On maximal relative projection constants

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• 作者：Simon Foucart^a ; ¹ ; ^{foucart@tamu.edu" class="auth_mail" title="E-mail the corresponding author} ; Lesław Skrzypek^b
• 关键词：Projection constants ; Seidel matrices ; Tight frames ; Equiangular lines ; Graphs ; Semicircle law
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 March 2017
• 年：2017
• 卷：447
• 期：1
• 页码：309-328
• 全文大小：489 K

文摘

This article focuses on the maximum of relative projection constants over all m-dimensional subspaces of the N-dimensional coordinate space equipped with the max-norm. This quantity, called maximal relative projection constant, is studied in parallel with a lower bound, dubbed quasimaximal relative projection constant. Exploiting alternative expressions for these quantities, we show how they can be computed when N is small and how to reverse the Kadec–Snobar inequality when N   does not tend to infinity. Precisely, we first prove that the (quasi)maximal relative projection constant can be lower-bounded by $c\sqrt{m}$, with c arbitrarily close to one, when N is superlinear in m  . The main ingredient is a connection with equiangular tight frames. By using the semicircle law, we then prove that the lower bound $c\sqrt{m}$ holds with c<1$c<1$ when N is linear in m.