Translation invariance and sampling theorem of wavelet

The sampling theorem for wavelet spaces built by Walter (1992) lacks the translation invariance except for Walterʹs weak translation invariant wavelet, i.e., Meyerʹs 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