Showing papers on "Fourier series published in 1986"

Improved Fourier coefficients for maps using phases from partial structures with errors

Abstract: Unrefined or partially refined models of macromolecules are generally incomplete and typically have large coordinate errors. It is shown that phase probability equations appropriate for a perfect partial structure lead to inaccurate estimates of phase probabilities in such cases. Therefore, it is necessary to use equations that have been derived allowing for errors in the partial structure. A method is given to estimate the parameter σA in these phase probability expressions from the observed and calculated structure factor amplitudes. From the variation of σA with resolution, one can estimate the mean coordinate error for the model. Electron density maps calculated using partial structure phases are biased towards the partial structure. When there are coordinate errors, a new expression for the non-centric Fourier coefficients [(2m|FN| - D|FcP|) exp(iαcP)] is required to suppress this model bias. Judged by correlation coefficients comparing electron density maps with the correct and the partial structure maps, the Fourier coefficients derived here are superior to others currently in use.

Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation

Abstract: The singularly perturbed differential-delay equation $$\varepsilon \dot x(t) = - x(t) + f(x(t - 1))$$ is studied. Existence of periodic solutions is shown using a global continuation technique based on degree theory. For small ɛ these solutions are proved to have a square-wave shape, and are related to periodic points of the mapping f:R→R.When f is not monotone the convergence of x(t) to the square-wave typically is not uniform, and resembles the Gibbs phenomenon of Fourier series.

The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass

Abstract: 1: The Elements of Analysis in the Eighteenth Century.- 2: The Lagrangian Calculus and Fourier Series.- 3: New Trends in Rigor.- 4: Complex Functions and Integration.- 5: The Convergence of Fourier Series.- 6: Riemann's Theory of Functions.- 7: The Arithmetization of Analysis.- Appendix: On the History of "Dirichlet's Principle.

Novel Properties Of The Fourier Decomposition Of The Sinogram

Abstract: The double Fourier decomposition of the sinogram is obtained by first taking the Fourier transform of each parallel-ray projection and then calculating the coefficients of a Fourier series with respect to angle for each frequency component of the transformed projections. The values of these coefficients may be plotted on a two-dimensional map whose coordinates are spatial frequency w (continuous) and angular harmonic number n (discrete). For |w| large, the Fourier coefficients on the line n=kw of slope k through the origin of the coefficient space are found to depend strongly on the contributions to the projection data that, for each view, come from a certain distance to the detector plane, where the distance is a linear function of k. The values of these coefficients depend only weakly on contributions from other distances from the detector. The theoretical basis of this property is presented in this paper and a potential application to emission computerized tomography is discussed.

Partial averaging approach to Fourier coefficient path integration

Abstract: The recently introduced method of partial averaging is developed into a general formalism for computing simple Cartesian path integrals. Examples of its application to both harmonic and anharmonic systems are given. For harmonic systems, where analytical results can be derived, both imaginary and complex time evolution is discussed. For two representative anharmonic systems, Monte Carlo path integral simulations of the imaginary time propagator (statistical density matrix) are presented. Connections with other Cartesian path integral techniques are stressed.

On the connection between Fourier coefficient and Discretized Cartesian path integration

Abstract: The relationship between so‐called Discretized and Fourier coefficient formulations of Cartesian path integration is examined. In particular, an intimate connection between the two is established by rewriting the Discretized formulation in a manifestly Fourier‐like way. This leads to improved understanding of both the limit behavior and the convergence properties of computational prescriptions based on the two formalisms. The performance of various prescriptions is compared with regard to calculation of on‐diagonal statistical density matrix elements for a number of prototypical 1‐d potentials. A consistent convergence order among these prescriptions is established.

An analytical function for absorption correction

Abstract: A new analytical function is proposed for absorption correction. It is expressed by surface harmonics with polar angles that specify the primary and secondary beam directions. This function has an advantage over Fourier expansion because it is rotationally invariant. Two empirical, methods are used to determine the expansion coefficients. One uses the intensity deviations of equivalent reflections, and the other uses the calculated intensities at the stage of structure refinement. The utility of the analytical function is demonstrated with a model and with actual data.

The Dynamics of a Rotor System with a Cracked Shaft

Abstract: A theoretical analysis of the dynamics of a rotor-bearing system with a transversely cracked rotor is presented. The rotating assembly is modeled using finite rotating shaft elements and the presence of a crack is taken into account by a rotating stiffness variation. This stiffness variation is a function of the rotor’s bending curvature at the crack location and is represented by a Fourier series expansion. The resulting parametrically excited system is nonlinear and is analyzed using a perturbation method coupled with an iteration procedure. The system equations are written in terms of complex variables and an associated computer code has been developed for simulation studies. Results obtained by this analysis procedure are compared with previous analytical and experimental work presented by Grabowski.

Fourier transforms of dini-lipschitz functions

Abstract: It is well known that if Lipschitz conditions of a certain order are imposed on a function f(x), then these conditions affect considerably the absolute convergence of the Fourier series and Fourier transforms of f. In general, if f(x) belongs to a certain function class, then the Lipschitz conditions have bearing as to the dual space to which the Fourier coefficients and transforms of f(x) belong. In the present work we do study the same phenomena for the wider Dini-Lipschitz class as well as for some other allied classes of functions.

The use of vector transforms in solving some electromagnetic scattering problems

Abstract: It is shown that there exists a class of vector transforms and vector series expansions that are independent of their scalar counterparts. They can be used to simplify the solutions of a class of vector electromagnetic scattering problems. The applications are shown of vector Fourier, Hankel, and Mathieu transforms and series expansions to the formal solutions of scattering by rectangular, circular and elliptical disks and open-ended waveguides. The problems solved are canonical problems, however, more complex problems for applications in microwaves, microstrip integrated circuits, geophysical probing, etc. can be solved in a similar fashion.

Fourier coefficients of Eisenstein series on non-congruence subgroups

Abstract: Let Γ be a subgroup of PSL2 (ℤ) of finite index. In this note we are concerned with the arithmetic nature of the Fourier coefficients of holomorphic Eisenstein series on Γ.

Calculation of geometric moments using Fourier plane intensities.

Abstract: A new system for calculating the geometric moments of an input image is presented. The system is based on a mathematical derivation that relates the geometric moments of the input image to the intensity of the Fourier transform of the image. Since optical systems are very efficient at obtaining Fourier transforms, an optical system is used to provide the transform of the input image in this design. An array of detectors is then used to sample the Fourier plane, and a digital postprocessor performs a differentiation process on these Fourier magnitude samples to obtain a vector of values which are combined in a predetermined fashion to provide the geometric moments of the original input function.

The Selection of Terms in an Orthogonal Series Density Estimator

Abstract: We show that Kronmal and Tarter's well-known rule for selecting the terms in an orthogonal series density estimator can lead to poor performance and even inconsistency in certain cases. These difficulties arise when the underlying density has a nonmonotone sequence of Fourier coefficients, as is likely to be the case with sharply peaked or multimodal distributions. We suggest a way of overcoming these shortcomings.

Modeling of two-dimensional random fields by parametric cepstrum

Abstract: A method is presented for parametric modeling of stationary random fields. The class of parametric models considered allows the most general elliptic field, and by linear constraints can include such special cases as isotropic, quarter plane, and separable fields. The technique, based on the cepstrum, has the great advantage of requiring only the use of fast Fourier transforms in the fitting process. Thus, unlike the fitting of two-dimensional autoregressions, no iteration is necessary. Other advantages are that any (Wiener) filters constructed from the fitted spectrum are guaranteed to be stable, and that the spectrum is guaranteed to be positive. Statistical tests for determining various special types of field from data are developed. The choice of model order is discussed as well.

Concepts of generalized bounded variation and the theory of Fourier series.

Abstract: The aim of this article is to unify a large part of present knowledge about the behaviour of Fourier series of functions of generalized bounded variation. The connections between various concepts are discussed, particular attention being payed to those due to Waterman and Chanturiya. Exploiting the existing interactions and utilizing the power of one or another approach to some typical questions of the Fourier theory, a number of previously unnoticed results are obtained in the course of this exposition.

Discrete Fourier Transforms and their Applications

Abstract: This text is designed to be a practial handbook on the evaluation and application of one of the major techniques for discrete signal processing. Knowledge of the discrete Fourier transform (DFT) and the ability to construct alogorithms based on the techniques of fast Fourier analysis are essential prerequisites for communications and cybernetics engineers. These methods are also of inestimable value to applied scientists in many other fields. The treatment given here is aimed specifically at such experimentalists and practitioners, and includes only such mathematical development as is necessary to give a feel for the significance of the methods, and to promote proficiency in its use. An introductory discourse on the general theory of Fourier series and transforms is followed by a thorough review of the properties and means of computation of the DFT. The fast Fourier transform is presented as a particularly efficient algorithm for DFT evaluation, and is described in some detail. Some applications of DFT's are discussed, and the book is rounded off with an introduction to discrete Hilbert transforms. Examples are provided throughout the text, and a full bibliography provides the basis for further study of the mathematical theory and specific areas of application.

Rotationally invariant correlation filtering for multiple images

Abstract: A method is presented for designing translation-invariant optical correlation filters that have a specified rotational response for each of several input images. The correlation filter is postulated to have the form of an infinite linear combination of the angular Fourier harmonics of the input images. The corresponding response of the optical system to rotations of the multiple input images is described by a vector–matrix convolution equation. The solution of this equation for the unknown correlation filter is presented in terms of Fourier series. Use of one term in the Fourier series gives the multiple circular-harmonic filter that provides a specified rotationally invariant response for each of the multiple input images. Applications to rotationally invariant discrimination are described, and examples are given.