Abstract:
We study a connection between nonstationary nonperiodic wavelets and periodic wavelets. The case of Parseval wavelet frames generated by the unitary extension principle is discussed. It is proved that the periodization of a nonstationary Parseval wavelet frame is a periodic Parseval wavelet frame. And conversely, a periodic Parseval wavelet frame generates a nonstationary wavelet system. There are infinitely many nonstationary systems corresponding to the same periodic wavelet. Under natural conditions on periodic scaling functions, among these nonstationary wavelet systems, there exists 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_j) = \lim_{j\to \infty}UC_H(\psi^0_j), $ where $UC_B$ and $UC_H$ are the Breitenberger and the Heisenberg uncertainty constants, $\psi_j \in L_2(\mathbb{T})$ and $\psi^0_j\in L_2(\mathbb{R})$ are periodic and nonstationary wavelet functions respectively.