Best multi-valued approximants via multi-designs

Let d=(dj)j∈Im∈Nm be a decreasing finite sequence of positive integers, and let α=(αi)i∈In be a finite and non-increasing sequence of positive weights. Given a family Φ0=(Fj0)j∈Im of Bessel sequences with Fj0={fi,j0}i∈Ik∈(Cdj)k for each 1≤j≤m, our main purpose in this work, is to characterize the best approximants of the m-tuple of frame operators of the elements of Φ0 in the set of the so-called (α,d)-designs. Specifically, in this work we completely characterize the minimizers of the Joint Frame Operator Distance (JFOD) function. Indeed, we show that local minimizers of the JFOD are also global and we obtain an algorithm to construct the optimal (α,d)-designs. 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.