Best multi-valued approximants via multi-designs
Let ${\mathbf d} =(d_j)_{j\in\mathbb{I}_m}\in \mathbb{N}^m$ be a decreasing finite sequence of positive integers, and let $α=(α_i)_{i\in\mathbb{I}_n}$ be a finite and non-increasing sequence of positive weights. Given a family $Φ^0=(\mathcal{F}_j^0)_{j\in\mathbb{I}_m}$ of Bessel sequences with $\mathcal{F}_j^0=\{f_{i,j}^0\}_{i\in \mathbb{I}_k}\in (\mathbb{C}^{d_j})^k$ for each $1\leq j\leq m$, our main purpose on this work is to characterize the best approximants of the $m$-tuple of frame operators of the elements of $Φ^0$ in the set $D(α,\mathbf d)$ of the so-called $(α,\mathbf d)$-designs, which are the $m$-tuples $Φ=(\mathcal{F}_j)_{j\in\mathbb{I}_m}$ such that each $\mathcal{F}_j=\{f_{i,j}\}_{i\in\mathbb{I}_n}$ is a finite sequence in $\mathbb{C}^{d_j}$, and $\sum_{j\in\mathbb{I}_m}\|f_{i,j}\|^2=α_i$ for $i\in\mathbb{I}_n$. Specifically, in this work we completely characterize the minimizers of the Joint Frame Operator Distance (JFOD) function: $Θ:D(α,\mathbf d)\to \mathbb{R}_{\geq 0} $ given by $$Θ(Φ)=\sum_{j=1}^m \| S_{\mathcal{F}_j} - S_{\mathcal{F}^0_j}\|_2^2 \,,$$ where $S_{\mathcal{F}}$ denotes the frame operator of $\mathcal{F}$ and $\|\cdot\|_2$ is the Frobenius norm. Indeed, we show that local minimizers of $Θ$ are also global and we obtain an algorithm to construct the optimal $(α,\mathbf d)$-desings. As an application of the main result, in the particular case that $m=1$, we also characterize global minimizers of a G-frames problem recently considered by He, Leng and Xu.