Optimal (α,d)-multi-completion of d-designs

Abstract

Given finite sequences d=(dj)j∈Im∈Nm and α=(αi)i∈In∈R>0n of dimensions and weights (where Ik={1,?,k}, for k∈N), we consider the set D(α,d) of (α,d)-designs, i.e. m-tuples Φ=(Fj)j∈Im such that each Fj={fi,j}i∈In∈(Cdj)n and ∑j∈Im‖fi,j‖2=αi for i∈In. In this work we solve the optimal (α,d)-completion problem of an initial d-design Φ0=(Fj0)j∈Im with Fj0∈(Cdj)k, for j∈Im. Explicitly, given an strictly convex function φ:[0,∞)→[0,∞), we compute the (α,d)-designs Φφop that are (local) minimizers of the joint convex potential Pφ(Φ0,Φ)=∑j∈Imtr(φ[S(Fj0,Fj)]) of the multi-completions (Φ0,Φ), among all (α,d)-designs Φ=(Fj)j∈Im; here S(Fj0,Fj) denotes the frame operator of the completed sequence (Fj0,Fj)∈(Cdj)k+n, for j∈Im. We obtain the geometrical and spectral features of these optimal (α,d)-multi-completions. We further show that the optimal (α,d)-designs Φφop as above do not depend on φ. We also consider some reformulations and applications of our main results in different contexts in frame theory. Finally, we describe a fast finite step algorithm for computing optimal multi-completions that becomes relevant for the applications of our results and present some numerical examples of optimal multi-completions with prescribed weights.