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

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

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