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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 :jJ}L 2 ( d ) is called a frame for L 2 ( d ) if there exist constants 0<AB<, such that, for all fL 2 ( d ), Af 2 2 jJ |f,e j | 2 Bf 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 ( 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.

MSC:
42C15General harmonic expansions, frames
42C40Wavelets and other special systems