Optimal \(\alpha,\mathfrak{d})\)-multi-completion of \(\mathfrak{d}\)-designs (Q2105119)

The present article deals with the completion problem of an initial \(\mathfrak{d}\)-design. The authors consider the set of (\(\alpha\), \(\mathfrak{d}\))-designs for given finite dimensions \(\mathfrak{d}=(d_j)\) and weights \(\alpha=(\alpha_i)\) together with a strictly convex function and compute the designs which are the minimizers of the joint convex potential. Some results concerning these are given in the paper. The authors examine the spectral features of these designs and obtain few results in this direction. Applications of optimal multi-completions are discussed in the context of frame theory. An algorithm for the computation of optimal completions is given and numerical computations of the algorithm are illustrated with the help of examples in the paper.