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.

ABSTRACT. Gibbs' phenomenon occurs for most orthogonal wavelet expansions. It is also shown to occur with many wavelet interpolating series, and a characterization is given. By introducing modifications in such a series, it can be avoided. However, some series that exhibit Gibbs' phenomenon for orthogonal series do not for the associated sampling series.

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Gibbs' phenomenon occurs for most orthogonal wavelet expansions in one dimension. It also exists in higher dimensions but fundamental concepts must be redefined. This is done for both separable and nonseparable wavelet expansions in severable variables.