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Showing papers on "Fourier series published in 1983"


Journal ArticleDOI
TL;DR: In this article, an energy principle is used to obtain the solution of the magnetohydrodynamic (MHD) equilibrium equation J×B−∇p=0 for nested magnetic flux surfaces that are expressed in the inverse coordinate representation x=x(ρ, ρ, π, σ, ω, φ, υ, τ, ϵ, ϳ, ς, ψ, ϩ, ϸ, ϴ, Ϡ, ϖ, ϓ, ό, ϐ, Ϻ, ϔ
Abstract: An energy principle is used to obtain the solution of the magnetohydrodynamic (MHD) equilibrium equation J×B−∇p=0 for nested magnetic flux surfaces that are expressed in the inverse coordinate representation x=x(ρ, θ, ζ). Here, θ are ζ are poloidal and toroidal flux coordinate angles, respectively, and p=p(ρ) labels a magnetic surface. Ordinary differential equations in ρ are obtained for the Fourier amplitudes (moments) in the doubly periodic spectral decomposition of x. A steepest‐descent iteration is developed for efficiently solving these nonlinear, coupled moment equations. The existence of a positive‐definite energy functional guarantees the monotonic convergence of this iteration toward an equilibrium solution (in the absence of magnetic island formation). A renormalization parameter λ is introduced to ensure the rapid convergence of the Fourier series for x, while simultaneously satisfying the MHD requirement that magnetic field lines are straight in flux coordinates. A descent iteration is also developed for determining the self‐consistent value for λ.

750 citations


Journal ArticleDOI
TL;DR: In this article, the amplitude, frequency, wavenumber and phase speed of an unstable deep-water wavetrain were measured using a Hilbert transform technique, showing that the modulation variables evolve from sinusoidal perturbations that are well described as slowly varying Stokes waves, through increasingly asymmetric modulations that finally result in very rapid jumps or phase reversals.
Abstract: Time series of amplitude, frequency, wavenumber and phase speed are measured in an unstable deep-water wavetrain using a Hilbert-transform technique. The modulation variables evolve from sinusoidal perturbations that are well described as slowly varying Stokes waves, through increasingly asymmetric modulations that finally result in very rapid jumps or ‘phase reversals’. These anomalies appear to correspond to the ‘crest pairing’ described by Ramamonjiarisoa & Mollo-Christensen (1979). The measurements offer a novel local description of the instability of deep-water waves which contrasts markedly with the description afforded by conventional Fourier decomposition. The measurements display very large local modulations in the phase speed, modulations that may not be anticipated from measurements of the phase speeds of individual Fourier components travelling (to leading order) at the linear phase speed (Lake & Yuen 1978).

147 citations


Journal ArticleDOI
M. Ram Murty1
TL;DR: The Sato-Tate conjecture for elliptic curves has been shown to be real for any normalized Hecke eigenform as discussed by the authors, and it has been known for a long time that the truth of this conjecture implies much about the oscillatory behaviour of the Fourier coefficients.
Abstract: a(p) = 2p~ @ ) cos 0(p). Since we know the truth of the Ramanujan-Petersson conjecture, it follows that the 0(p)'s are real. Inspired by the Sato-Tate conjecture for elliptic curves, Serre [14] conjectured that the 0(p)'s are uniformly distributed in the interval [0, rc] with respect to the 1 measure -sin2OdO. Following Serre, we shall refer to this as the Sato-Tate r~ conjecture, there being no room for confusion. It has been known for a long time that the truth of this conjecture implies much about the oscillatory behaviour of the Fourier coefficients. In particular, the following is implied by the Sato-Tate conjecture. Theorem 1. For any normalized Hecke eigenform,

117 citations


Book
01 Dec 1983
TL;DR: In this article, the Fourier coefficients and Kloosterman sums are used to compute the Poincare series and their Fourier series expansions, as well as analytic continuations and functional equations.
Abstract: Preliminaries.- Decompositions of G.- Integral representations of eigenfunctions.- Fourier coefficients and Kloosterman sums.- Computation of some integrals I.- Poincare series and their Fourier series expansions.- Computation of some integrals II.- Analytic continuations and functional equations.- Sum formulae (first form).- Sum formulae (second form).

107 citations


Journal ArticleDOI
TL;DR: In this article, the escape of a particle from a potential well is treated using a generalized Langevin equation (GLE) in the low friction limit, where the friction is represented by a memory kernel and the random noise is characterized by a finite correlation time.
Abstract: The escape of a particle from a potential well is treated using a generalized Langevin equation (GLE) in the low friction limit. The friction is represented by a memory kernel and the random noise is characterized by a finite correlation time. This non‐Markovian stochastic equation is reduced to a Smoluchowski diffusion equation for the action variable of the particle and explicit expressions are obtained for the drift and diffusion terms in this equation in terms of the Fourier coefficients of the deterministic trajectory (associated with the motion without coupling to the heat bath) and of the Fourier transform of the friction kernel. The latter (frequency dependent friction) determines the rate of energy exchange with the heat bath. The resulting energy (or action) diffusion equation is used to determine the rate of achieving the critical (escape) energy. Explicit expressions are obtained for a Morse potential. These results for the escape rate agree with those from stochastic trajectories based on the...

84 citations


01 Nov 1983
TL;DR: In this article, an energy principle is used to obtain the solution of the magnetohydrodynamic (MHD) equilibrium equation J×B−∇p=0 for nested magnetic flux surfaces that are expressed in the inverse coordinate representation x=x(ρ, ρ, π, σ, ω, φ, υ, τ, ϵ, ϳ, ς, ψ, ϩ, ϸ, ϴ, Ϡ, ϖ, ϓ, ό, ϐ, Ϻ, ϔ
Abstract: An energy principle is used to obtain the solution of the magnetohydrodynamic (MHD) equilibrium equation J×B−∇p=0 for nested magnetic flux surfaces that are expressed in the inverse coordinate representation x=x(ρ, θ, ζ). Here, θ are ζ are poloidal and toroidal flux coordinate angles, respectively, and p=p(ρ) labels a magnetic surface. Ordinary differential equations in ρ are obtained for the Fourier amplitudes (moments) in the doubly periodic spectral decomposition of x. A steepest‐descent iteration is developed for efficiently solving these nonlinear, coupled moment equations. The existence of a positive‐definite energy functional guarantees the monotonic convergence of this iteration toward an equilibrium solution (in the absence of magnetic island formation). A renormalization parameter λ is introduced to ensure the rapid convergence of the Fourier series for x, while simultaneously satisfying the MHD requirement that magnetic field lines are straight in flux coordinates. A descent iteration is also developed for determining the self‐consistent value for λ.

81 citations


Journal ArticleDOI
TL;DR: In this article, a simply supported rectangular plate with a symmetrically located crack parallel to one edge is considered and the problem is analyzed by means of finite Fourier transformation of discontinuous functions.

73 citations


Journal ArticleDOI
TL;DR: In this paper, a numerical method involving truncated Fourier series is presented for the calculation of properties of short-crested water waves over much of the nonlinear regime.
Abstract: A numerical method involving truncated Fourier series is presented for the calculation of properties of short‐crested water waves. The method produces accurate results over much of the nonlinear regime. In the case of infinite water depth, calculated results include a representative wave profile and the variation in frequency and energy densities with wave steepness and planform skewness.

51 citations



Journal ArticleDOI
David E. Aspnes1
TL;DR: In this article, the authors derived an approximate but closed form Fourier representation of a general branch point and used this result to show that critical point parameters are obtained more directly and with better separation from Fourier coefficients than from least-squares lineshape fitting to the spectra themselves.

41 citations


Book ChapterDOI
TL;DR: In this paper, the authors used the Fourier series fitting and the periodogram for the analysis of time-series data and used the cross spectrum to identify or select time-domain models.
Abstract: Publisher Summary The forerunners of modern spectral analysis have been Fourier series fitting techniques and the periodograms. The reason for the periodogram giving evidence of too many apparent cycles is explained by the low correlation between estimates at adjacent frequencies and the fact that it is an inconsistent estimator of the theoretical spectrum. Recently, spectral techniques have largely been out of favor by applied econometricians, although they are still used as one of the bundle of empirical techniques available for the analysis of time-series data. The theoretical aspects of the frequency-domain representations remain important when the properties of these various techniques are considered. The cross spectrum may also be used to identify or select time-domain models.

Journal ArticleDOI
TL;DR: In this article, the linear cable equation with uniform Poisson or white noise input current is employed as a model for the voltage across the membrane of a onedimensional nerve cylinder, which may sometimes represent the dendritic tree of a nerve cell.
Abstract: The linear cable equation with uniform Poisson or white noise input current is employed as a model for the voltage across the membrane of a onedimensional nerve cylinder, which may sometimes represent the dendritic tree of a nerve cell. From the Green's function representation of the solutions, the mean, variance and covariance of the voltage are found. At large times, the voltage becomes asymptotically wide-sense stationary and we find the spectral density functions for various cable lengths and boundary conditions. For large frequencies the voltage exhibits "1/f 3/2 noise". Using the Fourier series representation of the voltage we study the moments of the firing times for the diffusion model with numerical techniques, employing a simplified threshold criterion. We also simulate the solution of the stochastic cable equation by two different methods in order to estimate the moments and density of the firing time.

Journal ArticleDOI
TL;DR: In this paper, the Fourier expansion of the phase function and the reflection function of a semi-infinite, conservatively scattering atmosphere composed of cloud particles was investigated. But the analysis was limited to the case of very thin to very thick clouds.
Abstract: Computational results are presented for the separate terms in the Fourier expansion of the phase function and the reflection function of a semiinfinite, conservatively scattering atmosphere composed of cloud particles. The calculations involve successive applications of invariant imbedding, doubling, and asymptotic fitting methods to cover the range from very thin to very thick atmospheres. From the results, the ratio of the total reflection function to the first-order reflection function is determined as well as the number of terms required to describe the reflection function to an accuracy of 0.1 percent. The number of terms required depends strongly on the zenith angles of incidence and reflection as well as on details of the phase function. These results are compared with similar results obtained for a Henyey-Greenstein phase function having the same asymmetry factor as in the cloud model.

Book ChapterDOI
TL;DR: In this paper, the Fourier transform has been used for functional approximation and interpolation of stochastic processes, and it has proved of special use to statisticians concerned with stationary process data or concerned with the analysis of linear time-invariant systems.
Abstract: Publisher Summary The Fourier transform has proved of substantial use in most fields of science. It has proved of special use to statisticians concerned with stationary process data or concerned with the analysis of linear time-invariant systems. This chapter describes some of the uses and properties of Fourier transforms of stochastic processes. The Fourier transform turns up in the problems of functional approximation and interpolation. In seismic engineering, the Fourier transforms of observed strong motion records are taken as design inputs and corresponding responses of structures evaluated prior to construction. There are various classes of functions that may be viewed as subject to a harmonic analysis. Quite a different class of functions is provided by the realizations of stationary stochastic processes. Fourier transforms at distinct frequencies and based on nonintersecting data stretches may be approximated by independent normals. The variance of the approximating normal is proportional to the power spectrum of the series.

Journal ArticleDOI
TL;DR: In this paper, the diffraction of a solitary wave around a vertical surface-piercing circular cylinder is described to a first approximation, and the resulting forces on the cylinder are calculated.
Abstract: The diffraction of a solitary wave around a vertical surface-piercing circular cylinder is described to a first approximation, and the resulting forces on the cylinder are calculated. This represents a limiting case of a cnoidal wave diffraction solution given previously, and in order to treat this limit the incident wave is represented in terms of a Fourier integral rather than a Fourier series. The maximum force coefficient and the dimensionless runup depend on a single parameter which replaces the usual diffraction parameter. A criterion for diffraction to be significant for shallow water waves is established and depends upon both the diffraction parameter as well as the additional parmater used here.

Journal ArticleDOI
TL;DR: Several sampling representations for bandlimited and non-bandlimeted functions and their derivatives, as well as of the Hilbert transform and its derivatives are established in this paper, where the results are deduced from a general theorem which in turn is a consequence of the Parseval formula for Fourier series.

Book ChapterDOI
Serge Lang1
01 Jan 1983
TL;DR: In this paper, the authors define a space of functions such that any operation we want to make on improper integrals converges for functions in that space, and they define a set of functions in the space such that every operation we make on an improper integral converges with a function in this space.
Abstract: We are going to define a space of functions such that any operation we want to make on improper integrals converges for functions in that space.

Journal ArticleDOI
TL;DR: In this paper, the transient response of a straight fin composed of two different materials is analyzed and the inverse Laplace transform is solved by utilizing the Fourier series technique, which is shown that the conductivity ratio plays an important role on both the heat transfer rate and the time to reach the steady state.
Abstract: The transient response of a straight fin composed of two different materials is analyzed. The Laplace transformation and eigenfunction expansion methods are used in the analysis. The inverse Laplace transform is solved by utilizing the Fourier series technique. It is shown that the conductivity ratio plays an important role on both the heat transfer rate and the time to reach the steady state. However, the effect of diffusivity ratio is found to be insignificant on the transient response when the conductivities are constant.

Patent
21 Sep 1983
TL;DR: In this article, a process and apparatus for obtaining harmonic-free periodic signals in an incremental measuring system is disclosed in which a graduation having a graduation period P is scanned by at least six scanning elements (for the case in which the previously determined bandwidth N of the analog signals obtained in scanning the graduation N is equal to 3).
Abstract: A process and apparatus for obtaining harmonic-free periodic signals in an incremental measuring system is disclosed in which a graduation having a graduation period P is scanned by at least six scanning elements (for the case in which the previously determined bandwidth N of the analog signals obtained in scanning the graduation N is equal to 3). The periodic analog signals generated by the scanning elements are subjected to a Fourier analysis for the determination of the Fourier coefficients of the base or fundamental wave of the periodic analog signals. These Fourier coefficients are then evaluated as harmonic-free periodic signals for the formation of position measuring values.

Journal ArticleDOI
TL;DR: In this paper, a novel formula is proposed to determine the index profile of optical fibers or preforms from transverse interferograms, which can be calculated only from simple algebra using the Fourier coefficients of the fringe shift and a matrix independent of fiber parameters.
Abstract: A novel formula is proposed to determine the index profile of optical fibers or preforms from transverse interferograms Neither numerical differentiation nor an Abel transformation of the fringe shift is required Index profiles can be calculated only from simple algebra using the Fourier coefficients of the fringe shift and a matrix independent of fiber parameters

Journal ArticleDOI
TL;DR: In this article, Kolmogorov's theorem on the divergence of Fourier series of class L2 in a rearranged trigonometric system and some generalizations of it are discussed.
Abstract: CONTENTS § 1. Introduction § 2. Definitions and auxiliary results § 3. Kolmogorov's example of a trigonometric Fourier series that diverges almost everywhere § 4. Further results on divergent Fourier series § 5. Kolmogorov's theorem on the divergence of Fourier series of class L2 in a rearranged trigonometric system and some generalizations of it § 6. The Kolmogorov-Men'shov theorem on divergent Fourier series in an orthonormal system of sign functionsReferences


Journal ArticleDOI
TL;DR: In this paper, the authors give a new proof of a Hausdorff-Young theorem for amalgams on locally compact abelian groups and prove complementary results about amalgams and their Fourier transforms.
Abstract: Certain function spaces called amalgams have been used and studied in several recent papers on abstract harmonic analysis. In this paper, we give a new proof of a Hausdorff-Young theorem for amalgams on locally compact abelian groups. We also prove some complementary results about amalgams and their Fourier transforms, and in particular give simple proofs of some facts about the Fourier multipliers from certain spaces of functions with compact support intoA(G).

Journal ArticleDOI
TL;DR: In this article, a new interpretation of the self-imaging phenomenon using the Fourier plane of periodical objects is proposed, in which all properties of self-images may be described, in the Fresnel approximation, by the quadratic phase corrections of the object Fourier transform.
Abstract: A new interpretation of the self-imaging phenomenon using the Fourier plane of periodical objects is proposed. All properties of the self-images may be described, in the Fresnel approximation, by the quadratic phase corrections of the object Fourier transform. The angular dimensions of the self-images, as well as the notions of the constant of periodical field configuration and the self-image vergence, are introduced. They allow the characterization, in a uniform manner, of the field distribution in the whole space independently of the chosen self-image plane. The equivalency between the self-imaging phenomenon and the image defocusing by an optical system are considered. The general formulae for the harmonics analysis of the intensity distribution are derived.

Journal ArticleDOI
01 Jan 1983
TL;DR: In this paper, a direct proof is given that these conditions are equivalent; this can be used to simplify some of the proofs in those papers, and it can also be used for simplifying the proof in this paper.
Abstract: Recent papers have given two different conditions on pairs of nonnegative weight functions that insure that a Fourier transform norm inequality holds in R". With additional assumptions these conditions were also shown to be implied by the norm inequality. A direct proof is given here that these conditions are equivalent; this can be used to simplify some of the proofs in those papers.

Journal ArticleDOI
TL;DR: The role of the coefficients cn and dn (n ≥ 0) of φ and φ−1 respectivey, in prediction, filtering and control theory is well-knwn.
Abstract: Let f be the spectral density function of a purely nondeterministic stationary stochastic process and be the optimal (canonical) fator of f. The role of the coefficients cn and dn (n ≥ 0) of φ and φ−1 respectivey, in prediction, filtering and control theory is well-knwn. We show that the cn's and dn's can be obtained recursively in terms of the Fourier coefficients of log f. Also, recursive and updating formulae fr the kolmogorovwiener predictor similar to those Box-Jenkins are provided..

Journal ArticleDOI
TL;DR: In this article, Biot's consolidation of layered soil is solved by the finite layer (strip) method based on quasi-variational as well as least square approaches, and the results compare favorably with other techniques using intergral transforms.
Abstract: In this paper, Biot's consolidation of layered soil is solved by the finite layer (strip) method based on quasi‐variational as well as least square approaches. The two‐ (three‐) dimensional medium is idealized by as many strips (layers) as is necessary to achieve the required accuracy, and the formulation of the relevant matrices requires only simple mathematics involving polynomials and Fourier series, and therefore is much simpler to compare with other techniques using intergral transforms. Examples are computed and the results compare favorably with known solutions.

Journal ArticleDOI
TL;DR: In this article, the authors proposed a set of solvable Volterra equations with constant attenuation in a circular activity region followed by Fourier series expansion of period 2?.
Abstract: Measured projection values of activity distributions in tissue are degraded by the detector aperture function and the exponential attenuation along the path to the object boundary. The aperture function varies with the distance to the detector and turns the integration path into a "weighted cone" shape. Fortunately, summation of opposite views shows a nearly uniform response. Adding opposite projections reduces the problem to a deconvolution problem. Division of the projection values by the total attenuation along the path from the object boundary to the rotated axis yields an integral equation which is independent of the boundary. An additional assumption of constant attenuation in a circular activity region followed by Fourier series expansion of period 2? gives a set of solvable Volterra equations.


Journal ArticleDOI
D. E. Aspnes1, H. Arwin1
TL;DR: In this article, a method for determining critical point parameters from optical spectra in which digital filtering in real (energy) and reciprocal (Fourier-coefficient) space is treated on an equivalent basis is described.
Abstract: We describe a method for accurately determining critical point parameters from optical spectra in which digital filtering in real (energy) and reciprocal (Fourier-coefficient) space is treated on an equivalent basis. Experimental and theoretical line shapes are also filtered in parallel, thereby eliminating systematic errors that can arise in the standard approach in which only the data are processed. Real-space filtering is done using false data to isolate individual or groups of critical points in complicated spectra, to provide a more accurate representation of the data in reciprocal space, and to minimize the effects of end-point discontinuities and truncation errors on the Fourier coefficients calculated from these spectra. Reciprocal-space filtering is done by numerically differentiating the data to maximize the amplitudes of the Fourier coefficients carrying the critical point information, followed by truncating low- and high-order coefficients to minimize artifacts that are due to baseline effects and noise. The optimum order of differentiation (not necessarily integral) is determined from the coefficients themselves. We show that a least-squares regression (LSR) analysis of a restricted interval of equally weighted points in reciprocal space is equivalent to the LSR analysis of all data points equally weighted in real space, making LSR particularly useful for analyzing higher-derivative spectra, where the real-space line shapes rapidly approach zero outside the central structure. For a specific example discussed here, maximum accuracy is obtained if the data are analyzed in the form of a third derivative, as was previously concluded empirically from numerical processing in real space.