On a connection between nonstationary and periodic wavelets

We compare frameworks of nonstationary nonperiodic wavelets and periodic wavelets. We construct one system from another using periodization. There are infinitely many nonstationary systems corresponding to the same periodic wavelet. Under mild conditions on periodic scaling functions, among these nonstationary wavelet systems, we find a system such that its time-frequency localization is adjusted with an angular-frequency localization of an initial periodic wavelet system. Namely, we get the following equality $ \lim_{j\to \infty} UC_B(\psi^P_j) = \lim_{j\to \infty}UC_H(\psi^N_j), $ where $UC_B$ and $UC_H$ are the Breitenberger and the Heisenberg uncertainty constants, $\psi^P_j \in L_2(\mathbb{T})$ and $\psi^N_j\in L_2(\mathbb{R})$ are periodic and nonstationary wavelet functions respectively.