Explicit construction of non-stationary frames for $L^2$

Abstract

We show the existence of a family of frames of $L^2(\mathbb{R})$ which depend on a parameter $\alpha \in [0,1]$. If $\alpha =0$, we recover the usual Gabor frame, if $\alpha =1$ we obtain a frame system which is closely related to the so called DOST basis, first introduced by Stockwell and then analyzed by Battisti and Riba. If $\alpha \in (0,1)$, the frame system is associated to a so called $\alpha$-partitioning of the frequency domain. Restricting to the case $\alpha =1$, we provide a truly $n$-dimensional version of the DOST basis and an associated frame of $L^2({\mathbb{R}}^d)$.

28 pages, 7 figures