Title :

Ritt operators and convergence in the method of alternating projections

Author(s) :

Badea, Catalin [Auteur]
Université de Lille
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Seifert, David [Auteur]

Journal title :

Journal of Approximation Theory

Pages :

133-148

Publisher :

Elsevier

Publication date :

2016-05

ISSN :

0021-9045

English keyword(s) :

Iterative methods
best approximation
alternating projections
rate of convergence
Ritt operators
numerical range
resolvent condition
Friedrichs angle
unconditional convergence

HAL domain(s) :

Mathématiques [math]/Analyse fonctionnelle [math.FA]

English abstract : [en]

Given $N\ge2$ closed subspaces $M_1,\dotsc, M_N$ of a Hilbert space $X$, let $P_k$ denote the orthogonal projection onto $M_k$, $1\le k\le N$. It is known that the sequence $(x_n)$, defined recursively by $x_0=x$ and ...
Show more >Given $N\ge2$ closed subspaces $M_1,\dotsc, M_N$ of a Hilbert space $X$, let $P_k$ denote the orthogonal projection onto $M_k$, $1\le k\le N$. It is known that the sequence $(x_n)$, defined recursively by $x_0=x$ and $x_{n+1}=P_N\cdots P_1x_n$ for $n\ge0$, converges in norm to $P_Mx$ as $n\to\infty$ for all $x\in X$, where $P_M$ denotes the orthogonal projection onto $M=M_1\cap\dotsc\cap M_N$. Moreover, the rate of convergence is either exponentially fast for all $x\in X$ or as slow as one likes for appropriately chosen initial vectors $x\in X$. We give a new estimate in terms of natural geometric quantities on the rate of convergence in the case when it is known to be exponentially fast. More importantly, we then show that even when the rate of convergence is arbitrarily slow there exists, for each real number $\alpha>0$, a dense subset $X_\alpha$ of $X$ such that $\|x_n-P_Mx\|=o(n^{-\alpha})$ as $n\to\infty$ for all $x\in X_\alpha$. Furthermore, there exists another dense subset $X_\infty$ of $X$ such that, if $x\in X_\infty$, then $\|x_n-P_Mx\|=o(n^{-\alpha})$ as $n\to\infty$ for \emph{all} $\alpha>0$. These latter results are obtained as consequences of general properties of Ritt operators. As a by-product, we also strengthen the unquantified convergence result by showing that $P_M x$ is in fact the limit of a series which converges unconditionally.Show less >

Language :

Anglais

Popular science :

Non

ANR Project :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL