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Wavelet frames: Multiresolution analysis and extension principles. (English) Zbl 1036.42032
Debnath, Lokenath, Wavelet transforms and time-frequency signal analysis. Boston, MA: Birkhäuser (ISBN 0-8176-4104-1/hbk). Applied and Numerical Harmonic Analysis, 3-36 (2001).
Frame multiresolution analysis was introduced by J. J. Benedetto and S. Li [Appl. Comput. Harmon. Anal. 5, 389–427 (1998; Zbl 0915.42029)]. For \(f\in L^2(\mathbb R)\), let \((T_af)(x)=f(x-a)\) and \((D_af) (x)=a^{-1/2}f(\frac{x}{a}), a\neq 0\). An FMRA for \(L^2(\mathbb R)\) consists of a sequence of closed subspaces \(\{V_j \}_{j\in \mathbb Z} \subseteq L^2(\mathbb R)\) and a function \(\phi \in V_0\) such that (i) \( \cdots V_{-1} \subseteq V_0 \subseteq V_1 \cdots\), (ii) \(\overline{\cup_j V_j}= L^2 (\mathbb R)\) and \(\overline{\cap_j V_j}=0\), (iii) \(f\in V_j \Leftrightarrow [t\to f(2t)]\in V_{j+1}\), (iv) \(f\in V_0 \Rightarrow T_k f \in V_0, \;\forall k\in \mathbb Z\), and (v) \(\{T_k\phi\}_{k\in \mathbb Z}\) is a frame for \(V_0\).
The purpose of an FMRA is to construct wavelet frames for \(L^2(\mathbb R)\), i.e., frames of the type \(\{D_{2^j}T_k \psi\}_{j,k\in \mathbb Z}\); the obvious idea is to proceed as in the construction of an orthonormal basis via an MRA. Let \(W_j\) denote the orthogonal complement of \(V_j\) in \(V_{j+1}\). The main question is to find a wavelet \(\psi\) such that \(\{T_k \psi\}_{k\in \mathbb Z}\) is a frame for \(W_0\); this implies that \(\{D_{2^j}T_k \psi\}_{j,k\in \mathbb Z}\) is a frame for \(L^2(\mathbb R)\). Here the authors prove that the existence of such a function \(\psi\) depends solely on the “size” of the set \(\Gamma:= \{x\in [0,1]: \Phi(2x)=0, \Phi(x)>0, \Phi(x+\frac{1}{2})>0\},\) where \(\Phi(x)=\sum_{k\in \mathbb Z} | \hat{\phi}(x+k)| ^2\). In fact, there exists a function \(\psi \in L^2(\mathbb R)\) such that \(\{T_k \psi\}_{k\in \mathbb Z}\) is a frame for \(W_0\) if and only if \(\Gamma\) has vanishing Lebesgue measure. In case \(\Gamma\) has vanishing Lebesgue measure, the authors also show how to define a suitable function \(\psi\).
It is explained how overcompleteness of frames implies robustness against noise that appear for example via transmission. Furthermore, a new direct proof of Ron and Shen’s Unitary Extension Principle is given.
For the entire collection see [Zbl 0996.00017].

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
65T60 Numerical methods for wavelets