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Frames and generalized shift-invariant systems. (English) Zbl 1104.42018

Boggiatto, Paolo (ed.) et al., Pseudo-differential operators and related topics. Papers based on lectures given at the international conference, Växjö University, Sweden, June 22 to June 25, 2005. Basel: Birkhäuser (ISBN 3-7643-7513-2/hbk). Operator Theory: Advances and Applications 164, 193-209 (2006).
A countable family \(\{e_j: j \in J\}\subset L^2({\mathbb R}^d)\) is called a frame for \(L^2({\mathbb R}^d)\) if there exist constants \(0<A\leq B<\infty\), such that, for all \(f \in L^2({\mathbb R}^d)\), \(A\| f\| _2^2 \leq \sum_{j\in J} | \langle f, e_j \rangle | ^2 \leq B\| f\| _2^2\). We say that a frame is tight if \(A=B\).
The paper under review is based on a lecture given at the Växjö University, Sweden, in 2005. It provides a motivation for the study of frames in \(L^2({\mathbb R}^d)\) and their duals. Basic facts from the frame theory are followed by a discussion of 3 types of frames: Gabor frames, wavelet frames, and generalized shift-invariant frames. The aim of this paper is to present advantages of using tight frames over general frames.
For the entire collection see [Zbl 1086.47005].

MSC:

42C15 General harmonic expansions, frames
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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