Translation invariance and sampling theorem of wavelet
The sampling theorem for wavelet spaces built by Walter (1992) lacks the translation invariance except for Walter's weak translation invariant wavelet, i.e., Meyer's wavelet. Indeed, we must know a priori the shift offset a in the samples {f(n+a);n/spl isin/Z}; otherwise, the waveform cannot be recovered since the interpolation function is dependent on this offset. In this correspondence, we generalize our metric functional to metrize weak shiftability and find a somewhat surprising result that the B spline wavelets of order n/spl ges/3 are degenerate shiftable. Thus, we can recover approximately the waveform by double sampling without any information on shift offset a.