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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: In this paper, the degenerate scale problem in boundary element method for the two-dimensional Laplace equation is analytically studied in the continuous system by using degenerate kernels and Fourier series instead of using discrete system using circulants.
Abstract: The boundary integral equation approach has been shown to suffer a nonunique solution when the geometry is equal to a degenerate scale for a potential problem. In this paper, the degenerate scale problem in boundary element method for the two-dimensional Laplace equation is analytically studied in the continuous system by using degenerate kernels and Fourier series instead of using discrete system using circulants [Engng Anal. Bound. Elem. 25 (2001) 819]. For circular and multiply-connected domain problems, the rank-deficiency problem of the degenerate scale is solved by using the combined Helmholtz exterior integral equation formulation (CHEEF) concept. An additional constraint by collocating a point outside the domain is added to promote the rank of influence matrix. Two examples are shown to demonstrate the numerical instability using the singular integral equation for circular and annular domain problems. The CHEEF concept is successfully applied to overcome the degenerate scale and the error is suppressed in the numerical experiment.
58 citations
TL;DR: In this article, a general method for calculating the solution of the scalar wave equation for the field propagating through integrated optical devices is presented, which is capable of a three-dimensional description and of treating problems with reflected waves.
Abstract: A general method for calculating the solution of the scalar wave equation for the field propagating through integrated optical devices is presented. The method is capable of a three-dimensional description and of treating problems with reflected waves. It consists of dividing the device into a series of sections of axially uniform waveguides. The modes in each section are found by expansion of the field in a two-dimensional Fourier series and solving the associated matrix eigenvalue problem. Propagation is then described by relating the mode amplitudes of each section to the previous one. The amplitudes are related by a matrix that is the product of the eigenvector matrices of the two sections. The method is illustrated by the analysis of an adiabatic mode transformer, the coupling of light from a semiconductor laser through free space to a waveguide, and the propagation through an adiabatic 3 dB coupler and Y branch. >
58 citations
TL;DR: For a Schwartz function f on the plane and a non-zero v ∈ R 2 define the Hilbert transform of f in the direction v to be H v f(x) = p.v.
Abstract: For a Schwartz function f on the plane and a non-zero v ∈ R 2 define the Hilbert transform of f in the direction v to be H v f(x) = p.v. ∫ R f(x-vy)dy y. Let ζ be a Schwartz function with frequency support in the annulus 1 2. The L 2 estimate is sharp. The method of proof is based upon techniques related to the pointwise convergence of Fourier series. Indeed, our main theorem implies this result on Fourier series.
58 citations
TL;DR: In this article, the convergence and summability of non-harmonic Fourier series in the -norm on every segment is discussed. But the convergence is not shown to hold in the case of nonlinear systems of exponentials.
Abstract: CONTENTSIntroductionChapter I. Non-harmonic Fourier series (behaviour on the initial interval) ??1.1. Minimal systems of exponentials ??1.2. Expansions of functions in , ??1.3. Expansions of functions in Comments and supplements Chapter II. Non-harmonic Fourier series (the behaviour on the real axis) ??2.1. Extension of convergence of quasi-polynomials ??2.2. Continuation of functions from the initial interval ??2.3. Convergence and summability of non-harmonic Fourier series in the -norm () on every segmentComments and supplements Chapter III. Properties of the system ??3.1. Basis properties ??3.2. Angles between subspaces of exponentials Comments and supplements References
58 citations
TL;DR: In this article, the free vibration of laminated functionally graded (FG) spherical shells with general boundary conditions and arbitrary geometric parameters is studied based on the three-dimensional shell theory of elasticity and the energy based Rayleigh-Ritz procedure.
58 citations