LAUSR

Let d = ( d j ) j ∈ I m ∈ ℕ m $\mathbf d=(d_{j})_{j\in \mathbb {I}_{m}}\in \mathbb {N}^{m}$ be a finite sequence (of dimensions) and α = ( α i ) i ∈ I n $\alpha =(\alpha _{i})_{i\in \mathbb {I}_{n}}$ be a sequence of positive numbers (of weights), where I k = { 1 , … , k } $\mathbb {I}_{k}=\{1,\ldots ,k\}$ for k ∈ ℕ $k\in \mathbb {N}$ . We introduce the ( α , d )-designs, i.e., m -tuples Φ = ( F j ) j ∈ I m ${\Phi }=(\mathcal F_{j})_{j\in \mathbb {I}_{m}}$ such that F j = { f i j } i ∈ I n $\mathcal F_{j}=\{f_{ij}\}_{i\in \mathbb {I}_{n}}$ is a finite sequence in ℂ d j $\mathbb {C}^{d_{j}}$ , j ∈ I m $j\in \mathbb {I}_{m}$ , and such that the sequence of non-negative numbers ( ∥ f i j ∥ 2 ) j ∈ I m $(\|f_{ij}\|^{2})_{j\in \mathbb {I}_{m}}$ forms a partition of α i , i ∈ I n $i\in \mathbb {I}_{n}$ . 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 Φ op = ( F j op ) j ∈ I m ${\Phi }^{\text {op}}=(\mathcal {F}_{j}^{\text {op}})_{j\in \mathbb {I}_{m}}$ that are universally optimal; that is, for every convex function φ : [ 0 , ∞ ) → [ 0 , ∞ ) $\varphi :[0,\infty )\rightarrow [0,\infty )$ , then Φ op minimizes the joint convex potential induced by φ among ( α , d )-designs, namely ∑ j ∈ I m P φ ( F j op ) ≤ ∑ j ∈ I m P φ ( F j ) $ \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 Φ = ( F j ) j ∈ I m ${\Phi }=(\mathcal F_{j})_{j\in \mathbb {I}_{m}}$ , where P φ ( F ) = tr ( φ ( S F ) ) $\text {P}_{\varphi }(\mathcal F)=\text {tr}(\varphi (S_{\mathcal {F}}))$ ; 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, F j op $\mathcal {F}_{j}^{\text {op}}$ is a frame for ℂ d j $\mathbb {C}^{d_{j}}$ , for j ∈ I m $j\in \mathbb {I}_{m}$ . This corresponds to the existence of optimal encoding-decoding schemes for multitasking devices with energy restrictions.