Abstract
Even though the Shannon wavelet is a prototype of wavelets, it lacks condition on decay which most wavelets are assumed to have. By providing a sufficient condition to compute the size of Gibbs phenomenon for the Shannon wavelet series, we can see the overshoot is propotional to the jump at discontinuity. By comparing it with that of the Fourier series, we also see that these two have exactly the same Gibbs constant.
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References
J. Foster and F.B. Richard,A Gibbs phenomenon for piecewise linear approximation, The Am. Math. Month (1991), 47–49.
J.W. Gibbs,Letter to the editor, Nature59 (1899), 606.
S. Kelly,Gibbs’ phenomenon for wavelets, App. Comp. Harmon. Anal3 (1996), 72.
A.A. Michelson,Letter to the editor, Nature58 (1898), 544–545.
F.B. Richard,A Gibbs phenomenon for spline functions, J. Approx. Theory66 (1996), 334–351.
H.-T. Shim and H. Volkmer,On the Gibbs phenomenon for wavelet expansions, J. Approx. Theory84 (1995), 75–95.
G. G. Walter,Wavelets and Other Orthogonal Systems with Applications, CRC Press, Boca Raton, FL, 1994.
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Shim, HT. On Gibbs constant for the Shannon wavelet expansion. Korean J. Comp. & Appl. Math. 4, 469–473 (1997). https://doi.org/10.1007/BF03014493
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DOI: https://doi.org/10.1007/BF03014493