Frame completions with prescribed norms: local minimizers and applications
| Autor: | Demetrio Stojanoff, Noelia Belén Rios, Pedro Gustavo Massey |
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| Rok vydání: | 2017 |
| Předmět: |
Discrete mathematics
Matemática Matemáticas Applied Mathematics 010102 general mathematics purl.org/becyt/ford/1.1 [https] Matemática Aplicada 010103 numerical & computational mathematics 01 natural sciences purl.org/becyt/ford/1 [https] Computational Mathematics Frame completions Local minimum Frame (artificial intelligence) Computational Science and Engineering Majorization 0101 mathematics Convex potential Humanities CIENCIAS NATURALES Y EXACTAS Mathematics |
| Zdroj: | CONICET Digital (CONICET) Consejo Nacional de Investigaciones Científicas y Técnicas instacron:CONICET |
| ISSN: | 1572-9044 1019-7168 |
| Popis: | Let F0 = {fi}i∈In0 be a finite sequence of vectors in Cd and let a = (ai)i∈Ik be a finite sequence of positive numbers, where In = {1,...,n} for n ∈ N. We consider the completions of F0 of the form F = (F0, G) obtained by appending a sequence G = {gi}i∈Ik of vectors in Cd such that gi2 = ai for i ∈ Ik, and endow the set of completions with the metric d(F, F˜) = max{ gi − ˜gi : i ∈ Ik} where F˜ = (F0, G˜). In this context we show that local minimizers on the set of completions of a convex potential Pϕ, induced by a strictly convex function ϕ, are also global minimizers. In case that ϕ(x) = x2 then Pϕ is the so-called frame potential introduced by Benedetto and Fickus, and our work generalizes several well known results for this potential. We show that there is an intimate connection between frame completion problems with prescribed norms and frame operator distance (FOD) problems. We use this connection and our results to settle in the affirmative a generalized version of Strawn’s conjecture on the FOD. Facultad de Ciencias Exactas |
| Databáze: | OpenAIRE |
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