Summary: “Classical” wavelet are obtained by the action of particular countable subset of operators associated with the affine group on a function \(\psi\in L^2(\mathbb{R})\). More precisely, this set is the collection \(\{D_{2^j}T_k:j,k\in \mathbb{Z}\}\), where \(T_k\) is the translation by the integer \(k\) and \(D_{2^j}\) is the (unitary) dilation by \(2^j\). We thus obtain the discrete wavelet system. A. Ron and Z. Shen [J. Funct. Anal. 148, No. 2, 408–447 (1997; Zbl 0891.42018)] have shown that by interchanging and renormalizing “half” of the operators in this set one obtains an important collection of systems that can be considered “equivalent” to this affine system. In this paper we show that, in a precise sense, the choice of Ron and Shen is optimal.
For the entire collection see [Zbl 1077.42002].