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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.

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42C15 General harmonic expansions, frames
81R30 Coherent states
Full Text: DOI
[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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