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

BENAC, MARÍA JOSÉ; MASSEY, PEDRO; RUIZ, MARIANO; STOJANOFF, DEMETRIO

APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Año: 2023 vol. 62 p. 331 - 364

1063-5203

Given finite sequences $dd=(d_j)_{jinI_m}in N^m$ and $alpha=(alpha_i)_{iinI_n}in R_{>0}^n$ of dimensions and weights (where $I_k={1,ldots,k}$, for $kinN$), we consider the set $cD(alphacoma dd)$ of $(alphacoma dd)$-designs, i.e. $m$-tuples $Phi=(cF_j)_{jinI_m}$ such that each $cF_j={f_{i,j}}_{iinI_n}in (C^{d_j})^n$ and $$sum_{jinI_m}|f_{i,j}|^2=alpha_i peso{for} iinI_n,.$$In this work we solve the optimal $(alphacoma dd)$-completion problem of an initial $dd$-design $Phi^0=(cF_j^0)_{jinI_m}$ with $cF_j^0in (C^{d_j})^k$, for $jinI_m$. Explicitly, given an strictly convex function $arphi:[0,infty)ightarrow [0,infty)$, we compute the $(alphacoma dd)$-designs $Phi_arphi^{m op}$ that are (local) minimizers of the joint convex potential $${m P}_arphi(Phi^0coma Phi)=sum_{jinI_m} r(arphi[S_{(cF_j^0comacF_j)} ])$$ of the multi-completions $(Phi^0coma Phi)$, among all $(alphacoma dd)$-designs $Phi=(cF_j)_{jinI_m}$; here $S_{(cF_j^0comacF_j)}$ denotes the frame operator of the completed sequence $(cF_j^0comacF_j)in (C^{d_j})^{k+n}$, for $jinI_m$. We obtain the geometrical and spectral features of these optimal $(alphacoma dd)$-multi-completions. We further show that the optimal $(alphacoma dd)$-designs $Phi_arphi^{m op}$ as above do not depend on $arphi$. 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.