Abstract
Construction of interpolatory wavelets is an important topic in discrete signal processing. In classical wavelet sampling theories, interpolatory wavelets are constructed from Riesz bases in wavelet spaces. Since analytical expressions for such Riesz bases are generally complex or unavailable, it has been difficult to obtain suitable interpolatory wavelets in practice. In this paper, interpolatory scaling functions are used to determine and construct interpolatory wavelets. We first show that there may not exist interpolatory wavelets even when interpolatory scaling functions exist. Then, an inequality in terms of interpolatory scaling functions, denoted as the two-scale condition, is given as a necessary and sufficient condition for existence of interpolatory wavelets. Finally, based on the two-scale condition, a filter bank is constructed for obtaining interpolatory wavelets directly from interpolatory scaling functions. In examples, our theorems are applied to some typical wavelet spaces, demonstrating our construction algorithm for interpolatory wavelets.
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