The mathematical theory of wavelets (Q2772780)

A classical wavelet in \(L^2(R^n)\) is a function \(\psi \in L^2(R^n)\) such that the collection \(\{ \psi_{j,k}(x) = 2^{jn/2} \psi(2^j x - k) : j\in Z, k \in Z^n\}\) forms an orthonormal basis for \(L^2(R^n)\). It is closely related to the continuous wavelet transform \(W_\psi (f) (a,b) = \langle f, \psi_{a,b} \rangle\), where \(\psi_{a,b}(x) = |\det (a)|^{1/2} \psi(a^{-1}x -b)\), \((a,b) \in\text{Gl}(n,R) \times R^n\). The paper in review is a very interesting overview of different aspects of the ``mathematical theory of wavelets''. It is both a survey of existing results, accessible for people with only elementary knowledge of Fourier analysis, and a rigorous presentation of the newest results in this area. The paper starts with the studies of the Calderón admissibility condition for continuous wavelet transforms (Theorem (2.1)) and investigates different groups of admissible dilations and translations. It turns out that such characterizing conditions for continuous wavelets are closely related to characterizations of discrete wavelets. Several types of characterizations of wavelets are presented in this paper, both known works and new results (e.g., Theorem (3.2)). The latter are introduced through the study of properties of shift invariant spaces (cf., \textit{Z. Rzeszotnik} [``Calderón's condition and wavelets'', Collect. Math. 52, No. 2, 181-191 (2001; Zbl 0989.42017)]). Multiresolution analyses are introduced next and the paper ends with a survey of results related to the connectivity of wavelets (see also \textit{Wutam consortium}, [``Basic properties of wavelets'', J. Fourier Anal. Appl. 4, No. 4-5, 575-594 (1998; Zbl 0934.42024)]).NEWLINENEWLINEFor the entire collection see [Zbl 0972.00019].