Uncertainty principles, minimum uncertainty samplings and translations

It has been shown recently, that the conventional variance based uncertainty measure associated with the wavelet transform can be arbitrarily small. Hence, no global minimizer exists. In this paper we introduce a new discretization scheme in scale and time shifts, such that the total uncertainty of a corresponding function system has the lowest possible value. We also describe a generalized uncertainty principle inspired by the familiar uncertainty principle in time-frequency analysis. As an example we apply this concept to wavelet analysis, leading to a new affine uncertainty principle. We also introduce waveforms minimizing this principle. Furthermore, we remark that the uncertainty measure associated with this new principle allows for decay estimates of the ambiguity function (reproducing kernel) associated with the wavelet transform.