Compactly Supported (bi)orthogonal Wavelets

This paper provides several constructions of compactly supported wavelets generated by interpolatory refinable functions. It was shown in [D1] that there is no real compactly supported orthonormal symmetric dyadic refinable function, except the trivial case; and also shown in [L] and [GM] that there is no compactly supported interpolatory orthonormal dyadic refinable function. Hence, for the dyadic dilation case, compactly supported wavelets generated by interpolatory refinable functions have to be biorthogonal wavelets. The key step to construct the biorthogonal wavelets is to construct a compactly supported dual function for a given interpolatory refinable function. We provide two explicit iterative constructions of such dual functions with desired regularity. When the dilation factors are larger than 3, we provide several examples of compactly supported interpolatory orthonormal symmetric refinable functions from a general method. This leads to several examples of orthogonal symmetric (anti-symmetric) wavelets generated by interpolatory refinable functions