Optimal frame designs for multitasking devices with weight restrictions

Abstract

Let d = (d j ) j∈Im ∈ N m be a finite sequence (of dimensions) and α = (α i ) i∈In be a sequence of positive numbers (weights), where I k = {1, . . . , k} for k ∈ N. We introduce the (α , d)-designs i.e., families Φ = (F j ) j∈Im such that F j = {f ij } i∈In is a frame for C dj , j ∈ I m , and such that the sequence of non-negative numbers ( f ij 2 ) j∈Im forms a partition of α i , i ∈ I n . We show, by means of a finite-step algorithm, that there exist (α , d)-designs Φ op = (F op j ) that are universally optimal; that is, for every convex function ϕ : [0, ∞) → [0, ∞) then Φ op minimizes the joint convex potential induced by ϕ among (α , d)-designs, namely j∈Im

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