2019/08/01 by Ivan Feshchenko, Feshchenko, Ivan
Computer Science · Mathematics · #46C07 #47B15 #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics #math.FA #msc:46C07 #msc:47B15
paper · pdf · doi:10.48550/arxiv.1908.00531
arxiv created 2019/08/01 · openalex publication_date 2019/08/01 · arxiv updated 2019/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let H be a Hilbert space and H1,...,Hn be closed subspaces of H. Set H0:=H1∩ H2∩...∩ Hn and let Pk be the orthogonal projection onto Hk, k=0,1,...,n. The paper is devoted to the study of functions fn:[0,1]→ℝ defined by fn(c)=sup\‖Pn...P2 P1-P0‖ |cF(H1,...,Hn)\leqslant c\, c∈[0,1], where the supremum is taken over all systems of subspaces H1,...,Hn for which the Friedrichs number cF(H1,...,Hn) is less than or equal to c. Using the functions fn one can easily get an upper bound for the rate of convergence in the method of cyclic alternating projections. We will show that the problem of finding fn(c) is equivalent to a certain optimization problem on a subset of the set of Hermitian complex n× n matrices. Using the equivalence we find f3 and study properties of fn, n\geqslant 4. Moreover, we show that 1-an(1-c)-\widetildebn(1-c)2\leqslant fn(c)\leqslant 1-an(1-c)+bn(1-c)2 for all c∈[0,1], where an=2(n-1)sin2(π/(2n)), bn=6(n-1)2sin4(π/(2n)) and \widetildebn is some positive number.