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∈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⩾3 are degenerate shiftable. Thus, we can recover approximately the waveform by double sampling without any information on shift offset a