The Gibbs phenomenon bounds in wavelet approximations

Typical Gibbs phenomenon manifests itself by appearance of overshoots and undershoots around the jump discontinuities. It is well known that the first maximum and minimum of approximation of the signum function by the truncated Fourier integral is 1.17898 and 0.9028 respectively, which divided to the jump discontinuity correspond to 8.95% the Gibbs overshoot and to 4.86% undershoot. We proved that the Gibbs overshoot of any integral wavelet transform is less than 8.95%; if integral wavelet transform is defined by tight wavelet, then the Gibbs overshoot does not exceed 7.07%. When the integral wavelet transform is defined by the Shannon wavelet, the Gibbs overshoot and undershoot equals to 7.07% and 1.73%. We have found compactly supported orthogonal wavelet having filter length 10, that defines dyadic wavelet expansion having bigger the Gibbs overshoot then the one of the Fourier integral.