Showing papers on "Fourier series published in 2011"
TL;DR: In this article, an innovations state space modeling framework is introduced for forecasting complex seasonal time series such as those with multiple seasonal periods, high-frequency seasonality, non-integer seasonality and dual-calendar effects.
Abstract: An innovations state space modeling framework is introduced for forecasting complex seasonal time series such as those with multiple seasonal periods, high-frequency seasonality, non-integer seasonality, and dual-calendar effects. The new framework incorporates Box–Cox transformations, Fourier representations with time varying coefficients, and ARMA error correction. Likelihood evaluation and analytical expressions for point forecasts and interval predictions under the assumption of Gaussian errors are derived, leading to a simple, comprehensive approach to forecasting complex seasonal time series. A key feature of the framework is that it relies on a new method that greatly reduces the computational burden in the maximum likelihood estimation. The modeling framework is useful for a broad range of applications, its versatility being illustrated in three empirical studies. In addition, the proposed trigonometric formulation is presented as a means of decomposing complex seasonal time series, and it is show...
761 citations
01 Jan 2011
TL;DR: In this paper, the Fourier Transform on the Real Line is used to transform the real line into a Fourier series, and Wavelet Bases and Frames are used in applied harmonic analysis.
Abstract: ANHA Series Preface.- Preface.- General Notation.- Part I. A Primer on Functional Analysis .- Banach Spaces and Operator Theory.- Functional Analysis.- Part II. Bases and Frames.- Unconditional Convergence of Series in Banach and Hilbert Spaces.- Bases in Banach Spaces.- Biorthogonality, Minimality, and More About Bases.- Unconditional Bases in Banach Spaces.- Bessel Sequences and Bases in Hilbert Spaces.- Frames in Hilbert Spaces.- Part III. Bases and Frames in Applied Harmonic Analysis.- The Fourier Transform on the Real Line.- Sampling, Weighted Exponentials, and Translations.- Gabor Bases and Frames.- Wavelet Bases and Frames.- Part IV. Fourier Series.- Fourier Series.- Basic Properties of Fourier Series.- Part V. Appendices.- Lebesgue Measure and Integration.- Compact and Hilbert-Schmidt Operators.- Hints for Exercises.- Index of Symbols.- References.- Index.
345 citations
Proceedings Article•
07 Aug 2011TL;DR: The Fourier basis is described, a linear value function approximation scheme based on the Fourier series that performs well compared to radial basis functions and the polynomial basis, and is competitive with learned proto-value functions.
Abstract: We describe the Fourier basis, a linear value function approximation scheme based on the Fourier series. We empirically demonstrate that it performs well compared to radial basis functions and the polynomial basis, the two most popular fixed bases for linear value function approximation, and is competitive with learned proto-value functions.
313 citations
Book•
11 Nov 2011
TL;DR: The convergence and summability of Trigonometric Fourier series and their conjugates in the spaces Lp(T), p epsilon]0,+INFINITY[.
Abstract: Preface. Part 1: Simple Trigonometric Series. I. The Conjugation Operator and the Hilbert Transform. II. Pointwise Convergence and Summability of Trigonometric Series. III. Convergence and Summability of Trigonometric Fourier Series and Their Conjugates in the Spaces Lp(T), p epsilon]0,+INFINITY[. IV. Some Approximating Properties of Cesaro Means of the Series sigma[f] and sigma-bar[f]. Part 2: Multiple Trigonometric Series. I. Conjugate Functions and Hilbert Transforms of Functions of Several Variables. II. Convergence and Summability at a Point or Almost Everywhere of Multiple Trigonometric Fourier Series and Their Conjugates. III. Some Approximating Properties of n-Fold Cesaro Means of the Series sigman[f] and sigma-barn[f,B]. IV. Convergence and Summability of Multiple Trigonometric Fourier Series and Their Conjugates in the Spaces Lp(Tn), p epsilon]0,+INFINITY]. V. Summability of Series sigma2[f] and sigma-bar2[f,B]. Bibliography. Index.
91 citations
TL;DR: It is shown that a Fourier expansion of the exponential multiplier yields an exponential series that can compute high-accuracy values of the complex error function in a rapid algorithm that is efficient and practically convenient in numerical methods related to the spectral line broadening and other applications requiring errorfunction evaluation over extended input arrays.
88 citations
TL;DR: The results show that the property of a Boolean function having a concise Fourier representation is locally testable and an “implicit learning” algorithm is given that lets us test any subproperty of Fourier concision.
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 give the first efficient algorithms for testing whether a Boolean function has a sparse Fourier spectrum (small number of nonzero coefficients) and for testing whether the Fourier spectrum of a Boolean function is supported in a low-dimensional subspace of $\mathbb{F}_2^n$. 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. Building on these results, we give an “implicit learning” algorithm that lets us test any subproperty of Fourier concision. We also present some applications of these results to exact learning and decoding. Our technical contributions include new structural results about sparse Boolean functions and new analysis of the pairwise independent hashing of Fourier coefficients from [V. Feldman, P. Gopalan, S. Khot, and A. Ponnuswami, Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2006, pp. 563-576].
85 citations
TL;DR: A novel method to obtain the basic frequency of an unknown periodic signal with an arbitrary waveform, which can work online with no additional signal processing or logical operations and can be used for rhythmic robotic tasks.
Abstract: In this paper we present a novel method to obtain the basic frequency of an unknown periodic signal with an arbitrary waveform, which can work online with no additional signal processing or logical operations. The method originates from non-linear dynamical systems for frequency extraction, which are based on adaptive frequency oscillators in a feedback loop. In previous work, we had developed a method that could extract separate frequency components by using several adaptive frequency oscillators in a loop, but that method required a logical algorithm to identify the basic frequency. The novel method presented here uses a Fourier series representation in the feedback loop combined with a single oscillator. In this way it can extract the frequency and the phase of an unknown periodic signal in real time and without any additional signal processing or preprocessing. The method determines the Fourier series coefficients and can be used for dynamic Fourier series implementation. The proposed method can be used for the control of rhythmic robotic tasks, where only the extraction of the basic frequency is crucial. For demonstration several highly non-linear and dynamic periodic robotic tasks are shown, including also a task where an electromyography (EMG) signal is used in a feedback loop.
83 citations
TL;DR: There is no single "best" basis for the disk, but there are seven different strategies for computing spectrally-accurate approximations or differential equation solutions in a disk and the merits and flaws of each spectral option are laid out.
82 citations
TL;DR: The proposed Fourier continuation algorithm inherits many of the highly desirable properties arising from rapidly convergent Fourier expansions, including high-order convergence, essentially spectrally accurate dispersion relations, and much milder CFL constraints than those imposed by polynomial-based spectral methods.
76 citations
TL;DR: A Fourier series method for the acoustic analysis of a rectangular cavity with impedance boundary conditions arbitrarily specified on any of the walls, which can be simultaneously obtained from solving a standard matrix eigenvalue problem instead of iteratively solving a nonlinear transcendental equation as in the existing methods.
Abstract: A Fourier series method is proposed for the acoustic analysis of a rectangular cavity with impedance boundary conditions arbitrarily specified on any of the walls. The sound pressure is expressed as the combination of a three-dimensional Fourier cosine series and six supplementary two-dimensional expansions introduced to ensure (accelerate) the uniform and absolute convergence (rate) of the series representation in the cavity including the boundary surfaces. The expansion coefficients are determined using the Rayleigh–Ritz method. Since the pressure field is constructed adequately smooth throughout the entire solution domain, the Rayleigh–Ritz solution is mathematically equivalent to what is obtained from a strong formulation based on directly solving the governing equations and the boundary conditions. To unify the treatments of arbitrary nonuniform impedance boundary conditions, the impedance distribution function on each specified surface is invariantly expressed as a double Fourier series expansion so that all the relevant integrals can be calculated analytically. The modal parameters for the acoustic cavity can be simultaneously obtained from solving a standard matrix eigenvalue problem instead of iteratively solving a nonlinear transcendental equation as in the existing methods. Several numerical examples are presented to demonstrate the effectiveness and reliability of the current method for various impedance boundary conditions, including nonuniform impedance distributions.
TL;DR: In this paper, an analytical method is derived for determining the vibrations of two plates which are generally supported along the boundary edges, and elastically coupled together at an arbitrary angle by four types of coupling springs of arbitrary stiffnesses.
TL;DR: The problem of the maximization of the energy produced by a self reacting point absorber subject to motion restriction is addressed and the optimal control problem is reformulated as a non linear program where the properties of the cost function and of the constraint are affected by the choice of the basis functions.
TL;DR: In this paper, a three dimensional vibration analysis of nano-plates is studied by decoupling the field equations of Eringen theory, considering the small scale effect, the three dimensional equations of nonlocal elasticity are obtained.
TL;DR: Detailed analyses of four limiting factors of cryo electron microscopy, electron-beam tilt, inaccurate determination of defocus values, focus gradient through particles, and dynamic scattering of electrons are presented and strategies to cope with these factors are proposed.
Book•
14 Oct 2011
TL;DR: The Asymptotic Behavior of Generalized Functions and Integral Transforms Summability of Fourier Series and Integrals and Fourier series and integrals are studied.
Abstract: Asymptotic Behavior of Generalized Functions: S-Asymptotics F'g Quasi-Asymptotics in F' Applications of the Asymptotic Behavior of Generalized Functions: Asymptotic Behavior of Solutions to Partial Differential Equations Asymptotics and Integral Transforms Summability of Fourier Series and Integrals.
Book•
01 Jan 2011
TL;DR: The Dirac delta function as discussed by the authors is a generalised Dirac function with time-invariant linear systems and its derivatives, which is a type of generalised functions. Introduction to distributions Integration theory and generalized functions Solutions References Index.
Abstract: Results from elementary analysis The Dirac delta function Properties of the delta function and its derivatives Time-invariant linear systems The Laplace Transform Fourier Series and Fourier transforms Other types of generalised functions Introduction to distributions Integration theory NSA and generalised functions Solutions References Index.
Patent•
CGG1
TL;DR: In this article, an improved method for analyzing seismic data to obtain elastic attributes is disclosed in one embodiment, a reflectivity series is determined for at least one seismic trace of seismic data obtained for a subterranean formation, where the reflectivities series includes anisotropy properties of a formation.
Abstract: An improved method for analyzing seismic data to obtain elastic attributes is disclosed In one embodiment, a reflectivity series is determined for at least one seismic trace of seismic data obtained for a subterranean formation, where the reflectivity series includes anisotropy properties of a formation One or more synthetic seismic traces are obtained by convolving the reflectivity series with a source wavelet The one or more synthetic seismic traces are inverted to obtain elastic parameters estimates According to one aspect, the data inputs are angle-azimuth stacks According to another aspect, the data inputs are azimuthal Fourier coefficients, un, vn
TL;DR: In this article, the authors investigated the three-dimensional (3D) scattering of guided waves by a through-thickness cavity with irregular shape in an isotropic plate, where the scattered field is decomposed on the basis of Lamb and SH waves (propagating and non-propagation), and the amplitude of the modes is calculated by writing the nullity of the total stress at the boundary of the cavity.
TL;DR: To improve inpainting performances over large missing regions, a highly nonconvex generalization of the texture model is introduced and allows one to impose an arbitrary oscillation profile.
Abstract: This article presents a new adaptive framework for locally parallel texture modeling. Oscillating patterns are modeled with functionals that constrain the local Fourier decomposition of the texture. We first introduce a texture functional which is a weighted Hilbert norm. The weights on the local Fourier atoms are optimized to match the local orientation and frequency of the texture. This adaptive model is used to solve image processing inverse problems, such as image decomposition and inpainting. The local orientation and frequency of the texture component are adaptively estimated during the minimization process. To improve inpainting performances over large missing regions, we introduce a highly nonconvex generalization of our texture model. This new model constrains the amplitude of the texture and allows one to impose an arbitrary oscillation profile. Numerical results illustrate the effectiveness of the method.
TL;DR: In this article, the authors formulate the polarized RT equation in multi-dimensional media that takes into account the Hanle effect with angle-dependent PRD functions, which can be solved by any iterative method such as an approximate Lambda iteration or a Bi-Conjugate Gradient-type projection method provided we truncate the Fourier series to have a finite number of terms.
Abstract: To explain the linear polarization observed in spatially resolved structures in the solar atmosphere, the solution of polarized radiative transfer (RT) equation in multi-dimensional (multi-D) geometries is essential. For strong resonance lines, partial frequency redistribution (PRD) effects also become important. In a series of papers, we have been investigating the nature of Stokes profiles formed in multi-D media including PRD in line scattering. For numerical simplicity, so far we have restricted our attention to the particular case of PRD functions which are averaged over all the incident and scattered directions. In this paper, we formulate the polarized RT equation in multi-D media that takes into account the Hanle effect with angle-dependent PRD functions. We generalize here to the multi-D case the method for Fourier series expansion of angle-dependent PRD functions originally developed for RT in one-dimensional geometry. We show that the Stokes source vector S = (S{sub I} , S{sub Q} , S{sub U} ){sup T} and the Stokes vector I = (I, Q, U){sup T} can be expanded in terms of infinite sets of components S-tilde{sup (k)}, I-tilde{sup (k)}, respectively, k in [0, +{infinity}). We show that the components S-tilde{sup (k)} become independent of the azimuthal angle ({psi}) ofmore » the scattered ray, whereas the components I-tilde{sup (k)} remain dependent on {psi} due to the nature of RT in multi-D geometry. We also establish that S-tilde{sup (k)} and I-tilde{sup (k)} satisfy a simple transfer equation, which can be solved by any iterative method such as an approximate Lambda iteration or a Bi-Conjugate Gradient-type projection method provided we truncate the Fourier series to have a finite number of terms.« less
TL;DR: A new method is proposed which employs linear filtering stage coupled with adaptive filtering stage to remove drift and attenuation and outperforms the existing analytical integration method.
Abstract: Position sensing with inertial sensors such as accelerometers and gyroscopes usually requires other aided sensors or prior knowledge of motion characteristics to remove position drift resulting from integration of acceleration or velocity so as to obtain accurate position estimation. A method based on analytical integration has previously been developed to obtain accurate position estimate of periodic or quasi-periodic motion from inertial sensors using prior knowledge of the motion but without using aided sensors. In this paper, a new method is proposed which employs linear filtering stage coupled with adaptive filtering stage to remove drift and attenuation. The prior knowledge of the motion the proposed method requires is only approximate band of frequencies of the motion. Existing adaptive filtering methods based on Fourier series such as weighted-frequency Fourier linear combiner (WFLC), and band-limited multiple Fourier linear combiner (BMFLC) are modified to combine with the proposed method. To validate and compare the performance of the proposed method with the method based on analytical integration, simulation study is performed using periodic signals as well as real physiological tremor data, and real-time experiments are conducted using an ADXL-203 accelerometer. Results demonstrate that the performance of the proposed method outperforms the existing analytical integration method.
TL;DR: In this paper, the authors investigated the thermal instability in a rotating anisotropic porous layer with heat source using the extended Darcy model, which includes the time derivative and Coriolis term in the momentum equation.
Abstract: In this article, linear and nonlinear thermal instability in a rotating anisotropic porous layer with heat source has been investigated. The extended Darcy model, which includes the time derivative and Coriolis term has been employed in the momentum equation. The linear theory has been performed by using normal mode technique, while nonlinear analysis is based on minimal representation of the truncated Fourier series having only two terms. The criteria for both stationary and oscillatory convection is derived analytically. The rotation inhibits the onset of convection in both stationary and oscillatory modes. Effects of parameters on critical Rayleigh number has also been investigated. A weak nonlinear analysis based on the truncated representation of Fourier series method has been used to find the Nusselt number. The transient behavior of the Nusselt number has also been investigated by solving the finite amplitude equations using a numerical method. Steady and unsteady streamlines, and isotherms have been drawn to determine the nature of flow pattern. The results obtained during the analysis have been presented graphically.
TL;DR: In this article, a formula for the values of automorphic Green functions on the special rational 0-cycles (big CM points) attached to certain maximal tori in the Shimura varieties associated to rational quadratic spaces of signature (2d, 2).
Abstract: We give a formula for the values of automorphic Green functions on the special rational 0-cycles (big CM points) attached to certain maximal tori in the Shimura varieties associated to rational quadratic spaces of signature (2d, 2). Our approach depends on the fact that the Green functions in question are constructed as regularized theta lifts of harmonic weak Maass forms, and it involves the Siegel-Weil formula and the central derivatives of incoherent Eisenstein series for totally real fields. In the case of a weakly holomorphic form, the formula is an explicit combination of quantities obtained from the Fourier coefficients of the central derivative of the incoherent Eisenstein series. In the case of a general harmonic weak Maass form, there is an additional term given by the central derivative of a Rankin-Selberg type convolution.
TL;DR: In this paper, the authors considered the problem of a thermoelastic infinitely long hollow cylinder in the context of the theory of generalized thermo-elastic diffusion with one relaxation time, and the solution of the problem in the physical domain was obtained numerically using a numerical method for the inversion of the Laplace transform based on Fourier expansion techniques.
Abstract: In this work, we consider the problem of a thermoelastic infinitely long hollow cylinder in the context of the theory of generalized thermoelastic diffusion with one relaxation time. The outer surface of the cylinder is taken traction free and subjected to a thermal shock, while the inner surface is taken to be in contact with a rigid surface and is thermally insulated. Laplace transform techniques are used. The solution is obtained in the Laplace transform domain by using a direct approach. The solution of the problem in the physical domain is obtained numerically using a numerical method for the inversion of the Laplace transform based on Fourier expansion techniques. The temperature, displacement, stress and concentration as well as the chemical potential are obtained. Numerical computations are carried out and represented graphically.
TL;DR: A novel decoupling of the least-squares problem is demonstrated which results in two systems of equations, one of which may be solved quickly by means of fast Fourier transforms (FFTs) and another that is demonstrated to be well approximated by a low-rank system.
Abstract: A new algorithm is presented which provides a fast method for the computation of recently developed Fourier continuations (a particular type of Fourier extension method) that yield superalgebraically convergent Fourier series approximations of nonperiodic functions. Previously, the coefficients of an approximating Fourier series have been obtained by means of a regularized singular value decomposition (SVD)-based least-squares solution to an overdetermined linear system of equations. These SVD methods are effective when the size of the system does not become too large, but they quickly become unwieldy as the number of unknowns in the system grows. We demonstrate a novel decoupling of the least-squares problem which results in two systems of equations, one of which may be solved quickly by means of fast Fourier transforms (FFTs) and another that is demonstrated to be well approximated by a low-rank system. Utilizing randomized algorithms, the low-rank system is reduced to a significantly smaller system of equations. This new system is then efficiently solved with drastically reduced computational cost and memory requirements while still benefiting from the advantages of using a regularized SVD. The computational cost of the new algorithm in on the order of the cost of a single FFT multiplied by a slowly increasing factor that grows only logarithmically with the size of the system.
01 Jan 2011
TL;DR: In this paper, the authors used the tool of multiple Fourier series, built in the space L 2 and poitwise, for the mean-square approximation of multiple stochastic integrals.
Abstract: It is well known, that Ito stochastic differential equations (SDE) are adequate mathematical models of dynamic systems under the influence of random disturbances. One of the effective approaches to numerical integration of Ito SDE is an approach based on Taylor-Ito and Taylor-Stratonovich expansions. The most important feature of such expansions is presence in them of so called multiple Ito or Stratonovich stochastic integrals, which play the key role for solving the problem of numerical integration of Ito SDE. We successfully use the tool of multiple Fourier series, built in the space L2 and poitwise, for the mean-square approximation of multiple stochastic integrals.
TL;DR: In this article, a closed form solution for the natural frequencies of a rectangular simply supported nano-plate is obtained by using state-space method in the thickness direction and Fourier series in the in-plane directions.
Abstract: Vibration analysis of a nano-plate, based on three-dimensional theory of elasticity, is studied employing non-local continuum mechanics. By using state-space method in the thickness direction and Fourier series in the in-plane directions, a closed form solution for the natural frequencies of a rectangular simply supported nano-plate is obtained. To verify the accuracy of the present approach, numerical results are compared with the results available in the literature. The effect of the non-local parameter, aspect ratio, thickness-to-length ratio and half-wave numbers in the frequency behavior is examined.
TL;DR: In this article, a steady-state large-signal model of coupled-cavity traveling-wave tubes is described, in which the input and output signals are periodic functions of time that may be represented by Fourier series of finite length.
Abstract: We describe a steady-state large-signal model of coupled-cavity traveling-wave tubes in which the input and output signals are periodic functions of time that may be represented by Fourier series of finite length. The model includes both linear and nonlinear effects including circuit dispersion, reflections, intermodulation, and harmonic generation. The model uses a lumped element representation of the circuit and a 1-D disk model of the beam. Several favorable comparisons of model predictions with experimental measurements, including gain versus frequency and power transfer characteristics, are illustrated. The inclusion of nonlinear effects in this multifrequency model enables predictions of intermodulation products, as functions of the input power. An example of the computation of C3IM is illustrated.