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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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TL;DR: The truncated Fourier series of products of discontinuous functions that were used in the differential theory of gratings during the past 30 years are shown to be converging everywhere only at the limit of infinitely low modulated gratings, and a new formulation is proposed, applicable to any theory that requires the Fourier analysis of continuous products of discontinued periodic functions.
Abstract: Using theorems of Fourier factorization, a recent paper [J. Opt. Soc. Am. A 13, 1870 (1996)] has shown that the truncated Fourier series of products of discontinuous functions that were used in the differential theory of gratings during the past 30 years are not converging everywhere in TM polarization. They turn out to be converging everywhere only at the limit of infinitely low modulated gratings. We derive new truncated equations and implement them numerically. The computed efficiencies turn out to converge about as fast as in the TE-polarization case with respect to the number of Fourier harmonics used to represent the field. The fast convergence is observed on both metallic and dielectric gratings with sinusoidal, triangular, and lamellar profiles as well as with cylindrical and rectangular rods, and examples are shown on gratings with 100% modulation. The new formulation opens a new wide range of applications of the method, concerning not only gratings used in TM polarization but also conical diffraction, crossed gratings, three-dimensional problems, nonperiodic objects, rough surfaces, photonic band gaps, nonlinear optics, etc. The formulation also concerns the TE polarization case for a grating ruled on a magnetic material as well as gratings ruled on anisotropic materials. The method developed is applicable to any theory that requires the Fourier analysis of continuous products of discontinuous periodic functions; we propose to call it the fast Fourier factorization method.
165 citations
TL;DR: In this paper, the authors present an English translation of the Beurling Theorem, which states that a function and its Fourier transform cannot simultaneously be very small at infinity.
Abstract: There are many theorems known which state that a function and its Fourier transform cannot simultaneously be very small at infinity, such as various forms of the uncertainty principle and the basic results on quasianalytic functions. One such theorem is stated on page 372 in volume II of the collected works of Arne Beurling [1]. Although it is not in every respect the most precise result of its kind, it has a simplicity and generality which make it very attractive. The editors state that no proof has been preserved. However, in my files I have notes which I made when Arne Beurling explained this result to me during a private conversation some time during the years 1964---1968 when we were colleagues at the Institute for Advanced Study, I shall reproduce these notes here in English translation with onIy minor details added where my notes are too sketchy. Theorem. Let fELl (R) and assume that
163 citations
TL;DR: In this paper, a new formulation of eigenproblem for phononic crystals is developed and the convergence of the new formulation in the band-structure calculations is examined in detail and compared with that of the conventional plane wave expansion (CPWE) method.
162 citations
TL;DR: In this article, the spectral Fourier series is used to estimate the local fluid velocity at the instantaneous particle position, and various approximate methods are tested and comparisons made of both their accuracy and the computational effort required.
162 citations
TL;DR: A new class of search-free direction-of-arrival (DOA) estimation methods applicable to arrays of arbitrary geometry and shown to offer substantially improved performance-to-complexity tradeoffs as compared to the MS technique.
Abstract: In this paper, the problem of spectral search-free direction-of-arrival (DOA) estimation in arbitrary nonuniform sensor arrays is addressed. In the first part of the paper, we present a finite-sample performance analysis of the well-known manifold separation (MS) based root-MUSIC technique. Then, we propose a new class of search-free DOA estimation methods applicable to arrays of arbitrary geometry and establish their relationship to the MS approach. Our first technique is referred to as Fourier-domain (FD) root-MUSIC and is based on the fact that the spectral MUSIC function is periodic in angle. It uses the Fourier series to expand this function and reformulate the underlying DOA estimation problem as an equivalent polynomial rooting problem. Our second approach applies the zero-padded inverse Fourier transform to the FD root-MUSIC polynomial to avoid the polynomial rooting step and replace it with a simple line search. Our third technique refines the FD root-MUSIC approach by using weighted least-squares approximation to compute the polynomial coefficients. The proposed techniques are shown to offer substantially improved performance-to-complexity tradeoffs as compared to the MS technique.
162 citations