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Tight frames of compactly supported affine wavelets. (English) Zbl 0708.46020
Summary: This paper extends the class of orthonormal bases of compactly supported wavelets recently constructed by I. Daubechies [Commun. Pure Appl. Math. 41, 901–906 (1988; Zbl 0644.42026)]. For each integer $$N\geq 1$$, a family of wavelet functions $$\psi$$ having support [0,2N-1] is constructed such that $$\{\psi_{jk}(x)=2^{j/2} \psi (2^ jx-k)|$$ j,k$$\in {\mathbb Z}\}$$ is a tight frame of $$L^ 2({\mathbb R})$$, i.e., for every $$f\in L^ 2({\mathbb R})$$, $$f=c\sum_{jk}<\psi_{jk}| f>\psi_{jk}$$ for some $$c>0$$. This family is parametrized by an algebraic subset $$V_ N$$ of $${\mathbb R}^{4N}$$. Furthermore, for $$N\geq 2$$, 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.

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 42C15 General harmonic expansions, frames 81R30 Coherent states
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##### References:
 [1] DOI: 10.1137/0515056 · Zbl 0578.42007 · doi:10.1137/0515056 [2] DOI: 10.1063/1.526761 · Zbl 0571.22021 · doi:10.1063/1.526761 [3] DOI: 10.1063/1.527388 · Zbl 0608.46014 · doi:10.1063/1.527388 [4] DOI: 10.1007/BF01456326 · JFM 41.0469.03 · doi:10.1007/BF01456326 [5] DOI: 10.1002/cpa.3160410705 · Zbl 0644.42026 · doi:10.1002/cpa.3160410705
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