On the splitting trick and wavelet frame packets. (English) Zbl 0966.42024

Chen, Di-Rong

SIAM J. Math. Anal. 31, No. 4, 726-739 (2000).

Given a function \(\psi\in L^2(R)\), define the set of functions \(\psi_{j,k}(x)=2^{j/2}\psi(2^jx-k), \;j,k\in Z\). If \(\{\psi_{j,k}\}_{j,k\in Z}\) is a (wavelet) frame for \(L^2(R)\), every function \(f\in L^2(R)\) has a representation \(f= \sum_{j,k}c_{j,k}\psi_{j,k}\); in contrast to the representation via a basis, the coefficients might not be unique. Based on a function \(\psi\) generating a wavelet frame, the splitting trick (originally due to Daubechies) allows a whole library of frames to be constructed. Technically, the splitting trick replaces shifts of one function by shifts of two functions, with the double shift parameter. This interesting paper presents the splitting trick in general shift invariant spaces, and for multiple generators. The known algorithms for finding optimal representations in a wavelet packet also applies in this more general setting.

Reviewer: Ole Christensen (Lyngby)

MSC:

42C40

Nontrigonometric harmonic analysis involving wavelets and other special systems

Keywords:

frames; wavelets; wavelet frame; Riesz basis; shift-invariant space