Showing papers on "Fourier series published in 2003"

Classical and modern Fourier analysis

Abstract: Prolegomena 1 L p Spaces and Interpolation 2 Maximal Functions, Fourier Transform, and Distributions 3 Fourier Analysis on the Torus 4 Singular Integrals of Convolution Type 5 Littlewood-Paley Theory and Multipliers 6 Smoothness and Function Spaces 7 BMO and Carleson Measures 8 Singular Integrals of Nonconvolution Type 9 Weighted Inequalities 10 Boundedness and Convergence of Fourier Integrals Bibliography Index of Notation Index

Fourier Analysis: An Introduction

Abstract: Foreword vii Preface xi Chapter 1. The Genesis of Fourier Analysis 1 Chapter 2. Basic Properties of Fourier Series 29 Chapter 3. Convergence of Fourier Series 69 Chapter 4. Some Applications of Fourier Series 100 Chapter 5. The Fourier Transform on R 129 Chapter 6. The Fourier Transform on R d 175 Chapter 7. Finite Fourier Analysis 218 Chapter 8. Dirichlet's Theorem 241 Appendix: Integration 281 Notes and References 299 Bibliography 301 Symbol Glossary 305

Control of a multilevel converter using resultant theory

Abstract: In this work, a method is given to compute the switching angles in a multilevel converter to produce the required fundamental voltage while at the same time cancel out specified higher order harmonics. Specifically, a complete analysis is given for a seven-level converter (three dc sources), where it is shown that for a range of the modulation index m/sub I/, the switching angles can be chosen to produce the desired fundamental V/sub 1/=m/sub I/(s4V/sub dc///spl pi/) while making the fifth and seventh harmonics identically zero.

Photonic band structures solved by a plane-wave-based transfer-matrix method.

Abstract: Transfer-matrix methods adopting a plane-wave basis have been routinely used to calculate the scattering of electromagnetic waves by general multilayer gratings and photonic crystal slabs. In this paper we show that this technique, when combined with Bloch's theorem, can be extended to solve the photonic band structure for 2D and 3D photonic crystal structures. Three different eigensolution schemes to solve the traditional band diagrams along high-symmetry lines in the first Brillouin zone of the crystal are discussed. Optimal rules for the Fourier expansion over the dielectric function and electromagnetic fields with discontinuities occurring at the boundary of different material domains have been employed to accelerate the convergence of numerical computation. Application of this method to an important class of 3D layer-by-layer photonic crystals reveals the superior convergency of this different approach over the conventional plane-wave expansion method.

Fourier Analysis and Its Applications

Abstract: Introduction.- Preparations.- Laplace and Z Transforms.- Fourier Series.- L^2 Theory.- Separation of Variables.- Fourier Transforms.- Distributions.- Multi-Dimensional Fourier Analysis.- Appendix A: The ubiquitous convolution.- Appendix B: The Discrete Fourier Transform.- Appendix C: Formulae.- Appendix D: Answers to exercises.- Appendix E: Literature.

Almost Periodic Type Functions and Ergodicity

Abstract: 1 Almost periodic type functions.- 1.1.- 1.1.1 Numerical almost periodic functions.- 1.1.2 Uniform almost periodic functions.- 1.1.3 Vector-valued almost periodic functions.- 1.2 Asymptotically almost periodic functions.- 1.3 Weakly almost periodic functions.- 1.3.1 Vector-valued weakly almost periodic functions.- 1.3.2 Ergodic theorem.- 1.3.3 Invariant mean and mean convolution.- 1.3.4 Fourier series of WAV(?, H).- 1.3.5 Uniformly weakly almost periodic functions.- 1.4 Approximate theorem and applications.- 1.4.1 Numerical approximate theorem.- 1.4.2 Vector-valued approximate theorem.- 1.4.3 Unique decomposition theorem.- 1.5 Pseudo almost periodic functions.- 1.5.1 Pseudo almost periodic functions.- 1.5.2 Generalized pseudo almost periodic functions.- 1.6 Converse problems of Fourier expansions.- 1.7 Almost periodic type sequences.- 1.7.1 Almost periodic sequences.- 1.7.2 Other almost periodic type sequences.- 2 Almost periodic type differential equations.- 2.1 Linear differential equations.- 2.1.1 Ordinary differential equations.- 2.1.2 Abstract differential equations.- 2.1.3 Integration of almost periodic type functions.- 2.2 Partial differential equations.- 2.2.1 Dirichlet Problems.- 2.2.2 Parabolic equations.- 2.2.3 Second-order equations with gradient operators.- 2.3 Means, introversion and nonlinear equations.- 2.3.1 General theory of means and introversions.- 2.3.2 Applications to (weakly) almost periodic functions.- 2.3.3 Nonlinear differential equations.- 2.3.4 Implications of almost periodic type solutions.- 2.4 Regularity and exponential dichotomy.- 2.4.1 General theory of regularity.- 2.4.2 Stability of regularity.- 2.4.3 Almost periodic type solutions.- 2.5 Equations with piecewise constant argument.- 2.5.1 Exponential dichotomy for difference equations.- 2.5.2 Equations with piecewise constant argument.- 2.5.3 Almost periodic difference equations.- 2.6 Equations with unbounded forcing term.- 2.7 Almost periodic structural stability.- 2.7.1 Topological equivalence and structural stability.- 2.7.2 Exponential dichotomy and structural stability.- 3 Ergodicity and abstract differential equations.- 3.1 Ergodicity and regularity.- 3.1.1 Ergodicity and regularity.- 3.1.2 Solutions of almost periodic type equations.- 3.2 Ergodicity and nonlinear equations.- 3.3 Semigroup of operators and applications.- 3.3.1 Semigroup of operators.- 3.3.2 Almost periodic type solutions.- 3.4 Delay differential equations.- 3.4.1 Introduction of delay differential equations.- 3.4.2 Linear autonomous equations.- 3.4.3 Linear nonautonomous equations.- 3.5 Spectrum of functions.- 3.6 Abstract Cauchy Problems.- 3.6.1 Harmonic analysis of solutions.- 3.6.2 Asymptotic behavior of solutions.- 3.6.3 Mild solutions.- 3.6.4 Weakly almost periodic solutions.- 4 Ergodicity and averaging methods.- 4.1 Ergodicity and its properties.- 4.2 Quantitative theory.- 4.2.1 Introduction.- 4.2.2 Quantitative theory of averaging methods.- 4.2.3 Example and comments.- 4.3 Perturbations of noncritical linear systems.- 4.4 Qualitative theory of averaging methods.- 4.4.1 Almost periodic type solutions of nonlinear equations.- 4.4.2 Some examples.- 4.5 Averaging methods for functional equations.- 4.5.1 Averaging for functional differential equations.- 4.5.2 Averaging for delay difference equations.- Notations.

Introduction to the mathematics of medical imaging

Abstract: Preface to the second edition Preface How to use this nook Notational conventions 1. Measurements and modeling 2. Linear models and linear equations 3. A basic model for tomography 4. Introduction to the Fourier transform 5. Convolution 6. The radon transform 7. Introduction to Fourier series 8. Sampling 9. Filters 10. Implementing shift invariant filters 11. Reconstruction in X-ray tomography 12. Imaging artifacts in X-ray tomography 13. Algebraic reconstruction techniques 14. Magnetic resonance imaging 15. Probability and random variables 16. Applications of probability 17. Random processes A. Background material B. Basic analysis Bibliography Index.

Statistical convergence of multiple sequences

Abstract: We extend the concept of and basic results on statistical convergence from ordinary (single) sequences to multiple sequences of (real or complex) numbers. As an application to Fourier analysis, we obtain the following Theorem 3: (i) If $f \in L(\textrm{log}^{+} L)^{d-1}(\mathbb{T}^d)$, where $\mathbb{T}^d := [-\pi, \pi)^{d}$ is the d-dimensional torus, then the Fourier series of f is statistically convergent to $f({\bf t})$ at almost every ${\bf t} \in \mathbb{T}^d$; (ii) If $f \in C(\mathbb{T}^d)$, then the Fourier series of f is statistically convergent to $f ({\bf t})$ uniformly on $\mathbb{T}^d$.

Fourier and Laplace transforms

Abstract: This textbook presents in a unified manner the fundamentals of both continuous and discrete versions of the Fourier and Laplace transforms. These transforms play an important role in the analysis of all kinds of physical phenomena. As a link between the various applications of these transforms the authors use the theory of signals and systems, as well as the theory of ordinary and partial differential equations. The book is divided into four major parts: periodic functions and Fourier series, non-periodic functions and the Fourier integral, switched-on signals and the Laplace transform, and finally the discrete versions of these transforms, in particular the Discrete Fourier Transform together with its fast implementation, and the z-transform. This textbook is designed for self-study. It includes many worked examples, together with more than 120 exercises, and will be of great value to undergraduates and graduate students in applied mathematics, electrical engineering, physics and computer science

On Fourier coefficients of automorphic forms of symplectic groups

Abstract: In this paper we study certain properties of Fourier coefficients of cuspidal representations on symplectic groups. We prove that every cuspidal representation has a nontrivial Fourier coefficient with respect to a certain type of unipotent class.

Fourier's Law confirmed for a class of small quantum systems

Abstract: Within the Lindblad formalism we consider an interacting spin chain coupled locally to heat baths. We investigate the dependence of the energy transport on the type of interaction in the system as well as on the overall interaction strength. For a large class of couplings we find a normal heat conduction and confirm Fourier's Law. In a fully quantum mechanical approach linear transport behavior appears to be generic even for small quantum systems.

Volume averaging for determining the effective dispersion tensor: Closure using periodic unit cells and comparison with ensemble averaging

Abstract: In this work, we use the method of volume averaging to determine the effective dispersion tensor for a heterogeneous porous medium; closure for the averaged equation is obtained by solution of a concentration deviation equation over a periodic unit cell. Our purpose is to show how the method of volume averaging with closure can be rectified with the results obtained by other upscaling methods under particular conditions. Although this rectification is something that is generally believed to be true, there has been very little research that explores this issue explicitly. We show that under certain limiting (but mild) assumptions, the closure problem provides a Fourier series solution for the effective dispersion tensor. When second-order spatial stationarity is imposed on the velocity field, the method yields a simple Fourier series that converges to an integral form in the limit as the period of the unit cell approaches infinity. This limiting result is identical to the quasi-Fickian forms that have been developed previously via ensemble averaging by Deng et al. [1993] and recently by Fiori and Dagan [2000] except in the definition of the averaging operation. As a second objective we have conducted a numerical study to evaluate the influence of the size of the averaging volume on the effective dispersion tensor and its volume averaged statistics. This second objective is complimentary in many ways to recent research reported by Rubin et al. [1999] (via ensemble averaging) and by Wang and Kitanidis [1999] (via volume averaging) on the block-averaged effective dispersion tensor. The variability of the effective dispersion tensor from realization to realization is assessed by computing the volume-averaged effective dispersion tensor for an ensemble of finite fields with the same (ensemble) statistics. Ensembles were generated using three different sizes of unit cells. All three unit cell sizes yield similar results for the value of the mean effective dispersion tensor. However, the coefficient of variation depends strongly upon the size of the unit cell, and our results are consistent with those developed by Fiori [1998] from the ensemble averaging perspective. This implies that in applications the actual value of the effective dispersion tensor may be significantly different than expected on the basis of unconditioned hydraulic conductivity statistics, and this variation should be considered when applying macrodispersion to real-world systems.

Fast, High-Order, High-Frequency Integral Methods for Computational Acoustics and Electromagnetics

Abstract: We review a set of algorithms and methodologies developed recently for the numerical solution of problems of scattering by complex bodies in three-dimensional space. These methods, which are based on integral equations, high-order integration, Fast Fourier Transforms and highly accurate high-frequency integrators, can be used in the solution of problems of electromagnetic and acoustic scattering by surfaces and penetrable scatterers — even in cases in which the scatterers contain geometric singularities such as comers and edges. All of the solvers presented here exhibit high-order convergence, they run on low memories and reduced operation counts, and they result in solutions with a high degree of accuracy. In particular, our approach to direct solution of integral equations results in algorithms that can evaluate accurately in a personal computer scattering from hundred-wavelength-long objects — a goal, otherwise achievable today only by super-computing. The high-order high-frequency methods we present, in turn, are efficient where our direct methods become costly, thus leading to an overall computational methodology which is applicable and accurate throughout the electromagnetic spectrum.

An introduction to basic Fourier series

Abstract: Foreword. Preface. 1: Introduction. 2: Basic Exponential and Trigonometric Functions. 3: Addition Theorems. 4: Some Expansions and Integrals. 5: Introduction of Basic Fourier Series. 6: Investigation of Basic Fourier Series. 7: Completeness of Basic Trigonometric Systems. 8: Improved Asymptotics of Zeros. 9: Some Expansions in Basic Fourier Series. 10: Basic Bernoulli and Euler Polynomials and Numbers and q-Zeta Function. 11: Numerical Investigation of Basic Fourier Series. 12: Suggestions for Further Work. Appendix A: Selected Summation and Transformation Formulas and Integrals. A.1. Basic Hypergeometric Series. A.2. Selected Summation Formulas. A.3. Selected Transformation Formulas. A.4. Some Basic Integrals. Appendix B: Some Theorems of Complex Analysis. B.1. Entire Functions. B.2. Lagrange Inversion Formula. B.3. Dirichlet Series. B.4. Asymptotics. Appendix C: Tables of Zeros of Basic Sine and Cosine Functions. Appendix D: Numerical Examples of Improved Asymptotics. Appendix E: Numerical Examples of Euler-Rayleigh Method. Appendix F: Numerical Examples of Lower and Upper Bounds. Bibliography. Index.

On sums of Fourier coefficients of cusp forms

Abstract: The summatory function of $t_j(n^2)$ is estimated, where $H_j(s) = \sum_{n=1}^\infty t_j(n)n^{-s}$ is the Hecke series of a non-holomorphic cusp form. The analogous problem of holomorphic cusp forms is also treated.

Fourier-expansion solution of atom distributions in a crystal using X-ray standing waves.

Abstract: Term-by-term Fourier-expansion series, each made up of components having element-specific phases and amplitudes acquired with x-ray standing wave measurements on successive orders of Bragg reflections, are used to reconstruct impurity atom distributions in muscovite mica with respect to the (001) lattice without a priori assumptions on their structures.

An additive problem in the Fourier coefficients of cusp forms

Abstract: We establish an estimate on sums of shifted products of Fourier coefficients coming from holomorphic or Maass cusp forms of arbitrary level and nebentypus. These sums are analogous to the binary additive divisor sum which has been studied extensively. As an application we derive, extending work of Duke, Friedlander and Iwaniec, a subconvex estimate on the critical line for L-functions associated to character twists of these cusp forms.

Computing the Dynamics of Complex Singularities of Nonlinear PDEs

Abstract: A two-step strategy is proposed for the computation of singularities in nonlinear PDEs. The first step is the numerical solution of the PDE using a Fourier spectral method; the second step involves numerical analytical continuation into the complex plane using the epsilon algorithm to sum the Fourier series. Test examples include the inviscid Burgers and nonlinear heat equations as well as a transport equation involving the Hilbert transform. Numerical results, including Web animations that show the dynamics of the singularities in the complex plane, are presented.

Stability Analysis of a Hydrodynamic Journal Bearing With Rotating Herringbone Grooves

Abstract: This paper presents an analytical method to investigate the stability of a hydrodynamic journal bearing with rotating herringbone grooves. The dynamic coefficients of the hydrodynamic journal bearing are calculated using the FEM and the perturbation method. The linear equations of motion can be represented as a parametrically excited system because the dynamic coefficients have time-varying components due to the rotating grooves, even in the steady state. Their solution can be assumed as a Fourier series expansion so that the equations of motion can be rewritten as simultaneous algebraic equations with respect to the Fourier coefficients. Then, stability can be determined by solving Hill's infinite determinant of these algebraic equations. The validity of this research is proved by the comparison of the stability chart with the time response of the whirl radius obtained from the equations of motion. This research shows that the instability of the hydrodynamic journal bearing with rotating herringbone grooves increases with increasing eccentricity and with decreasing groove number, which play the major roles in increasing the average and variation of stiffness coefficients, respectively. It also shows that a high rotational speed is another source of instability by increasing the stiffness coefficients without changing the damping coefficients.

Coupling between finite elements and boundary elements for the numerical simulation of induction heating processes using a harmonic balance method

Abstract: For the modeling of induction heating processes, strongly coupled magnetodynamic and thermal problems can be solved together within the same finite element. This is called the direct method. In this case, the electromagnetic quantities are expressed through Fourier series according to the harmonic balance method. In this paper, each harmonic is calculated in the whole space by using the coupling between finite elements and boundary elements. Especially suitable when moving parts are involved and because the mesh of air is unnecessary, it is shown that this coupling is still successful if the direct method is used. At the end, the efficiency of this approach is illustrated with an example.

A Simplified Fourier Method for Nonhydrostatic Mountain Waves

Abstract: A previously derived approximation to the standard Fourier integral technique for linear mountain waves is extended to include nonhydrostatic effects in a background flow with height-dependent wind and stratification. The approximation involves using ray theory to simplify the vertical eigenfunctions. The generalization to nonhydrostatic waves requires special treatment for resonant modes and caustics. Resonant modes are handled with a small amount of damping, and caustics are handled with a uniformly valid approximation involving the Airy function. This method is developed for both two- and three-dimensional flows, and its results are shown to compare well with an exact analytical result for two-dimensional mountain waves and with a numerical simulation for two- and three-dimensional mountain waves.

Electromagnetic beam diffraction by a finite lamellar structure: an aperiodic coupled-wave method

Abstract: We have developed a new formulation of the coupled-wave method (CWM) to handle aperiodic lamellar structures, and it will be referred to as the aperiodic coupled-wave method (ACWM). The space is still divided into three regions, but the fields are written by use of their Fourier integrals instead of the Fourier series. In the modulated region the relative permittivity is represented by its Fourier transform, and then a set of integrodifferential equations is derived. Discretizing the last system leads to a set of ordinary differential equations that is reduced to an eigenvalue problem, as is usually done in the CWM. To assess the method, we compare our results with three independent formalisms: the Rayleigh perturbation method for small samples, the volume integral method, and the finite-element method.