Cardinal multiwavelets and the sampling theorem

This paper considers the classical Shannon sampling theorem in multiresolution spaces with scaling functions as interpolants. As discussed by Xia and Zhang (1993), for an orthogonal scaling function to support such a sampling theorem, the scaling function must be cardinal. They also showed that the only orthogonal scaling function that is both cardinal and of compact support is the Haar function, which has only 1 vanishing moment and is not continuous. This paper addresses the same question, but in the multiwavelet context, where the situation is different. This paper presents the construction of orthogonal multiscaling functions that are simultaneously cardinal, of compact support, and have more than one vanishing moment. The scaling functions thereby support a Shannon-like sampling theorem. Such wavelet bases are appealing because the initialization of the discrete wavelet transform (prefiltering) is the identity operator-the projection of a function onto the scaling space is given by its samples.