论文标题
通过多设计的最佳多价近似值
Best multi-valued approximants via multi-designs
论文作者
论文摘要
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正权重序列。给定一个家庭$φ^0 =（\ mathcal {f} _j^0）_ {j \ in \ Mathbb {i} _m} $，带有$ \ mathcal {f} _j^0 = \ {f_ {f_ {f_} i， \ Mathbb {i} _k} \ in（\ Mathbb {C}^{d_j}）所谓的$（α，\ mathbf d）$ - 设计，是$ m $ -tuples $φ=（\ mathcal {f} _j）_ {j \ in \ mathbb {i} _m} $ $ \ MATHCAL {f} _j = \ {f_ {f_ {i，j} \} _ {i \ in \ mathbb {i} _n} $是$ \ Mathbb {c}^{d_j} $中的有限顺序$ \ sum_ {j \ in \ mathbb {i} _m} \ | f_ {i，j} \ |^2 =α_i$ for $ i \ in \ in \ mathbb {i} _n $。具体而言，在这项工作中，我们完全表征了关节框架操作员距离（JFOD）函数的最小化器：$θ：d（α，\ mathbf d）\ to \ Mathbb {r} _ {\ geq 0} $给定$θ（φ） s _ {\ Mathcal {f} _J} - s _ {\ Mathcal {f}^0_j} \ | _2^2 \ ,, $ s _ {\ nathcal {f}} $ s $ s {\ nathcal {f}} $表示$ \ nathcal的框架$ \ natercal的框架|规范。实际上，我们表明$θ$的本地最小化也是全球的，我们获得了一种算法来构建最佳$（α，\ mathbf d）$ - dests。作为主要结果的应用，在特定情况下，我们还表征了He，Leng和Xu最近考虑的G框架问题的全球最小化器。
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.
