A New Class of Dual Fusion Frames and Implementation

Fusion frames as an important generalization of discrete frames that are completely different in duality. One reason for the technicalities of fusion frame duals is the mismatch of the coefficient spaces of different systems. In order to compensate this, a mapping between the coefficient Hilbert spaces is needed. Hence, it seems natural to modify the fusion frame operator by this map to be compatible with the duality in fusion frames (specially, Gv{a}vruc{t}a duals). For fusion frames $W$ and $V$, we define the alternate cross-frame operator by $L_{W,V}^M=T_WMT_V^*$, where $M:sumoplus V_i o sum oplus W_i$. If $W=V$ and $M={pi_{W_i}S_W^{-1}}_{iin I}$, we call it the alternate frame operator. We will establish a pioneer dual class of duals, called essential duals, employing the alternate frame operator. Finally in numerical experiments we provide some examples which compare the reconstruction formula via Gv{a}vruc{t}a and essential duals.