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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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[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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