Fourier series

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

On the Calculation of Stationary Solutions of Multi-Dimensional Fokker–Planck Equations by Orthogonal Functions

Abstract: In this paper, nonlinear stochastic systems are investigatedvia associated Fokker–Planck equations. Their stationary solutions arecalculated by expansions into orthogonal functions, e.g. especiallyadjusted polynomials and Fourier series. The weighting functions of thenew polynomials are obtained by the application of the stochasticaveraging method. The proposed analysis is demonstrated with severalexamples. The first one is a two-dimensional problem of nonlinearoscillators driven by white noise. The second one describes two-massoscillators with independent coloured noise excitations leading tosix-dimensional probability density functions. The next example ispresenting a system driven by both harmonic and stochastic excitationleading to three-dimensional probability density functions. Finally,oscillators with dry friction characteristics are examined.

Interpreting azimuthal Fourier coefficients for anisotropic and fracture parameters

Abstract: Amplitude variation with offset and azimuth (AVOAz) analysis can be separated into two separate parts: amplitude variation with offset (AVO) analysis and amplitude variation with azimuth (AVAz) analysis. Useful information about fractures and anisotropy can be obtained just by examining the AVAz. The AVAz can be described as a sum of sinusoids of different periodicities, each characterized by its magnitude and phase. This sum is mathematically equivalent to a Fourier series, and hence the coefficients describing the AVAz response are azimuthal Fourier coefficients (FCs). This FC parameterization is purely descriptive. The aim of this paper is to help the interpreter understand what these coefficients mean in terms of anisotropic and fracture parameters for the case of P-wave reflectivity using a linearized approximation. The FC representation is valid for general anisotropy. However, to gain insight into the significance of FCs, more restrictive assumptions about the anisotropy or facture system m...

Dirichlet series in the theory of Siegel modular forms

Abstract: We are concerned with Dirichlet series which appear in the Fourier expansion of the non-analytic Eisenstein series on the Siegel upper half space Hm of degree m. In the case of m = 2 Kaufhold [1] evaluated them. Here we treat the general cases by a different method.

Forecasting the turning time of stock market based on Markov-Fourier grey model

Abstract: This paper presents an integration prediction method including grey model (GM), Fourier series, and Markov state transition, known as Markov-Fourier grey model (MFGM), to predict the turning time of Taiwan weighted stock index (TAIEX) for increasing the forecasting accuracy. There are two parts of forecast. The first one is to build an optimal grey model from a series of data, the other uses the Fourier series to refine the residuals produced by the mentioned model. Finally, the Markov state scheme is used for predicting the possibility of location results to promote the intermediate results generated by the Fourier grey model (FGM). It is evident that the proposed approach gets the better result performance than that of the other methods.