TIGHT FRAMES OF COMPACTLY SUPPORTED AFFINE WAVELETS

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 (χ) = 2 j/2ψ(2jχ - k) \j,k∈Z} is a tight frame of L 2(R), i.e., for every f∈L 2(R), f= cΣjk 〈ψjk |f〉ψjk for some c>0. This family is parametrized by an algebraic subset VN of R 4N. 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. © 1990 American Institute of Physics.