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

TL;DR

This paper constructs a family of non-stationary frames for L^2(R) that interpolate between Gabor frames and DOST-based frames, providing new tools for time-frequency analysis with a parameterized approach.

Contribution

It introduces a parameter-dependent family of frames for L^2(R), connecting Gabor frames and DOST bases, and extends the construction to higher dimensions for alpha=1.

Findings

Existence of a family of frames depending on alpha in [0,1]

Special cases recover Gabor frames and DOST basis

Extension to n-dimensional L^2(R^d) for alpha=1

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 $L^2(\mathbb{R}^d)$.