Fourier series

About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.

A New Type of ‘X-Ray Microscope’

Abstract: A STANDARD method of X-ray analysis consists in measuring the strength of the diffracted beams corresponding to a series of reflections around a crystal zone (for example, the reflections with indices hol around the b axis), and then forming a double Fourier series with the amplitudes F as coefficients.

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 generalized Peierls–Nabarro model

Abstract: In the original Peierls-Nabarro model the core structure of a dislocation is determined as the solution of an integrodifferential equation. This equation describes the balance between the forces resulting from the deformation of two elastic half-spaces and from a one-dimensional periodic lattice potential acting across the glide plane. A method is described here which allows the core structure and core energy to be obtained for a straight dislocation with arbitrary Burgers vector in an arbitrary glide plane in a crystal of arbitrary anisotropy, for which the displacement potential is represented by a two-dimensional Fourier series. This is accomplished by describing the internal displacements by appropriate trial functions with a set of free parameters whose value is then determined by minimizing the total energy. The method is applied to obtain the core configuration of a screw dislocation dissociated in a {111} plane of a f.c.c. lattice.

Activation detection in functional MRI using subspace modeling and maximum likelihood estimation

Abstract: A statistical method for detecting activated pixels in functional MRI (fMRI) data is presented. In this method, the fMRI time series measured at each pixel is modeled as the sum of a response signal which arises due to the experimentally controlled activation-baseline pattern, a nuisance component representing effects of no interest, and Gaussian white noise. For periodic activation-baseline patterns, the response signal is modeled by a truncated Fourier series with a known fundamental frequency but unknown Fourier coefficients. The nuisance subspace is assumed to be unknown. A maximum likelihood estimate is derived for the component of the nuisance subspace which is orthogonal to the response signal subspace. An estimate for the order of the nuisance subspace is obtained from an information theoretic criterion. A statistical test is derived and shown to be the uniformly most powerful (UMP) test invariant to a group of transformations which are natural to the hypothesis testing problem. The maximal invariant statistic used in this test has an F distribution. The theoretical F distribution under the null hypothesis strongly concurred with the experimental frequency distribution obtained by performing null experiments in which the subjects did not perform any activation task. Applications of the theory to motor activation and visual stimulation fMRI studies are presented.