Compactly Supported (bi)orthogonal Wavelets Generated by Interpolatory Reenable Functions

This paper provides several constructions of compactly supported wavelets generated by interpolatory reenable functions. It was shown in D1] that there is no real compactly supported orthonormal symmetric dyadic reenable function, except the trivial case; and also shown in L] and GM] that there is no compactly supported interpolatory orthonormal dyadic reenable function. Hence, for the dyadic dilation case, compactly supported wavelets generated by interpolatory reenable 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 reenable 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 reenable functions from a general method. This leads to several examples of orthogonal symmetric (anti-symmetric) wavelets generated by interpolatory reenable functions.