Fourier series

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

Random Fourier Series with Applications to Harmonic Analysis. (AM-101), Volume 101

Abstract: In this book the authors give the first necessary and sufficient conditions for the uniform convergence a.s. of random Fourier series on locally compact Abelian groups and on compact non-Abelian groups. They also obtain many related results. For example, whenever a random Fourier series converges uniformly a.s. it also satisfies the central limit theorem. The methods developed are used to study some questions in harmonic analysis that are not intrinsically random. For example, a new characterization of Sidon sets is derived. The major results depend heavily on the Dudley-Fernique necessary and sufficient condition for the continuity of stationary Gaussian processes and on recent work on sums of independent Banach space valued random variables. It is noteworthy that the proofs for the Abelian case immediately extend to the non-Abelian case once the proper definition of random Fourier series is made. In doing this the authors obtain new results on sums of independent random matrices with elements in a Banach space. The final chapter of the book suggests several directions for further research.

Fast polarization modulated ellipsometer using a microprocessor system for digital Fourier analysis

Abstract: A new type of spectroscopic automatic ellipsometer using a piezobirefringent element for polarization modulation at 50 kHz is described. Instead of lock‐in amplifiers the data‐acquisition system consists of a 12.8‐MHz digital sampling of the detected signal with a high word rate 8‐bit ADC, followed by on line Fourier transformation of the accumulated data with a short instruction cycle (∼200 ns) microprocessor, driven by a commercial microcomputer. The absolute minimum time required for measuring one set of Fourier coefficients is thus reduced to the modulation period of 20 μs. For digital error reduction purposes and signal‐to‐noise ratio improvement a basic 5 ms sequence of 256 accumulated periods per point is chosen. At this data‐acquisition rate a precision of 5×10−4 is obtained. Further accumulation over 10 s leads to 10−5 precision capability. A detailed analysis of various sources of inaccuracy leads to an estimate of 0.5° maximum systematic error on the ellipsometric angles and ψ and Δ. Applicatio...

Numerical Solution of the Boltzmann Equation I: Spectrally Accurate Approximation of the Collision Operator

Abstract: In this paper we show that the use of spectral Galerkin methods for the approximation of the Boltzmann equation in the velocity space permits one to obtain spectrally accurate numerical solutions at a reduced computational cost. We prove that the spectral algorithm preserves the total mass and approximates with infinite-order accuracy momentum and energy. Consistency of the method is also proved, and a stability result for a smoothed positive scheme is given. We demonstrate that the Fourier coefficients associated with the collision kernel of the equation have a very simple structure and in some cases can be computed explicitly. Numerical examples for homogeneous test problems in two and three dimensions confirm the advantages of the method.

Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials

Abstract: A new technique is proposed for the recovery of optical phase from intensity information. The method is based on the decomposition of the transport-of-intensity equation into a series of Zernike polynomials. An explicit matrix formula is derived, expressing the Zernike coefficients of the phase as functions of the Zernike coefficients of the wave-front curvature inside the aperture and the Fourier coefficients of the wave-front boundary slopes. Analytical expressions are given, as well as a numerical example of the corresponding phase retrieval matrix. This work lays the basis for an effective algorithm for fast and accurate phase retrieval.

Periodograms for Multiband Astronomical Time Series

Abstract: This paper introduces the multiband periodogram, a general extension of the well-known Lomb-Scargle approach for detecting periodic signals in time-domain data In addition to advantages of the Lomb-Scargle method such as treatment of non-uniform sampling and heteroscedastic errors, the multiband periodogram significantly improves period finding for randomly sampled multiband light curves (eg, Pan-STARRS, DES and LSST) The light curves in each band are modeled as arbitrary truncated Fourier series, with the period and phase shared across all bands The key aspect is the use of Tikhonov regularization which drives most of the variability into the so-called base model common to all bands, while fits for individual bands describe residuals relative to the base model and typically require lower-order Fourier series This decrease in the effective model complexity is the main reason for improved performance We use simulated light curves and randomly subsampled SDSS Stripe 82 data to demonstrate the superiority of this method compared to other methods from the literature, and find that this method will be able to efficiently determine the correct period in the majority of LSST's bright RR Lyrae stars with as little as six months of LSST data A Python implementation of this method, along with code to fully reproduce the results reported here, is available on GitHub