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This paper extends the class of orthonormal bases of compactly supported wavelets recently constructed by Daubechies Commun. Pure Appl. Math. 41, 909 (1988). For each integer N≥1, a family of wavelet functions ψ having support 0, 2N−1 is constructed such that ψjk (x) =2j/2ψ (2jx−k) kj, k∈Z is a tight frame of L2 (R), i. e. , for every f∈L2 (R), f=c∑jk 〈ψjk‖f〉ψjk for some c0. This family is parametrized by an algebraic subset VN of R4N. Furthermore, for N≥2, a proper algebraic subset WN of VN is specified such that all points in VN outside of WN yield orthonormal bases. The relationship between these tight frames and the theory of group representations and coherent states is discussed.
Wayne Lawton (Wed,) studied this question.