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- W194148862 abstract "numerical method used to solve nonlinear partial differential equations (PDEs), is an example of such a method [12]. The method approximates the Fourier coefficients of the solution of a PDE. The Fourier coefficients are then used to calculate an approx imation to the solution. The accurate reconstruction of the solution requires that the positions of the discontinuities of the solution be known [5]. In this paper we discuss techniques for using a function's Fourier coefficients to determine the locations and sizes of the jump discontinuities of the function. At first glance the spectral representation of the signal?the Fourier series or trans form associated with the signal?does not seem to be the ideal place to look for information about discontinuities in the signal. When a signal is discontinuous the con vergence of the Fourier series or transform associated with the signal is not uniform; in such cases the Gibbs phenomenon [11] appears and truncating the series after any finite number of terms always leads to 0(1) oscillations in the reconstructed signal. (For a nice, detailed treatment of the Gibbs see [6].) Considering the question again, however, one realizes that if a discontinuity is characterized by a phenomenon, then the existence of the discontinuity is indeed encoded in the coefficients. The question becomes how to effectively decode the discontinuity. One does not do this by directly summing the series?one uses the spec tral representation in a somewhat different way to concentrate the function about the discontinuity. In what follows, we explain how this is done. We restrict ourselves to pe riodic (or compactly supported) functions and only consider Fourier series. (Those in terested in seeing a more general theory of concentration factors are referred to [3,4].) Much of the information in this article is well known [3, 4]. The use of the Euler" @default.
- W194148862 created "2016-06-24" @default.
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- W194148862 date "2008-06-01" @default.
- W194148862 modified "2023-10-02" @default.
- W194148862 title "Edge Detection Using Fourier Coefficients" @default.
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- W194148862 doi "https://doi.org/10.1080/00029890.2008.11920557" @default.
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