Showing papers on "Fourier series published in 2008"

A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions

Abstract: Here we develop an option pricing method for European options based on the Fourier-cosine series and call it the COS method. The key insight is in the close relation of the characteristic function with the series coefficients of the Fourier-cosine expansion of the density function. In most cases, the convergence rate of the COS method is exponential and the computational complexity is linear. Its range of application covers underlying asset processes for which the characteristic function is known and various types of option contracts. We will present the method and its applications in two separate parts. The first one is this paper, where we deal with European options in particular. In a follow-up paper we will present its application to options with early-exercise features.

FFTs on the Rotation Group

Abstract: We discuss an implementation of an efficient algorithm for the numerical computation of Fourier transforms of bandlimited functions defined on the rotation group SO(3). The implementation is freely available on the web. The algorithm described herein uses O(B 4) operations to compute the Fourier coefficients of a function whose Fourier expansion uses only (the O(B 3)) spherical harmonics of degree at most B. This compares very favorably with the direct O(B 6) algorithm derived from a basic quadrature rule on O(B 3) sample points. The efficient Fourier transform also makes possible the efficient calculation of convolution over SO(3) which has been used as the analytic engine for some new approaches to searching 3D databases (Funkhouser et al., ACM Trans. Graph. 83–105, [2003]; Kazhdan et al., Eurographics Symposium in Geometry Processing, pp. 167–175, [2003]). Our implementation is based on the “Separation of Variables” technique (see, e.g., Maslen and Rockmore, Proceedings of the DIMACS Workshop on Groups and Computation, pp. 183–237, [1997]). In conjunction with techniques developed for the efficient computation of orthogonal polynomial expansions (Driscoll et al., SIAM J. Comput. 26(4):1066–1099, [1997]), our fast SO(3) algorithm can be improved to give an algorithm of complexity O(B 3log 2 B), but at a cost in numerical reliability. Numerical and empirical results are presented establishing the empirical stability of the basic algorithm. Examples of applications are presented as well.

A First Course in Fourier Analysis

Abstract: This book provides a meaningful resource for applied mathematics through Fourier analysis. It develops a unified theory of discrete and continuous (univariate) Fourier analysis, the fast Fourier transform, and a powerful elementary theory of generalized functions and shows how these mathematical ideas can be used to study sampling theory, PDEs, probability, diffraction, musical tones, and wavelets. The book contains an unusually complete presentation of the Fourier transform calculus. It uses concepts from calculus to present an elementary theory of generalized functions. FT calculus and generalized functions are then used to study the wave equation, diffusion equation, and diffraction equation. Real-world applications of Fourier analysis are described in the chapter on musical tones. A valuable reference on Fourier analysis for a variety of students and scientific professionals, including mathematicians, physicists, chemists, geologists, electrical engineers, mechanical engineers, and others.

Higher order asymptotic homogenization and wave propagation in periodic composite materials

Abstract: We present an application of the higher order asymptotic homogenization method (AHM) to the study of wave dispersion in periodic composite materials. When the wavelength of a travelling signal becomes comparable with the size of heterogeneities, successive reflections and refractions of the waves at the component interfaces lead to the formation of a complicated sequence of the pass and stop frequency bands. Application of the AHM provides a long-wave approximation valid in the low-frequency range. Solution for the high frequencies is obtained on the basis of the Floquet–Bloch approach by expanding spatially varying properties of a composite medium in a Fourier series and representing unknown displacement fields by infinite plane-wave expansions. Steadystate elastic longitudinal waves in a composite rod (one-dimensional problem allowing the exact analytical solution) and transverse anti-plane shear waves in a fibre-reinforced composite with a square lattice of cylindrical inclusions (two-dimensional problem) are considered. The dispersion curves are obtained, the pass and stop frequency bands are identified.

Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations

Abstract: For classes of symplectic and symmetric time-stepping methods— trigonometric integrators and the Stormer–Verlet or leapfrog method—applied to spectral semi-discretizations of semilinear wave equations in a weakly non-linear setting, it is shown that energy, momentum, and all harmonic actions are approximately preserved over long times. For the case of interest where the CFL number is not a small parameter, such results are outside the reach of standard backward error analysis. Here, they are instead obtained via a modulated Fourier expansion in time.

Discovering ordered phases of block copolymers: new results from a generic Fourier-space approach.

Abstract: A generic Fourier-space approach to solve the self-consistent field theory of block copolymers is developed. This approach is based on the fact that, for any computational box with periodic boundary conditions, all spatially varying functions are spanned by the Fourier series determined by the size and shape of the box. The method reproduces all known diblock copolymer phases. The application of this method to a model "frustrated" triblock copolymer leads to a phase diagram with a number of new phases. Furthermore, the capability of the method to reproduce experimentally observed structures is demonstrated using the knitting pattern of triblock copolymers.

Sturm-Liouville theory and its applications

Abstract: Inner Product Space- The Sturm-Liouville Theory- Fourier Series- Orthogonal Polynomials- Bessel Functions- The Fourier Transformation- The Laplace Transformation

Vibroacoustic behavior of clamp mounted double-panel partition with enclosure air cavity

Abstract: A theoretical study on the vibroacoustic performance of a rectangular double-panel partition clamp mounted in an infinite acoustic rigid baffle is presented. With the clamped boundary condition taken into account by the method of modal function, a double Fourier series solution to the dynamic response of the structure is obtained by employing the weighted residual method (i.e., the Galerkin method). The double series solution can be considered as the exact solution of the problem, as the structural and acoustic-structural coupling effects are fully accounted for and the solution converges numerically. The accuracy of the theoretical predictions is checked against existing experimental data, with good agreement achieved. The influence of several key parameters on the sound isolation capability of the double-panel configuration is then systematically studied, including panel dimensions, thickness of air cavity, elevation angle, and azimuth angle of incidence sound. The present method is suitable for double-panel systems of finite or infinite extent and is applicable for both low- and high-frequency ranges. With these merits, the proposed method compares favorably with a number of other approaches, e.g., finite element method, boundary element method, and statistical energy analysis method.

Modeling Human Multichannel Perception and Control Using Linear Time-Invariant Models

Abstract: This paper introduces a two-step identification method of human multichannel perception and control. In the first step, frequency response functions are identified using linear time-invariant models. The analytical predictions of bias and variance in the estimated frequency response functions are validated using Monte Carlo simulations of a closed-loop control task and contrasted to a conventional method using Fourier coefficients. For both methods, the analytical predictions are reliable, but the linear time-invariant method has lower bias and variance than Fourier coefficients. It is further shown that the linear time-invariant method is more robust to higher levels of pilot remnant. Finally, both methods were successfully applied to experimental data from closed-loop control tasks with pilots.

A modified collocation Trefftz method for the inverse Cauchy problem of Laplace equation

Abstract: We consider an inverse problem for Laplace equation by recovering the boundary value on an inaccessible part of a circle from an overdetermined data on an accessible part of that circle. The available data are assumed to have a Fourier expansion, and thus the finite terms truncation plays a role of regularization to perturb the ill-posedness of this inverse problem into a well-posed one. Hence, we can apply a modified indirect Trefftz method to solve this problem and then a simple collocation technique is used to determine the unknown coefficients, which is named a modified collocation Trefftz method. The results may be useful to detect the corrosion inside a pipe through the measurements on a partial boundary. Numerical examples show the effectiveness of the new method in providing an excellent estimate of unknown data from the given data under noise.

Classes of Degenerate Elliptic Operators in Gelfand-Shilov Spaces

Abstract: We propose a novel approach for the study of the uniform regularity and the decay at infinity for Shubin type pseudo-differential operators which are globally hypoelliptic but not necessarily globally and even locally elliptic. The basic idea is to use the special role of the Hermite functions for the characterization of inductive and projective Gelfand-Shilov spaces. In this way we transform the problem to infinite dimensional linear systems on S Banach spaces of sequences by using Fourier series expansion with respect to the Hermite functions. As applications of our general results we obtain new theorems for global hypoellipticity for classes of degenerate operators in tensorized generalizations of Shubin spaces and in inductive and projective Gelfand-Shilov spaces.

From high oscillation to rapid approximation I: Modified Fourier expansions

Abstract: In this paper we consider a modification of the classical Fourier expansio n, whereby in ( 1, 1) the sinnx functions are replaced by sin�(n 1 )x, n � 1. This has a number of important advantages in the approximation of analytic, nonperiodic functions. In particular, expansion coefficients decay like O n 2 � , rather than like O n 1 � . We explore theoretical features of these modified Fourier expansions, prove suitable versions of Fej´

Long-Time Analysis of Nonlinearly Perturbed Wave Equations Via Modulated Fourier Expansions

Abstract: A modulated Fourier expansion in time is used to show long-time near-conservation of the harmonic actions associated with spatial Fourier modes along the solutions of nonlinear wave equations with small initial data. The result implies the long-time near-preservation of the Sobolev-type norm that specifies the smallness condition on the initial data.

Analytical Solution for Rotor Eddy-Current Losses in a Slotless Permanent-Magnet Motor: The Case of Current Sheet Excitation

Abstract: We present an analytical solution for the magnetic field, induced eddy currents, and the corresponding losses generated in the rotor of a slotless permanent-magnet (PM) motor. The field excitation is a current sheet placed at the stator interior surface. The solution is based on an analytical solution of the diffusion equation using double Fourier series and in the complex domain. The first and the second Fourier series correspond to the time and space harmonics, respectively. We have analyzed the effect of each time harmonic in detail, and verified the results with FEM software.

Identification of the class of initial data for the insensitizing control of the heat equation

Abstract: This paper is devoted to analyze the class of initial data that can be insensitized for the heat equation. This issue has been extensively addressed in the literature both in the case of complete and approximate insensitization (see [19] and [1], respectively). But in the context of pure insensitization there are very few results identifying the class of initial data that can be insensitized. This is a delicate issue which is related to the fact that insensitization turns out to be equivalent to suitable observability estimates for a coupled system of heat equations, one being forward and the other one backward in time. The existing Carleman inequalities techniques can be applied but they only give interior information of the solutions, which hardly allows identifying the initial data because of the strong irreversibility of the equations involved in the system, one of them being an obstruction at the initial time $t=0$ and the other one at the final one $t=T$. In this article we consider different geometric configurations in which the subdomains to be insensitized and the one in which the external control acts play a key role. We show that, under rather restrictive geometric restrictions, initial data in a class that can be characterized in terms of a summability condition of their Fourier coefficients with suitable weights, can be insensitized. But, the main result of the paper, which might seem surprising, shows that this fails to be true in general, so that even the first eigenfunction of the system can not be insensitized. This result is similar to those obtained in the context of the null controllability of the heat equation in unbounded domains in [14] where it is shown that smooth and compactly supported initial data may not be controlled. Our proofs combine the existing observability results for heat equations obtained by means of Carleman inequalities, energy and gaussian estimates and Fourier expansions.

A Locally Exact Homogenization Theory for Periodic Microstructures With Isotropic Phases

Abstract: Elements of the homogenization theory are utilized to develop a new micromechanics approach for unit cells of periodic heterogeneous materials based on locally exact elasticity solutions. The interior inclusion problem is exactly solved by using Fourier series representation of the local displacement field. The exterior unit cell periodic boundary-value problem is tackled by using a new variational principle for this class of nonseparable elasticity problems, which leads to exceptionally fast and well-behaved convergence of the Fourier series coefficients. Closed-form expressions for the homogenized moduli of unidirectionally reinforced heterogeneous materials are obtained in terms of Hill’s strain concentration matrices valid under arbitrary combined loading, which yield homogenized Hooke’s law. Homogenized engineering moduli and local displacement and stress fields of unit cells with offset fibers, which require the use of periodic boundary conditions, are compared to corresponding finite-element results demonstrating excellent correlation.

A two dimensional problem for a transversely isotropic generalized thermoelastic thick plate with spatially varying heat source

Abstract: This paper deals with a two dimensional problem for a transversely isotropic thick plate having heat source. The upper surface of the plate is stress free with prescribed surface temperature while the lower surface of the plate rests on a rigid foundation and is thermally insulated. The study is carried out in the context of generalized thermoelasticity proposed by Green and Naghdi. The governing equations for displacement and temperature fields are obtained in Laplace–Fourier transform domain by applying Laplace and Fourier transform techniques. The inversion of double transform has been done numerically. The numerical inversion of Laplace transform is done by using a method based on Fourier Series expansion technique. Numerical computations have been done for magnesium (Mg) and the results are presented graphically. The results for an isotropic material (Cu) have been deduced numerically and presented graphically to compare with those of transversely isotropic material (Mg).

Two-dimensional problem of a two-temperature generalized thermoelastic half-space subjected to ramp-type heating

Abstract: In this work, we will consider a half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in the context of the theory of two-temperature generalized thermoelasticity. A linear temperature ramping function is used to more realistically model thermal loading of the half-space surface. The medium is assumed initially quiescent. Laplace and Fourier transform techniques are used to obtain the general solution for any set of boundary conditions. The general solution obtained is applied to a specific problem of a half-space subjected to ramp-type heating. The inverse Fourier transforms are obtained analytically while the inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the ramping parameter of heating.

Semi-Classical Wavefront Set and Fourier Integral Operators

Abstract: Here we define and prove some properties of the semi-classical wavefront set. We also define and study semi-classical Fourier integral operators and prove a generalization of Egorov's theorem to manifolds of different dimensions.