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 (α, 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 α_i, \(i\in \mathbb {I}_{n}\). We characterize the existence of (α, d)-designs with prescribed properties in terms of majorization relations. We show, by means of a finite step algorithm, that there exist (α, d)-designs \({\Phi }^{\text {op}}=(\mathcal {F}_{j}^{\text {op}})_{j\in \mathbb {I}_{m}}\) that are universally optimal; that is, for every convex function \(\varphi :[0,\infty )\rightarrow [0,\infty )\), then Φ^op minimizes the joint convex potential induced by φ among (α, d)-designs, namely$ \sum \limits_{j\in \mathbb I_{m}}\text {P}_{\varphi }(\mathcal F_{j}^{\text {op}})\leq \sum \limits_{j\in \mathbb I_{m}}\text {P}_{\varphi }(\mathcal F_{j}) $for every (α, d)-design \({\Phi }=(\mathcal F_{j})_{j\in \mathbb {I}_{m}}\), where \(\text {P}_{\varphi }(\mathcal F)=\text {tr}(\varphi (S_{\mathcal {F}}))\); in particular, Φ^op minimizes both the joint frame potential and the joint mean square error among (α, d)-designs. We show that in this case, \(\mathcal {F}_{j}^{\text {op}}\) 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.

中文翻译：

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具有重量限制的多任务设备的最佳框架设计

令\（\ mathbf d =（d_ {j}）_ {j \ in \ mathbb {I} _ {m}} \\\ mathbb {N} ^ {m} \中​​）是一个有限序列（维数），并且\（\ alpha =（\ alpha _ {i}）_ {i \ in \ mathbb {I} _ {n}} \）是一个正数序列（权重），其中\（\ mathbb {I} _ {k} = \ {1，\ ldots，k \} \表示\（k \ in \ mathbb {N} \）。我们引入（α，d）设计，即m-元组\（{\ Phi} =（\ mathcal F_ {j}）_ {j \ in \ mathbb {I} _ {m}} \\）这样\（\ mathcal F_ {j} = \ {f_ {ij} \} _ {i \ in \ mathbb {I} _ {n}} \\）是\（\ mathbb {C} ^ {d_ { j}} \），\（j \ in \ mathbb {I} _ {m} \），并且使得非负数的序列\（（\ | F_ {IJ} \ | ^ {2}）_ {Ĵ\在\ mathbb {I} _ {米}} \）形成的分区α_我，\（i \ in \ mathbb {I} _ {n} \）中。我们根据主要关系描述了具有规定性质的（α，d）设计的存在。我们通过有限步算法证明存在（α，d）设计\（{\ Phi} ^ {\ text {op}} =（\ mathcal {F} _ {j} ^ {\ text {op}}）_ {j \ in \ mathbb {I} _ {m}} \\）中的）；即，对于每一个凸函数\（\ varphi：[0，\ infty）\ RIGHTARROW [0，\ infty）\） ，则Φ^运使（α，d）设计中由φ引起的关节凸电位最小化，即$ \ sum \ limits_ {j \ in \ mathbb I_ {m}} \ text {P} _ {\ varphi}（\ mathcal F_ {j } ^ {\文本{OP}}）\当量\总和\ limits_ {Ĵ\在\ mathbb I_ {M}} \文本{P} _ {\ varphi}（\ mathcal F_ {Ĵ}）$每隔（α，d）-设计\（{\ Phi} =（\ mathcal F_ {j}）_ {j \ in \ mathbb {I} _ {m}} \\），其中\（\ text {P} _ {\ varphi }（\ mathcal F）= \ text {tr}（\ varphi（S _ {\ mathcal {F}}））\） ; 特别是，Φ^运算最小化两者的联合帧电位和之间的关节均方误差（α，d）-designs。我们证明在这种情况下，\（\ mathcal {F} _ {j} ^ {\ text {op}} \）是\（\ mathbb {C} ^ {d_ {j}} \\）的框架，对于\（j \ in \ mathbb {I} _ {m} \）。这对应于具有能量限制的多任务设备的最佳编码-解码方案的存在。