Representations of Almost-Periodic Functions Using Generalized Shift-Invariant Systems in \mathbb{R}^{d}

The problem of estimating the AP-norm of univariate almost-periodic functions using a Gabor system or a wavelet system was studied by several authors, and culminated in the characterization given in a recent paper of the present author. The present article unifies and generalizes the various efforts via the study of this problem in the context of (multivariate) generalized shift-invariant (GSI) systems. The main result shows that the sought-for norm estimation of the AP functions is valid if and only if the given GSI system is an $L_{2}({\mathbb{R}}^{d})$ -frame. Moreover, the frame bounds of the system are also the sharpest bounds in our estimation.