Tight frames of compactly supported affine wavelets

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Abstract
This paper extends the class of orthonormal bases of compactly supported wavelets recently constructed by Daubechies [Commun. Pure Appl. Math. 4 1, 909 (1988)]. For each integer N≥1, a family of wavelet functions ψ having support [0,2N−1] is constructed such that {ψ j k (x)=2 j/2ψ(2 j x−k) k j,k∈Z} is a tight frame of L 2(R), i.e., for every f∈L 2(R), f=c∑ j k 〈ψ j k ‖f〉ψ j k for some c>0. This family is parametrized by an algebraic subset V N of R 4N . Furthermore, for N≥2, a proper algebraic subset W N of V N is specified such that all points in V N outside of W N yield orthonormal bases. The relationship between these tight frames and the theory of group representations and coherent states is discussed.

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