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Fourier series
About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.
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IBM1
TL;DR: In this paper, the phase shift method was used to solve the migration of zero-offset seismic records with laterally invariant velocities, and the migration process was solved very accurately by the phase-shift method.
Abstract: Accurate methods for the solution of the migration of zero-offset seismic records have been developed. The numerical operations are defined in the frequency domain. The source and recorder positions are lowered by means of a phase shift, or a rotation of the phase angle of the Fourier coefficients. For applications with laterally invariant velocities, the equations governing the migration process are solved very accurately by the phase-shift method. The partial differential equations considered include the 15 degree equation, as well as higher order approximations to the exact migration process. The most accurate migration is accomplished by using the asymptotic equation, whose dispersion relation is the same as that of the full wave equation for downward propagating waves. These equations, however, do not account for the reflection and transmission effects, multiples, or evanescent waves. For comparable accuracy, the present approach to migration is expected to be computationally more efficient than finite-difference methods in general.
885 citations
01 Dec 2005
TL;DR: In this paper, the authors present a model of many-chain systems with Fourier series and transforms, which they call Fourier Series and Transforms (FST) and Gaussian Integrals and Probability Theory.
Abstract: 1. Introduction 2. Ideal Chain Models 3. Single Chains in External Fields 4. Models of Many-Chain Systems 5. Self-Consistent Field Theory 6. Beyond Mean-Field Theory A. Fourier Series and Transforms B. Gaussian Integrals and Probability Theory C. Calculus of Functionals D. Complex Langevin Theory
857 citations
TL;DR: In this article, an improved procedure for numerical inversion of Laplace transforms is proposed based on accelerating the convergence of the Fourier series obtained from the inversion integral using the trapezoidal rule.
Abstract: An improved procedure for numerical inversion of Laplace transforms is proposed based on accelerating the convergence of the Fourier series obtained from the inversion integral using the trapezoidal rule. When the full complex series is used, at each time-value the epsilon-algorithm computes a .(trigonometric) Pade approximation which gives better results than existing acceleration methods. The quotient-difference algorithm is used to compute the coefficients of the corresponding continued fraction, which is evaluated at each time-value, greatly improving efficiency. The convergence of the continued fraction can in turn be accelerated, leading to a further improvement in accuracy.
844 citations
Book•
01 Jan 1990
TL;DR: In this paper, the authors present a probabilistic model for convexity in the Euclidean plane and a model of the massless damped spring, which is based on the Euler-Lagrange equations.
Abstract: 1. Probabilistic Modelling: Pros and Cons. Preliminary considerations. Probabilistic modelling in mechanics. Reliability of structures. Sensitivity of failure probability. Some quotations on the limitations of probabilistic methods. 2. Mathematics of Convexity. Convexity and Uncertainty. What is convexity? Geometric convexity in the Euclidean plane. Algebraic convexity in Euclidean space. Convexity in function spaces. Set-convexity and function-convexity. The structure of convex sets. Extreme points and convex hulls. Extrema of linear functions on convex sets. Hyperplane separation of convex sets. Convex models. 3. Uncertain Excitations. Introductory examples. The massless damped spring. Excitation sets. Maximum responses. Measurement optimization. Vehicle vibration. Introduction. The vehicle model. Uniformly bounded substrate profiles. Extremal responses on uniformly bounded substrates. Duration of acceleration excursions on uniformly bounded substrates. Substrate profiles with bounded slopes. Isochronous obstacles. Solution of the Euler-Lagrange equations. Seismic excitation. Vibration measurements. Introduction. Damped vibrations: full measurement. Example: 2-dimensional measurement. Damped vibrations: partial measurement. Transient vibrational acceleration. 4. Geometric Imperfections. Dynamics of thin bars. Introduction. Analytical formulation. Maximum deflection. Duration above a threshold. Maximum integral displacements. Impact loading of thin shells. Introduction. Basic equations. Extremal displacement. Numerical example. Buckling of thin shells. Introduction. Bounded Fourier coefficients: first-order analysis. Bounded Fourier coefficients: second-order analysis. Uniform bounds on imperfections. Envelope bounds on imperfections. Estimates of the knockdown factor. First and second-order analyses. 5. Concluding Remarks. Bibliography. Index.
801 citations