On a connection between nonstationary and periodic wavelets

TL;DR

This paper explores the relationship between nonstationary and periodic wavelets, demonstrating how one can be constructed from the other and analyzing their localization properties in the time-frequency domain.

Contribution

It introduces a method to connect nonstationary and periodic wavelet systems via periodization, revealing the equivalence of their localization limits under certain conditions.

Findings

Constructed a nonstationary wavelet system from a periodic wavelet system.

Established the equality of localization limits for nonstationary and periodic wavelets.

Showed the existence of infinitely many nonstationary systems corresponding to a single periodic wavelet.

Abstract

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.