Showing papers on "Fourier series published in 2009"

Classical Fourier Analysis

Abstract: Preface.- 1. Lp Spaces and Interpolation.- 2. Maximal Functions, Fourier Transform, and Distributions.- 3. Fourier Series.- 4. Topics on Fourier Series.- 5. Singular Integrals of Convolution Type.- 6. Littlewood-Paley Theory and Multipliers.- 7. Weighted Inequalities.- A. Gamma and Beta Functions.- B. Bessel Functions.- C. Rademacher Functions.- D. Spherical Coordinates.- E. Some Trigonometric Identities and Inequalities.- F. Summation by Parts.- G. Basic Functional Analysis.- H. The Minimax Lemma.- I. Taylor's and Mean Value Theorem in Several Variables.- J. The Whitney Decomposition of Open Sets in Rn.- Glossary.- References.- Index.

Théorie analytique de la chaleur

Abstract: French mathematician Joseph Fourier's Theorie Analytique de la Chaleur was originally published in 1822. In this groundbreaking study, arguing that previous theories of mechanics advanced by such outstanding scientists as Archimedes, Galileo, Newton and their successors did not explain the laws of heat, Fourier set out to study the mathematical laws governing heat diffusion and proposed that an infinite mathematical series may be used to analyse the conduction of heat in solids: this is now known as the 'Fourier Series'. His work paved the way for modern mathematical physics. This book will be especially useful for mathematicians who are interested in trigonometric series and their applications, and it is reissued simultaneously with Alexander Freeman's English translation, The Analytical Theory of Heat, of 1878.

Linear and nonlinear waves

Abstract: The study of waves can be traced back to antiquity where philosophers, such as Pythagoras (c.560-480 BC), studied the relation of pitch and length of string in musical instruments. However, it was not until the work of Giovani Benedetti (1530-90), Isaac Beeckman (1588-1637) and Galileo (1564-1642) that the relationship between pitch and frequency was discovered. This started the science of acoustics, a term coined by Joseph Sauveur (1653-1716) who showed that strings can vibrate simultaneously at a fundamental frequency and at integral multiples that he called harmonics. Isaac Newton (1642-1727) was the first to calculate the speed of sound in his Principia. However, he assumed isothermal conditions so his value was too low compared with measured values. This discrepancy was resolved by Laplace (1749-1827) when he included adiabatic heating and cooling effects. The first analytical solution for a vibrating string was given by Brook Taylor (1685-1731). After this, advances were made by Daniel Bernoulli (1700-82), Leonard Euler (1707-83) and Jean d’Alembert (1717-83) who found the first solution to the linear wave equation, see section (3.2). Whilst others had shown that a wave can be represented as a sum of simple harmonic oscillations, it was Joseph Fourier (1768-1830) who conjectured that arbitrary functions can be represented by the superposition of an infinite sum of sines and cosines now known as the Fourier series. However, whilst his conjecture was controversial and not widely accepted at the time, Dirichlet subsequently provided a proof, in 1828, that all functions satisfying Dirichlet’s conditions (i.e. non-pathological piecewise continuous) could be represented by a convergent Fourier series. Finally, the subject of classical acoustics was laid down and presented as a coherent whole by John William Strutt (Lord Rayleigh, 1832-1901) in his treatise Theory of Sound. The science of modern acoustics has now moved into such diverse areas as sonar, auditoria, electronic amplifiers, etc.

Direction-of-Arrival Estimation for Nonuniform Sensor Arrays: From Manifold Separation to Fourier Domain MUSIC Methods

Abstract: In this paper, the problem of spectral search-free direction-of-arrival (DOA) estimation in arbitrary nonuniform sensor arrays is addressed. In the first part of the paper, we present a finite-sample performance analysis of the well-known manifold separation (MS) based root-MUSIC technique. Then, we propose a new class of search-free DOA estimation methods applicable to arrays of arbitrary geometry and establish their relationship to the MS approach. Our first technique is referred to as Fourier-domain (FD) root-MUSIC and is based on the fact that the spectral MUSIC function is periodic in angle. It uses the Fourier series to expand this function and reformulate the underlying DOA estimation problem as an equivalent polynomial rooting problem. Our second approach applies the zero-padded inverse Fourier transform to the FD root-MUSIC polynomial to avoid the polynomial rooting step and replace it with a simple line search. Our third technique refines the FD root-MUSIC approach by using weighted least-squares approximation to compute the polynomial coefficients. The proposed techniques are shown to offer substantially improved performance-to-complexity tradeoffs as compared to the MS technique.

mathematical-and-experimental-modeling-of-physical-and-biological-processes

Abstract: Introduction: The Iterative Modeling Process Modeling and Inverse Problems Mechanical Vibrations Inverse Problems Mathematical and Statistical Aspects of Inverse Problems Probability and Statistics Overview Parameter Estimation or Inverse Problems Computation of sigman, Standard Errors, and Confidence Intervals Investigation of Statistical Assumptions Statistically Based Model Comparison Techniques Mass Balance and Mass Transport Introduction Compartmental Concepts Compartment Modeling General Mass Transport Equations Heat Conduction Motivating Problems Mathematical Modeling of Heat Transfer Experimental Modeling of Heat Transfer Structural Modeling: Force/Moments Balance Motivation: Control of Acoustics/Structural Interactions Introduction to Mechanics of Elastic Solids Deformations of Beams Separation of Variables: Modes and Mode Shapes Numerical Approximations: Galerkin's Method Energy Functional Formulation The Finite Element Method Experimental Beam Vibration Analysis Beam Vibrational Control and Real-Time Implementation Introduction Controllability and Observability of Linear Systems Design of State Feedback Control Systems and State Estimators Pole Placement (Relocation) Problem Linear Quadratic Regulator Theory Beam Vibrational Control: Real-Time Feedback Control Implementation Wave Propagation Fluid Dynamics Fluid Waves Experimental Modeling of the Wave Equation Size-Structured Population Models Introduction: A Motivating Application A Single Species Model (Malthusian Law) The Logistic Model A Predator/Prey Model A Size-Structured Population Model The Sinko-Streifer Model and Inverse Problems Size Structure and Mosquitofish Populations Appendix A: An Introduction to Fourier Techniques Fourier Series Fourier Transforms Appendix B: Review of Vector Calculus References appear at the end of each chapter.

A radial basis function method for the shallow water equations on a sphere

Abstract: The paper derives the first known numerical shallow water model on the sphere using radial basis function (RBF) spatial discretization, a novel numerical methodology that does not require any grid or mesh. In order to perform a study with regard to its spatial and temporal errors, two nonlinear test cases with known analytical solutions are considered. The first is a global steady-state flow with a compactly supported velocity field, while the second is an unsteady flow where features in the flow must be kept intact without dispersion. This behaviour is achieved by introducing forcing terms in the shallow water equations. Error and time stability studies are performed, both as the number of nodes are uniformly increased and the shape parameter of the RBF is varied, especially in the flat basis function limit. Results show that the RBF method is spectral, giving exceptionally high accuracy for low number of basis functions while being able to take unusually large time steps. In order to put it in the context of other commonly used global spectral methods on a sphere, comparisons are given with respect to spherical harmonics, double Fourier series and spectral element methods.

Adaptive backstepping dynamic surface control for systems with periodic disturbances using neural networks

Abstract: This paper addresses the adaptive neural network tracking control problem for a class of strict-feedback systems with unknown non-linearly parameterised and time-varying disturbed function of known periods. Radial basis function neural network and Fourier series expansion are combined into a new function approximator to model each suitable disturbed function in systems. Dynamic surface control approach is used to solve the problem of ‘explosion of complexity’ in backstepping design procedure. The uniform boundedness of all closed-loop signals is guaranteed. The tracking error is proved to converge to a small residual set around the origin. A simulation example is provided to illustrate the effectiveness of the control scheme designed.

Handbook of Fourier Analysis & Its Applications

Abstract: 1. Introduction 2. Fundamentals of Fourier Analysis 3. Fourier Analysis in Systems Theory 4. Fourier Transforms in Probability, Random Variables and Stochastic Processes 5. The Sampling Theory 6. Generalizations of the Sampling Theorem 7. Noise and Error Effects 8. Multidimensional Signal Analysis 9. Time-Frequency Representations 10. Signal Recovery 11. Signal and Image Synthesis: Alternating Projections Onto Convex Sets 12. Mathematical Morphology and Fourier Analysis on Time Sales 13. Applications 14. Appendices 15. Reference

Fourier Theoretic Probabilistic Inference over Permutations

Abstract: Permutations are ubiquitous in many real-world problems, such as voting, ranking, and data association. Representing uncertainty over permutations is challenging, since there are n! possibilities, and typical compact and factorized probability distribution representations, such as graphical models, cannot capture the mutual exclusivity constraints associated with permutations. In this paper, we use the "low-frequency" terms of a Fourier decomposition to represent distributions over permutations compactly. We present Kronecker conditioning, a novel approach for maintaining and updating these distributions directly in the Fourier domain, allowing for polynomial time bandlimited approximations. Low order Fourier-based approximations, however, may lead to functions that do not correspond to valid distributions. To address this problem, we present a quadratic program defined directly in the Fourier domain for projecting the approximation onto a relaxation of the polytope of legal marginal distributions. We demonstrate the effectiveness of our approach on a real camera-based multi-person tracking scenario.

Real-Time Estimation of Power System Frequency Using Nonlinear Least Squares

Abstract: This paper presents a nonlinear least squares method for measuring the power system frequency, wherein the voltage at the measurement point is modeled by using the Fourier series. The estimation of the fundamental frequency is a nonlinear problem in this formulation and is solved by performing a 1-D search over the range of allowed frequency variation. The voltage signal is used for frequency estimation because it is typically less distorted than the line current, resulting in computational efficiency. The robustness of this algorithm with respect to change in various parameters is studied through simulation and the results are validated by hardware implementation using a Virtex IV field-programmable gate array. An application of this algorithm to a shunt active power filter is also presented.

Rotational Invariance Based on Fourier Analysis in Polar and Spherical Coordinates

Abstract: In this paper, polar and spherical Fourier analysis are defined as the decomposition of a function in terms of eigenfunctions of the Laplacian with the eigenfunctions being separable in the corresponding coordinates. The proposed transforms provide effective decompositions of an image into basic patterns with simple radial and angular structures. The theory is compactly presented with an emphasis on the analogy to the normal Fourier transform. The relation between the polar or spherical Fourier transform and the normal Fourier transform is explored. As examples of applications, rotation-invariant descriptors based on polar and spherical Fourier coefficients are tested on pattern classification problems.

Representation Theorems in Hardy Spaces

Abstract: Preface 1. Fourier series 2. Abel-Poisson means 3. Harmonic functions in the unit disc 4. Logarithmic convexity 5. Analytic functions in the unit disc 6. Norm inequalities for the conjugate function 7. Blaschke products and their applications 8. Interpolating linear operators 9. The Fourier transform 10. Poisson integrals 11. Harmonic functions in the upper half plane 12. The Plancherel transform 13. Analytic functions in the upper half plane 14. The Hilbert transform on R A. Topics from real analysis B. A panoramic view of the representation theorems Bibliography Index.

Analytical modeling for transient probe response in pulsed eddy current testing

Abstract: An improved analytical model by the Fourier method for transient eddy current response is presented. In this work, an alternative approach is considered to solve the harmonic eddy current problem by the reflection and transmission theory of electromagnetic waves, thus a more concise closed-form expression is expected to be obtained. To reduce the inherent Gibbs phenomenon, a harmonic order-dependent decreasing factor is employed to weight the Fourier series (FS) representation. It is shown that the developed model is promising to be used as a fast and accurate analytical solver for the transient probe response and is helpful to gain a deep insight into pulsed eddy current (PEC) testing.

Forecasting the turning time of stock market based on Markov-Fourier grey model

Abstract: This paper presents an integration prediction method including grey model (GM), Fourier series, and Markov state transition, known as Markov-Fourier grey model (MFGM), to predict the turning time of Taiwan weighted stock index (TAIEX) for increasing the forecasting accuracy. There are two parts of forecast. The first one is to build an optimal grey model from a series of data, the other uses the Fourier series to refine the residuals produced by the mentioned model. Finally, the Markov state scheme is used for predicting the possibility of location results to promote the intermediate results generated by the Fourier grey model (FGM). It is evident that the proposed approach gets the better result performance than that of the other methods.

A variation norm Carleson theorem

Abstract: We strengthen the Carleson-Hunt theorem by proving $L^p$ estimates for the $r$-variation of the partial sum operators for Fourier series and integrals, for $p>\max\{r',2\}$. Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.

Light curve analysis of Variable stars using Fourier decomposition and Principal component analysis

Abstract: Context. Ongoing and future surveys of variable stars will require new techniques to analyse their light curves as well as to tag objects according to their variability class in an automated way. Aims. We show the use of principal component analysis (PCA) and Fourier decomposition (FD) method as tools for variable star light curve analysis and compare their relative performance in studying the changes in the light curve structures of pulsating Cepheids and in the classification of variable stars. Methods. We have calculated the Fourier parameters of 17 606 light curves of a variety of variables, e.g., RR Lyraes, Cepheids, Mira Variables and extrinsic variables for our analysis. We have also performed PCA on the same database of light curves. The inputs to the PCA are the 100 values of the magnitudes for each of these 17 606 light curves in the database interpolated between phase 0 to 1. Unlike some previous studies, Fourier coefficients are not used as input to the PCA. Results. We show that in general, the first few principal components (PCs) are enough to reconstruct the original light curves compared to the FD method where 2 to 3 times more parameters are required to satisfactorily reconstruct the light curves. The computation of the required number of Fourier parameters on average needs 20 times more CPU time than the computation of the required number of PCs. Therefore, PCA does have some advantages over the FD method in analysing the variable stars in a larger database. However, in some cases, particularly in finding the resonances in fundamental mode (FU) Cepheids, the PCA results show no distinct advantages over the FD method. We also demonstrate that the PCA technique can be used to classify variables into different variability classes in an automated, unsupervised way, a feature that has immense potential for larger databases in the future.

Asymptotic properties of the maximum likelihood estimator for stochastic parabolic equations with additive fractional brownian motion

Abstract: A parameter estimation problem is considered for a diagonalizable stochastic evolution equation using a finite number of the Fourier coefficients of the solution. The equation is driven by additive noise that is white in space and fractional in time with the Hurst parameter H ≥ 1/2. The objective is to study asymptotic properties of the maximum likelihood estimator as the number of the Fourier coefficients increases. A necessary and sufficient condition for consistency and asymptotic normality is presented in terms of the eigenvalues of the operators in the equation.

Modulated Fourier expansions and heterogeneous multiscale methods

Abstract: We show that, for highly oscillatory ordinary differential equation problems, the modulated Fourier expansion approach can be advantageously used to understand and analyse the heterogeneous multiscale methods introduced by E, Engquist and their co-workers.

Learning and Smoothed Analysis

Abstract: We give a new model of learning motivated by smoothed analysis (Spielman and Teng, 2001). In this model, we analyze two new algorithms, for PAC-learning DNFs and agnostically learning decision trees, from random examples drawn from a constant-bounded product distributions. These two problems had previously been solved using membership queries (Jackson, 1995; Gopalan et al, 2005). Our analysis demonstrates that the "heavy" Fourier coefficients of a DNF suffice to recover the DNF. We also show that a structural property of the Fourier spectrum of any boolean function over "typical" product distributions. In a second model, we consider a simple new distribution over the boolean hypercube, one which is symmetric but is not the uniform distribution, from which we can learn O(log n)-depth decision trees in polynomial time.

Modeling of FACTS Devices Based on SPWM VSCs

Abstract: In this contribution, two voltage-source converter (VSC) models based on Fourier series and hyperbolic tangent function are proposed. The models are described in detail. The proposed models can be used for fast simulation in the time domain of power-electronic devices based on sinusoidal pulse-width modulation VSCs; the undesirable error introduced by the high rates in the commutation instants are removed; even though the harmonic distortion coming from the converter is taken into account. The switching instants in the Fourier model are approximated in a closed form, and an iterative algorithm based on the Newton-Raphson method is developed for the exact calculation of the switching instants. The hyperbolic tangent model does not need the calculation of the switching instants as in the case of the Fourier model. The proposed models are validated against the solution obtained with the power system blockset of Simulink and with PSCAD/EMTDC.

Special cycles on unitary Shimura varieties II: global theory

Abstract: We introduce moduli spaces of abelian varieties which are arithmetic models of Shimura varieties attached to unitary groups of signature (n-1, 1). We define arithmetic cycles on these models and study their intersection behaviour. In particular, in the non-degenerate case, we prove a relation between their intersection numbers and Fourier coefficients of the derivative at s=0 of a certain incoherent Eisenstein series for the group U(n, n). This is done by relating the arithmetic cycles to their formal counterpart from Part I via non-archimedean uniformization, and by relating the Fourier coefficients to the derivatives of representation densities of hermitian forms. The result then follows from the main theorem of Part I and a counting argument.

On the Marcinkiewicz-Fejér means of double Fourier series with respect to the Walsh-Kaczmarz system

Abstract: The main aim of this paper is to prove that the maximal operator of Marcinkiewicz-Fejer means of double Fourier series with respect to the Walsh-Kaczmarz system is bounded from the dyadic Hardy-Lorentz space H pq into Lorentz space L pq for every p > 2/3 and 0 < q ≦ ∞. As a consequence, we obtain the a.e. convergence of Marcinkiewicz-Fejer means of double Fourier series with respect to the Walsh-Kaczmarz system. That is, σ n ( f, x 1 , x 2 ) → ( x 1 , x 2 ) a.e. as n → ∞.

Testing Fourier Dimensionality and Sparsity

Abstract: We present a range of new results for testing properties of Boolean functions that are defined in terms of the Fourier spectrum. Broadly speaking, our results show that the property of a Boolean function having a concise Fourier representation is locally testable. We first give an efficient algorithm for testing whether the Fourier spectrum of a Boolean function is supported in a low-dimensional subspace of ${\mathbb F}_2^n$ (equivalently, for testing whether f is a junta over a small number of parities). We next give an efficient algorithm for testing whether a Boolean function has a sparse Fourier spectrum (small number of nonzero coefficients). In both cases we also prove lower bounds showing that any testing algorithm -- even an adaptive one -- must have query complexity within a polynomial factor of our algorithms, which are nonadaptive. Finally, we give an "implicit learning" algorithm that lets us test any sub-property of Fourier concision. Our technical contributions include new structural results about sparse Boolean functions and new analysis of the pairwise independent hashing of Fourier coefficients from [12].