Showing papers on "Fourier series published in 2014"

Upper and Lower Bounds for Stochastic Processes : Modern Methods and Classical Problems

Abstract: 0. Introduction.- 1. Philosophy and Overview of the Book.- 2. Gaussian Processes and the Generic Chaining.- 3. Random Fourier Series and Trigonometric Sums, I. - 4. Matching Theorems I.- 5. Bernouilli Processes.- 6. Trees and the Art of Lower Bounds.- 7. Random Fourier Series and Trigonometric Sums, II.- 8. Processes Related to Gaussian Processes.- 9. Theory and Practice of Empirical Processes.- 10. Partition Scheme for Families of Distances.- 11. Infinitely Divisible Processes.- 12. The Fundamental Conjectures.- 13. Convergence of Orthogonal Series Majorizing Measures.- 14. Matching Theorems, II: Shor's Matching Theorem. 15. The Ultimate Matching Theorem in Dimension => 3.- 16. Applications to Banach Space Theory.- 17. Appendix: What this Book is Really About.- 18. Appendix: Continuity.- References. Index.

Advection modes by optimal mass transfer.

Abstract: Classical model reduction techniques approximate the solution of a physical model by a limited number of global modes. These modes are usually determined by variants of principal component analysis. Global modes can lead to reduced models that perform well in terms of stability and accuracy. However, when the physics of the model is mainly characterized by advection, the nonlocal representation of the solution by global modes essentially reduces to a Fourier expansion. In this paper we describe a method to determine a low-order representation of advection. This method is based on the solution of Monge-Kantorovich mass transfer problems. Examples of application to point vortex scattering, Korteweg--de Vries equation, and hurricane Dean advection are discussed.

Exact boundary controllability of 1-d parabolic and hyperbolic degenerate equations ∗

Abstract: The boundary controllability of a class of one-dimensional degenerate equations is studied in this paper. The novelty of this work is that the control acts through the part of the boundary where degeneracy occurs. First we consider a class of degenerate hyperbolic equations. Then, we prove sharp observability estimates for these equations using nonharmonic Fourier series. The transmutation method yields a result for the corresponding class of degenerate parabolic equations.

Using Fourier differential quadrature method to analyze transverse nonlinear vibrations of an axially accelerating viscoelastic beam

Abstract: In this paper, a Fourier expansion-based differential quadrature (FDQ) method is developed to analyze numerically the transverse nonlinear vibrations of an axially accelerating viscoelastic beam. The partial differential nonlinear governing equation is discretized in space region and in time domain using FDQ and Runge–Kutta–Fehlberg methods, respectively. The accuracy of the proposed method is represented by two numerical examples. The nonlinear dynamical behaviors, such as the bifurcations and chaotic motions of the axially accelerating viscoelastic beam, are investigated using the bifurcation diagrams, Lyapunov exponents, Poincare maps, and three-dimensional phase portraits. The bifurcation diagrams for the in-plane responses to the mean axial velocity, the amplitude of velocity fluctuation, and the frequency of velocity fluctuation are, respectively, presented when other parameters are fixed. The Lyapunov exponents are calculated to further identify the existence of the periodic and chaotic motions in the transverse nonlinear vibrations of the axially accelerating viscoelastic beam. The conclusion is drawn from numerical simulation results that the FDQ method is a simple and efficient method for the analysis of the nonlinear dynamics of the axially accelerating viscoelastic beam.

On the Numerical Stability of Fourier Extensions

Abstract: An effective means to approximate an analytic, nonperiodic function on a bounded interval is by using a Fourier series on a larger domain. When constructed appropriately, this so-called Fourier extension is known to converge geometrically fast in the truncation parameter. Unfortunately, computing a Fourier extension requires solving an ill-conditioned linear system, and hence one might expect such rapid convergence to be destroyed when carrying out computations in finite precision. The purpose of this paper is to show that this is not the case. Specifically, we show that Fourier extensions are actually numerically stable when implemented in finite arithmetic, and achieve a convergence rate that is at least superalgebraic. Thus, in this instance, ill-conditioning of the linear system does not prohibit a good approximation. In the second part of this paper we consider the issue of computing Fourier extensions from equispaced data. A result of Platte et al. (SIAM Rev. 53(2):308---318, 2011) states that no method for this problem can be both numerically stable and exponentially convergent. We explain how Fourier extensions relate to this theoretical barrier, and demonstrate that they are particularly well suited for this problem: namely, they obtain at least superalgebraic convergence in a numerically stable manner.

Overview of Analytical Models for the Design of Linear and Planar Motors

Abstract: In this paper, an overview of analytical techniques for the modeling of linear and planar permanent-magnet motors is given. These models can be used complementary to finite element analysis for fast evaluations of topologies, but they are indispensable for the design of magnetically levitated planar motors and other coreless multi-degrees of freedom motors, which are applied in (ultra) high-precision applications. The analytical methods describe the magnetic fields based on magnetic surface charges and Fourier series in 2-D and 3-D.

Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications

Abstract: In this paper, the initial-boundary-value problems for the one-dimensional linear and non-linear fractional diffusion equations with the Riemann-Liouville time-fractional derivative are analyzed. First, a weak and a strong maximum principles for solutions of the linear problems are derived. These principles are employed to show uniqueness of solutions of the initial-boundary-value problems for the non-linear fractional diffusion equations under some standard assumptions posed on the non-linear part of the equations. In the linear case and under some additional conditions, these solutions can be represented in form of the Fourier series with respect to the eigenfunctions of the corresponding Sturm-Liouville eigenvalue problems.

Fractional Fourier Series with Applications

Abstract: In this paper, we introduce conformable fractional Fourier series. We use such series to solve certain partial fractional differential equations.

Local Fractional Function Decomposition Method for Solving Inhomogeneous Wave Equations with Local Fractional Derivative

Abstract: We propose the local fractional function decomposition method, which is derived from the coupling method of local fractional Fourier series and Yang-Laplace transform. The forms of solutions for local fractional differential equations are established. Some examples for inhomogeneous wave equations are given to show the accuracy and efficiency of the presented technique.

Toward lattice fractional vector calculus

Abstract: An analog of fractional vector calculus for physical lattice models is suggested. We use an approach based on the models of three-dimensional lattices with long-range inter-particle interactions. The lattice analogs of fractional partial derivatives are represented by kernels of lattice long-range interactions, where the Fourier series transformations of these kernels have a power-law form with respect to wave vector components. In the continuum limit, these lattice partial derivatives give derivatives of non-integer order with respect to coordinates. In the three-dimensional description of the non-local continuum, the fractional differential operators have the form of fractional partial derivatives of the Riesz type. As examples of the applications of the suggested lattice fractional vector calculus, we give lattice models with long-range interactions for the fractional Maxwell equations of non-local continuous media and for the fractional generalization of the Mindlin and Aifantis continuum models of gradient elasticity.

Negative index Jacobi forms and quantum modular forms

Abstract: In this paper, we consider the Fourier coefficients of a special class of meromorphic Jacobi forms of negative index considered by Kac and Wakimoto. Much recent work has been done on such coefficients in the case of Jacobi forms of positive index, but almost nothing is known for Jacobi forms of negative index. In this paper we show, from two different perspectives, that their Fourier coefficients have a simple decomposition in terms of partial theta functions. The first perspective uses the language of Lie super algebras, and the second applies the theory of elliptic functions. In particular, we find a new infinite family of rank-crank type partial differential equations generalizing the famous example of Atkin and Garvan. We then describe the modularity properties of these coefficients, showing that they are ‘mixed partial theta functions’, along the way determining a new class of quantum modular partial theta functions which is of independent interest. In particular, we settle the final cases of a question of Kac concerning modularity properties of Fourier coefficients of certain Jacobi forms.MSC 11F03; 11F22; 11F37; 11F50

On the role of total variation in compressed sensing

Abstract: This paper considers the problem of recovering a one or two dimensional discrete signal which is approximately sparse in its discrete gradient from an incomplete subset of its discrete Fourier coefficients which have been corrupted with noise. We prove that in order to obtain a reconstruction which is robust to noise and stable to inexact gradient sparsity of order $s$ with high probability, it suffices to draw $\mathcal{O}(s \log N)$ of the available Fourier coefficients uniformly at random. However, we also show that if one draws $\mathcal{O}(s \log N)$ samples in accordance to a particular distribution which concentrates on the low Fourier frequencies, then the stability bounds which can be guaranteed are optimal up to $\log$ factors. Finally, we prove that in the one dimensional case where the underlying signal is gradient sparse and its sparsity pattern satisfies a minimum separation condition, then to guarantee exact recovery with high probability, for some $M

A Stability Barrier for Reconstructions from Fourier Samples

Abstract: We prove that any stable method for resolving the Gibbs phenomenon---that is, recovering high-order accuracy from the first $m$ Fourier coefficients of an analytic and nonperiodic function---can converge at best root-exponentially fast in $m$. Any method with faster convergence must also be unstable, and in particular, exponential convergence implies exponential ill-conditioning. This result is analogous to a recent theorem of Platte, Trefethen, and Kuijlaars concerning recovery from pointwise function values on an equispaced $m$-grid. The main step in our proof is an estimate for the maximal behavior of a polynomial of degree $n$ with bounded $m$-term Fourier series. This estimate is related to a conjecture of Hrycak and Grochenig. In the final part of the paper we discuss the implications of our main theorem to polynomial-based interpolation and least-squares approaches for overcoming the Gibbs phenomenon.

On the partial sums of Vilenkin-Fourier series

Abstract: The aim of this paper is to investigate weighted maximal operators of partial sums of Vilenkin-Fourier series. Also, the obtained results we use to prove approximation and strong convergence theorems on the martingale Hardy spaces Hp, when 0 < p ≤ 1.

Adaptive backstepping sliding mode control of flexible ball screw drives with time-varying parametric uncertainties and disturbances

Abstract: This paper presents a method to model and design servo controllers for flexible ball screw drives with dynamic variations. A mathematical model describing the structural flexibility of the ball screw drive containing time-varying uncertainties and disturbances with unknown bounds is proposed. A mode-compensating adaptive backstepping sliding mode controller is designed to suppress the vibration. The time-varying uncertainties and disturbances represented in finite-term Fourier series can be estimated by updating the Fourier coefficients through function approximation technique. Adaptive laws are obtained from Lyapunov approach to guarantee the convergence and stability of the closed loop system. The simulation results indicate that the tracking accuracy is improved considerably with the proposed scheme when the time-varying parametric uncertainties and disturbances exist.

One-Dimensional Spectral Analysis of Complex PWM Waveforms Using Superposition

Abstract: This paper takes a new look at the mechanisms underlying the double-edge pulse-width modulation (PWM) process. It presents a novel way of deriving equations for the spectrum of double-edge PWM using basic mathematical techniques. In the process the underlying nonlinearities that generate the PWM sidebands are identified. Unlike the classical double Fourier series approach, the proposed method of deriving the PWM spectrum does not require the construction of the so-called unit cell. The interaction between this new model of the pulse-width modulator and the regular sampling process is studied, and generalized equations for the Fourier transforms of regularly sampled PWM waveforms are derived. A general solution to the important question of what happens to the PWM spectrum when the PWM reference consists of a summation of signals is presented. It is shown that the addition of reference signals in the time domain results in a double convolution of the PWM sidebands in the frequency domain. As an application of this result, it is shown how new analytic equations for the harmonics of third-harmonic injection PWM and space vector modulation can easily be derived. Finally, the new theoretical results are benchmarked against results from the well-established double Fourier series method.

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.