Let $\mathbf d=(d_j)_{ j\in\mathbb I_m}\in\mathbb N^m$ be a finite sequence (of dimensions) and $\alpha=(\alpha_i)_{ i\in\mathbb I_n}$ be a sequence of positive numbers (of weights), where $\mathbb I_k=\{ 1,\ldots,k\}$ for $k\in\mathbb N$. We introduce the $(\alpha\, , \,\mathbf d)$-designs i.e., $m$-tuples $\Phi=(\mathcal F_j)_{ j\in\mathbb I_m}$ such that $\mathcal F_j=\{ f_{ ij}\}_{ i\in\mathbb I_n}$ is a finite sequence in $\mathbb C^{ d_j}$, $j\in\mathbb I_m$, and such that the sequence of non-negative numbers $(\|f_{ ij}\|^2)_{ j\in\mathbb I_m}$ forms a partition of $\alpha_i$, $i\in\mathbb I_n$. We characterize the existence of $(\alpha\, , \, \mathbf d)$-designs with prescribed properties in terms of majorization relations. We show, by means of a finite-step algorithm, that there exist $(\alpha\, , \, \mathbf d)$-designs $\Phi^{ \rm op}=(\mathcal F_j^{ \rm op})_{ j\in\mathbb I_m}$ that are universally optimal; that is, for every convex function $\varphi:[0,\infty)\rightarrow [0,\infty)$ then $\Phi^{ \rm op}$ minimizes the joint convex potential induced by $\varphi$ among $(\alpha\, , \, \mathbf d)$-designs, namely $$ \sum_{ j\in\mathbb I_m}\text{ P}_\varphi(\mathcal F_j^{ \rm op})\leq \sum_{ j\in \mathbb I_m}\text{ P}_\varphi(\mathcal F_j) $$ for every $(\alpha\, , \, \mathbf d)$-design $\Phi=(\mathcal F_j)_{ j\in\mathbb I_m}$, where $\text{ P}_\varphi(\mathcal F)=tr(\varphi(S_{ \mathcal F}))$; in particular, $\Phi^{ \rm op}$ minimizes both the joint frame potential and the joint mean square error among $(\alpha\, , \, \mathbf d)$-designs. We show that in this case $\mathcal F_j^{ \rm op}$ is a frame for $\mathbb C^{ d_j}$, for $j\in\mathbb I_m$. This corresponds to the existence of optimal encoding-decoding schemes for multitasking devices with energy restrictions.
Categories: math.FA