Perfect-translation-invariant customizable complex discrete wavelet transform (Q2868198)

The authors consider wavelet frame constructions exhibiting certain translation invariance properties. In particular they construct translation invariant wavelet frame operators starting from a basic set of bandlimited wavelets such that the Fourier transforms of these wavelets fulfill a covering condition closely resembling the painless construction given in [\textit{I. Daubechies} et al., J. Math. Phys. 27, 1271--1283 (1986; Zbl 0608.46014)]. The frames are self-dual, if the above-mentioned set is finite, an interpolation property of the frame construction is shown, when starting from a discretely sampled signal. The proofs of the theorems heavily rely on periodization arguments and the Cauchy-Schwarz inequality.NEWLINENEWLINEThere exist many contributions to the problem of constructing wavelet frames, whose \textit{analysis operators} exhibit an approximate translation invariance (cf. [\textit{I. Selesnick} et al., ``The dual-tree complex wavelet transform'', IEEE Signal Processing Magazine 22, No. 6, 123--151 (2005)] or [\textit{J. Neumann} and \textit{G. Steidl}, Int. J. Wavelets Multiresolut. Inf. Process. 3, No. 1, 43--65 (2005; Zbl 1073.94004)]). These papers contribute to signal analysis and signal classification, the complexity of wavelets is essential there.NEWLINENEWLINEIn contrast to what the title of the paper at hand might suggest, it does not focus on this direction of research, due to the translation invariance of \textit{frame operators} mentioned above. Actually, translation invariance of the frame operators is a corollary of the fact that the frame operators act multiplicatively in the Fourier domain. The latter statement is essential for constructing the frames described in the paper. Moreover, also the complexity of the wavelets is more a kind of by-product and not an essential ingredient. It would have been desirable, to include explicit choices of wavelets going beyond well-known examples and to describe specific application settings.