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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].

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