On the Gibbs Phenomenon for Wavelet Expansions

Abstract

It is shown that a Gibbs phenomenon occurs in the wavelet expansion of a function with a jump discontinuity at 0 for a wide class of wavelets. Additional results are provided on the asymptotic behavior of the Gibbs splines and on methods to remove the Gibbs phenomenon.

Gibbs' phenomenon for sampling series and what to do about it

“…Nonetheless, both cases can lead to Gibbs' phenomenon for functions with jump discontinuities at 0 and the overshoot calculated. Indeed in [14] it was shown that the overshoot is exactly the same as for Fourier series in the case of orthogonal approximations. We can also calculate it for sampling series.…”

“…It was shown by Kelly [9] to occur under certain conditions for orthogonal wavelet approximations. Shim and Volkmer [14] then showed that these conditions for Gibbs' phenomenon to exist are satisfied for all reasonable wavelets.…”

“…The Shannon system, although it serves as a prototype, does not satisfy the hypotheses of the theorems about Gibbs' phenomenon in [13] and [14]. The formulae, however, are rather simple and may be used to show directly that Gibbs occurs for both sampling series and orthogonal series.…”

Approximation-theoretic analysis of translation invariant wavelet expansions

“…The study of Gibbs phenomenon for wavelet approximations is more recent [13,14]. Suppose the function to be approximated has a discontinuity at the origin.…”

“…scaled by some constant, and the Gibbs artifacts are solely due to for some ² ¶ [13,14]. This makes sense:…”