects of both element meshsize and truncation order of series for the finite element approximation are derived. Numerical experiments are also presented to illustrate the efficiency and accuracy of the numerical scheme. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Cauchy problem connected with the Helmholtz equation. We investigate the denseness of Fourier–Bessel functions and propose a numerical method for approximating the solution to the problem. The convergence and stability are analyzed with truncated singular value decomposition method. Numerical experiments are also presented to show the effectiveness of our method. [ABSTRACT FROM AUTHOR]

The most important factor in planning and operating water distribution systems is satisfying consumer demand. This means continuously providing users with quality water in adequate volumes at reasonable pressure, thus ensuring reliable water distribution. In recent years, the application of statistical, machine learning, and artificial intelligence methodologies has been fostered for water demand forecasting. However, there is still room for improvement; and new challenges regarding on-line predictive models for water demand have appeared. This work proposes applying support vector regression, as one of the currently better machine learning options for short-term water demand forecasting, to build a base prediction. On this model, a Fourier time series process is built to improve the base prediction. This addition produces a tool able to eliminate many of the errors and much of the bias inherent in a fixed regression structure when responding to new incoming time series data. The final hybrid process is validated using demand data from a water utility in Franca, Brazil. Our model, being a near real-time model for water demand, may be directly exploited in water management decision-making processes. [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: Two-dimensional hypersingular equations over a disc are considered. A spectral method is developed, using Fourier series in the azimuthal direction and orthogonal polynomials in the radial direction. The method is proved to be convergent. Then, Tranter’s method is discussed. This method was devised in the 1950s to solve certain pairs of dual integral equations. It is shown that this method is also convergent because it leads to the same algebraic system as the spectral method. [Copyright &y& Elsevier]

Abstract: The description of the electron wavefunctions in atoms is generalized to the fractional Fourier series. This method introduces a continuous and infinite number of chirp basis sets with linear variation of the frequency to expand the wavefunctions, in which plane-waves are a special case. The chirp characteristics of each basis set can be adjusted through a single parameter. Thus, the basis set cutoff can be optimized variationally. The approach is tested with the expansion of the electron wavefunctions in atoms, and it is shown that chirp basis sets substantially improve the convergence in the description of the electron density. We have found that the natural oscillations of the electron core states are efficiently described in chirp-waves. [Copyright &y& Elsevier]

Abstract: Functions that are smooth but non-periodic on a certain interval possess Fourier series that lack uniform convergence and suffer from the Gibbs phenomenon. However, they can be represented accurately by a Fourier series that is periodic on a larger interval. This is commonly called a Fourier extension. When constructed in a particular manner, Fourier extensions share many of the same features of a standard Fourier series. In particular, one can compute Fourier extensions which converge spectrally fast whenever the function is smooth, and geometrically fast if the function is analytic, much the same as the Fourier series of a smooth/analytic and periodic function. With this in mind, the purpose of this paper is to describe, analyse and explain the observation that Fourier extensions, much like classical Fourier series, also have excellent resolution properties for representing oscillatory functions. The resolution power, or required number of degrees of freedom per wavelength, depends on a user-controlled parameter and, as we show, it varies between 2 and . The former value is optimal and is achieved by classical Fourier series for periodic functions, for example. The latter value is the resolution power of algebraic polynomial approximations. Thus, Fourier extensions with an appropriate choice of parameter are eminently suitable for problems with moderate to high degrees of oscillation. [Copyright &y& Elsevier]

Abstract: We consider the solutions of the one-dimensional heat equation in an unbounded domain with initial conditions of the form . This includes as a particular case the logistic-normal integral, which corresponds to . Such initial conditions appear in stochastic calculus problems, and the numerical simulation of short-rate interest rate models and credit models with log-normally distributed short rates and hazard rates respectively. We show that the solutions at time can be computed exactly on a grid of equidistant points of width in terms of the solutions of the heat equation with initial condition . The exact results on the grid can be used as nodes for a precise interpolation. Series representation of the solutions can be obtained by an application of the Poisson summation formula. [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: This paper introduces a new technique for the localization of discontinuity points from spectral data. Through this work, we will be able to detect discontinuity points of a -periodic piecewise smooth function from its Fourier coefficients. This could be useful in detecting edges and reducing the effects of the Gibbs phenomenon which appears near discontinuities and affects signal restitution. Our approach consists in moving from a discontinuity point detection problem to a pole detection problem, then adapting the quotient-difference (qd) algorithm in order to detect those discontinuity points. [Copyright &y& Elsevier]

Abstract: An efficient construction of two non-classical families of orthogonal polynomials is presented in the paper. The so-called half-range Chebyshev polynomials of the first and second kinds were first introduced by Huybrechs in Huybrechs (2010) . Some properties of these polynomials are also shown. Every integrable function can be represented as an infinite series of sines and cosines of these polynomials, the so-called half-range Chebyshev–Fourier (HCF) series. The second part of the paper is devoted to the efficient computation of derivatives and multiplication of the truncated HCF series, where two matrices are constructed for this purpose: the differentiation and the multiplication matrix. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with the error analysis for the finite element solution of the two-dimensional exterior Neumann boundary value problem in acoustics. In particular, we establish explicit priori error estimates in and - norms including both the effect of the truncation of the DtN mapping and that of the numerical discretization. To apply the finite element method (FEM) to the exterior problem, the original boundary value problem is reduced to an equivalent nonlocal boundary value problem via a Dirichlet-to-Neumann (DtN) mapping represented in terms of the Fourier expansion series. We discuss essential features of the corresponding variational equation and its modification due to the truncation of the DtN mapping in appropriate function spaces. Numerical tests are presented to validate our theoretical results. [Copyright &y& Elsevier]

Abstract: Combining Fourier series expansion with recursive matrix formulas, new reliable algorithms to compute the periodic, non-negative, definite stabilizing solutions of the periodic Riccati and Lyapunov matrix differential equations are proposed in this paper. First, periodic coefficients are expanded in terms of Fourier series to solve the time-varying periodic Riccati differential equation, and the state transition matrix of the associated Hamiltonian system is evaluated precisely with sine and cosine series. By introducing the Riccati transformation method, recursive matrix formulas are derived to solve the periodic Riccati differential equation, which is composed of four blocks of the state transition matrix. Second, two numerical sub-methods for solving Lyapunov differential equations with time-varying periodic coefficients are proposed, both based on Fourier series expansion and the recursive matrix formulas. The former algorithm is a dimension expanding method, and the latter one uses the solutions of the homogeneous periodic Riccati differential equations. Finally, the efficiency and reliability of the proposed algorithms are demonstrated by four numerical examples. [Copyright &y& Elsevier]

Abstract: We discuss the numerical computation of the cosine lemniscate function and its inverse, the lemniscate integral. These were previously studied by Bernoulli, Euler, Gauss, Abel, Jacobi and Ramanujan. We review general elliptic formulas for this special case and provide some novelties. We show that a Fourier series by Ramanujan converges twice as fast as the standard elliptic cosine Fourier series specialized to this case. The so-called imbricate series, however, converges geometrically fast over the entire complex plane. We derive two new expansions. The rational-plus-Fourier series converges much faster than Ramanujan’s: for real : each term is asymptotically 12,400 times smaller than its immediate predecessor: where is the elliptic nome, is the complete elliptic integral of the first kind for a modulus and . The rational imbricate series is uniformly valid over the complex plane, but converges twice as fast as the sech-imbricate series: where is the period in both the real and imaginary directions. We devise a new approximation for the lemniscate integral for real argument as the arccosine of a Chebyshev series and show that 17 terms yield about 15 digits of accuracy. For complex argument, we show that the lemniscate integral can be found to near machine precision (assumed as sixteen decimal digits) by computing the roots of a polynomial of degree thirteen. Alternatively, Newton’s iteration converges in three iterations with an initialization, accurate to four decimal places, that is the chosen root of a cubic equation. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a new kind of algebraic–trigonometric blended spline curve, called curves, generated over the space . The new curves not only inherit most properties of usual cubic B-spline curves in polynomial space, but also enjoy some other advantageous properties for modeling. For given control points, the shape of the new curves can be adjusted by using the parameters and . When the control points and the parameters are chosen appropriately, the new curves can represent some conics and transcendental curves. In addition, we present methods of constructing an interpolation -spline curve and an -spline curve which is tangent to the given control polygon. The generation of tensor product surfaces by these new spline curves is straightforward. Many properties of the curves can be easily extended to the surfaces. The new surfaces can exactly represent the rotation surfaces as well as the surfaces with elliptical or circular sections. [ABSTRACT FROM AUTHOR]

Abstract: An uncertainty inequality for the Fourier–Dunkl series, introduced by the authors in [Ó. Ciaurri, J.L. Varona, A Whittaker–Shannon–Kotel’nikov sampling theorem related to the Dunkl transform, Proc. Amer. Math. Soc. 135 (2007) 2939–2947], is proved. This result is an extension of the classical uncertainty inequality for the Fourier series. [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: For most orthogonal systems and their corresponding Fourier series, the study of the almost everywhere convergence for functions in requires very complicated research, harder than in the case of the mean convergence. For instance, for trigonometric series, the almost everywhere convergence for functions in is the celebrated Carleson theorem, proved in 1966 (and extended to by Hunt in 1967). In this paper, we take the system (with being the Bessel function of the first kind and of the order ), which is orthonormal in , and whose Fourier series are the so-called Fourier–Neumann series. We study the almost everywhere convergence of Fourier–Neumann series for functions in and we show that, surprisingly, the proof is relatively simple (inasmuch as the mean convergence has already been established). [Copyright &y& Elsevier]

Abstract: We consider quadrature formulas of high degree of precision for the computation of the Fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials. In particular, we show the uniqueness of a multiple node formula for the Fourier–Tchebycheff coefficients given by Micchelli and Sharma and construct new Gaussian formulas for the Fourier coefficients of a function, based on the values of the function and its derivatives. [Copyright &y& Elsevier]

Abstract: Highly oscillatory integrals require special techniques for their effective evaluation. Various studies have been conducted to find computational methods for evaluating such integrals. In this paper we present an efficient numerical method to evaluate a class of generalised Fourier integrals (on a line or a square) with integrands of the form , under the assumption that in the domain of integration, both and are sufficiently smooth and that does not have any stationary/critical points. Numerical analysis and results are given to illustrate the effectiveness of our method for computing generalised Fourier integrals. [Copyright &y& Elsevier]

Abstract: We discuss the effect of a particular sequence acceleration method, the process, on the partial sums of Fourier series. We show that for a very general class of functions, this method fails on a dense set of points; not only does it not speed up convergence, it turns the sequence of partial sums into a sequence with multiple limit points. [Copyright &y& Elsevier]

Abstract: Recently, Brezinski has proposed to use Wynn''s -algorithm in order to reduce the Gibbs phenomenon for partial Fourier sums of smooth functions with jumps, by displaying very convincing numerical experiments. In the present paper we derive analytic estimates for the error corresponding to a particular class of hypergeometric functions, and obtain the rate of column convergence for such functions, possibly perturbed by another sufficiently differentiable function. We also analyze the connection to Padé–Fourier and Padé–Chebyshev approximants, including those recently studied by Kaber and Maday. [Copyright &y& Elsevier]

Abstract: We study boundary value problems of the form on and on the boundary , with either Dirichlet or Neumann boundary conditions, where is a smooth bounded domain in and the data are distributions. This problem has to be first properly reformulated and, for practical applications, it is of crucial importance to obtain the continuity of the solution in terms of f and g. For , taking advantage of the fact that is harmonic on , we provide four formulations of this boundary value problem (one using nontangential limits of harmonic functions, one using Green functions, one using the Dirichlet-to-Neumann map, and a variational one); we show that these four formulations are equivalent. We provide a similar analysis for and discuss the roles of f and g, which turn to be somewhat interchangeable in the low regularity case. The weak formulation is more convenient for numerical approximation, whereas the nontangential limits definition is closer to the intuition and easier to check in concrete situations. We extend the weak formulation to polygonal domains using weighted Sobolev spaces. We also point out some new phenomena for the “concentrated loads” at the vertices in the polygonal case. [Copyright &y& Elsevier]

Abstract: We address design of a numerical algorithm for solving the linear system arising in numerical inversion of Laplace transforms in real case [L. D’Amore, A. Murli, Regularization of a Fourier series method for the Laplace transform inversion with real data, Inverse Problems 18 (2002) 1185–1205]. The matrix has a condition number that grows almost exponentially and the singular values decay gradually towards zero. In such a case, because of this intrinsic strong instability, the main difficulty of any numerical computation is the ability of discovering at run time, only using data, what is the maximum attainable accuracy on the solution. In this paper, we use GMRES with the aim of relating the current residuals to the maximum attainable accuracy of the approximate solution by using a suitable stopping rule. We prove that GMRES stops after, at most, as many iterations as the number of the largest eigenvalues (compared to the machine epsilon). We use a split preconditioner that symmetrically precondition the initial system. By this way, the largest eigenvalue dynamically provides the estimate of the condition number of the matrix. [Copyright &y& Elsevier]

Abstract: We shall first consider the random Dirichlet partitioning of the interval into n fragments at temperature Using calculus for Dirichlet integrals, pre-asymptotic versions of the Ewens sampling formulae from finite Dirichlet partitions follow up. From these preliminaries, straightforward proofs of the usual sampling formulae from random proportions with Poisson–Dirichlet distribution can be obtained, while considering the Kingman limit , , with . In this manuscript, the Gibbs version of the Dirichlet partition with symmetric selection is considered. By use of similar series expansion calculus for Dirichlet integrals, closed-form expressions of Ewens sampling formulae in the presence of selection are obtained; special types of Bell polynomials are shown to be involved. [Copyright &y& Elsevier]

Abstract: We relate iterates of the infinitesimal generator of a Markov process to space–time harmonic functions. First, we develop the theory for a general Markov process and create a family a space–time martingales. Next, we investigate the special class of subordinators. Combinatorics results on space–time harmonic polynomials and generalized Stirling numbers are developed and interpreted from a probabilistic point of view. Finally, we introduce the notion of pairs of subordinators in duality, investigate the implications on the associated martingales and consider some explicit examples. [Copyright &y& Elsevier]

Abstract: Quadratic trigonometric polynomial curves with local bias are presented in this paper. The quadratic trigonometric polynomial curves have continuity with a non-uniform knot vector and any value of the bias, while the quadratic B-spline curves have continuity. The changes of a local bias parameter will only affect two curve segments. With the bias parameters, the quadratic trigonometric polynomial curves can move locally toward or against a control vertex. A quadratic trigonometric Bézier curve is also introduced as special case of the given trigonometric polynomial curves. [Copyright &y& Elsevier]

For integrable functions f defined on the interval [−π,π], we denote the partial sums of the corresponding Fourier series by Sn(f) (n=0,1,2,…). In this paper, we deal with the problem of bounding supn||Sn||, where ||·|| denotes an operator norm induced by a weighted L2-norm for functions f on [−π,π]. For weight functions w belonging to a class introduced by Helson and Szego¨ (Ann. Mat. Pura Appl. 51 (1960) 107), we present explicit upper bounds for supn||Sn|| in terms of w.The authors were led to the problem of deriving explicit upper bounds for supn||Sn||, by the need for such bounds in an analysis related to the Kreiss matrix theorem—a famous result in the fields of linear algebra and numerical analysis. Accordingly, the present paper highlights the relevance of bounds on supn||Sn|| to these fields. [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]

When a function is smooth but not smoothly periodic with a particular period, and nonetheless is represented by partial sums of a Fourier series calculated using that period, the well-known Gibbs phenomenon defeats uniform convergence of the sums, and convergence is slow. In recent years, several workers have developed methods for recovering accurate and fast converging representations for functions in this situation. These efforts have not concentrated on bounds for the operators corresponding to the methods, and thus have not explicitly proven robustness in the presence of noise. In this paper we present a method for which explicit bounds are established for the operator. The method is, in effect, least-squares fitting of the given Fourier coefficients by the coefficients of polynomial splines with appropriate discontinuities. We obtain bounds by exact calculations of projections in spline spaces, using a computer algebra system. We give examples of the method and two other published methods working with noisy data. [Copyright &y& Elsevier]

We consider the problem of approximating a nonnegative function from the knowledge of its first Fourier coefficients. Here, we analyze a method introduced heuristically in a paper by Borwein and Huang (SIAM J. Opt. 5 (1995) 68–99), where it is shown how to construct cheaply a trigonometric or algebraic polynomial whose exponential is close in some sense to the considered function. In this note, we prove that approximations given by Borwein and Huang''s method, in the trigonometric case, can be related to a nonlinear constrained optimization problem, and their convergence can be easily proved under mild hypotheses as a consequence of known results in approximation theory and spectral properties of Toeplitz matrices. Moreover, they allow to obtain an improved convergence theorem for best entropy approximations. [Copyright &y& Elsevier]

In this paper we examine the possibility of using the Eta functions as a new base for high quality approximations of oscillatory functions with slowly varying weights. We focus on the least squares and piecewise least squares approximation of such functions and compare the results obtained by using Eta-based sets of functions with those obtained by means of the Legendre polynomials and Fourier series. We find out that the accuracies from these are more or less equivalent for small frequencies but they exhibit different behaviors when the frequency is increased: the accuracy worsens for the Legendre polynomials and Fourier series base but it remains bounded for the new base, in accordance with the known properties of the Eta functions. Such an advantage makes the new base quite attractive for being used in many other mathematical contexts where highly oscillatory functions are involved. [ABSTRACT FROM AUTHOR]

We study the convergence properties of the two-dimensional Rigorous Coupled Wave Approach (RCWA) for s -polarized monochromatic incident light. The RCWA is widely used to solve electromagnetic boundary-value problems where the relative permittivity varies periodically in one direction, i.e., scattering by a grating. This semi-analytical approach expands all the electromagnetic field phasors as well as the relative permittivity as Fourier series in the spatial variable along the direction of periodicity, and also replaces the relative permittivity with a stairstep approximation along the direction normal to the direction of periodicity. Thus, there is error due to Fourier truncation and also due to the approximation of grating permittivity. We prove that the RCWA is a Galerkin scheme, which allows us to employ techniques borrowed from the Finite Element Method to analyze the error. An essential tool is a Rellich identity that shows that certain continuous problems have unique solutions that depend continuously on the data with a continuity constant having explicit dependence on the relative permittivity. We prove that the RCWA converges with an increasing number of retained Fourier modes and with a finer approximation of the grating interfaces. Numerical results show that our convergence results for increasing the number of retained Fourier modes are seen in practice, while our estimates of convergence in slice thickness are pessimistic. [ABSTRACT FROM AUTHOR]