Localization of operator-valued frames
Lukas Köhldorfer, Peter Balazs
TL;DR
The article develops an intrinsic localization framework for operator-valued frames by demanding their g-Gram matrix $G_T$ belong to a spectral Banach algebra ${\mathcal A}$. It proves that localization is preserved under duality and introduces associated (quasi-)Banach spaces $\mathcal{H}^p_{\omega}(T^d,T)$ and $\mathcal{H}^{\infty}_{\omega}(T^d,T)$ in which reconstruction series converge unconditionally. The framework is applied to irregular Gabor g-frames, establishing polynomial-type off-diagonal decay in operator-valued Gram matrices and defining well-posed coorbit-type spaces $\mathcal{H}^p_m(\widetilde{\mathcal{G}},\mathcal{G})$ with stable reconstruction. Overall, the work connects localized operator-valued frame theory with coorbit spaces and time-frequency analysis, enabling robust reconstruction across a range of Banach spaces and irregular sampling patterns.
Abstract
We introduce a localization concept for operator-valued frames, where the quality of localization is measured by the associated operator-valued Gram matrix belonging to some suitable Banach algebra. We prove that intrinsic localization of an operator-valued frame is preserved by its canonical dual. Moreover, we show that the series associated to the perfect reconstruction of an operator-valued frame converges not only in the underlying Hilbert space, but also in a whole class of associated (quasi-)Banach spaces. Finally, we apply our results to irregular Gabor g-frames.