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- W202156287 abstract "The Gibbs phenomenon refers to the lack of uniform convergence which occurs in many orthogonal basis approximations to piecewise smooth functions. This lack of uniform convergence manifests itself in spurious oscillations near the points of discontinuity and a low order of convergence away from the discontinuities. Here we describe a numerical procedure for overcoming the Gibbs phenomenon called the inverse wavelet reconstruction method. The method takes the Fourier coef cients of an oscillatory partial sum and uses them to construct the wavelet coef cients of a non-oscillatory wavelet series. Index TermsGibbs phenomenon, Inverse wavelet reconstruction, Inverse polynomial reconstruction. Fourier and orthogonal polynomial series are known for their highly accurate expansions for smooth functions. In fact it is known that the more derivatives a function has, the faster the approximation will converge. However, when a function possesses jump-discontinuities the approximation will fail to converge uniformly. In addition, spurious oscillations will cause a loss of accuracy throughout the entire domain. This lack of uniform convergence is known as the Gibbs phenomenon. Methods for post-processing approximations which suffer from the Gibbs phenomenon include the Gegenbauer reconstruction method of Gottlieb and Shu [7,9], the method of Pade approximants due to Driscoll and Fornberg [2], the method of spectral molli ers due to Gottlieb and Tadmor [8] and Tadmor and Tanner [18, 19], the inverse polynomial reconstruction method of Shizgal and Jung [13,14,15,16,17], and the Freund polynomial reconstruction method of Gelb and Tanner [6]. These reconstruction methods can be combined with an effective method for edge-detection developed by Gelb and Tadmor [3,4,5,6], to yield an exponentially accurate reconstruction of the original function. In this paper we describe a new numerical method for overcoming the Gibbs phenomenon following the work of Shizgal and Jung, called the Inverse Wavelet Reconstruction method. We begin with a brief review of the essential de nitions of wavelets which we will need. Recall that a wavelet is a function 2 L (R) satisfying: Z 1" @default.
- W202156287 created "2016-06-24" @default.
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- W202156287 date "2008-01-01" @default.
- W202156287 modified "2023-09-27" @default.
- W202156287 title "Inverse Wavelet Reconstruction for Resolving the Gibbs Phenomenon" @default.
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