Ritt operators and convergence in the method of alternating projections

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• 作者：Catalin Badea^a ; ^{badea@math.univ-lille1.fr" class="auth_mail" title="E-mail the corresponding author} ; David Seifert^b ; ^{david.seifert@sjc.ox.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：41A65 (47J25 ; 47A10 ; 47A12 ; 47A25)
• 刊名：Journal of Approximation Theory
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：205
• 期：Complete
• 页码：133-148
• 全文大小：249 K

文摘

Given N≥2$N\ge 2$ closed subspaces M₁,…,M_N$M_1,\dots ,M_N$ of a Hilbert space X$X$, let P_k$P_k$ denote the orthogonal projection onto M_k$M_k$, 1≤k≤N$1\le k\le N$. It is known that the sequence (x_n)$\left(x_n\right)$, defined recursively by x₀=x$x_0=x$ and x_n+1=P_N⋯P₁x_n$x_{n+1}=P_N\cdots P_1{x}_n$ for n≥0$n\ge 0$, converges in norm to P_Mx$P_{M}x$ as n→∞$n\to \infty$ for all x∈X$x\in X$, where P_M$P_M$ denotes the orthogonal projection onto M=M₁∩…∩M_N$M=M_1\cap \dots \cap M_N$. Moreover, the rate of convergence is either exponentially fast for all x∈X$x\in X$ or as slow as one likes for appropriately chosen initial vectors x∈X$x\in X$. 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 α>0$\alpha >0$, a dense subset X_α$X_{\alpha }$ of X$X$ such that ‖x_n−P_Mx‖=o(n^−α)$\Vert x_n-P_{M}x\Vert =o\left(n^{-\alpha }\right)$ as n→∞$n\to \infty$ for all x∈X_α$x\in X_{\alpha }$. Furthermore, there exists another dense subset X_∞$X_{\infty }$ of X$X$ such that, if x∈X_∞$x\in X_{\infty }$, then ‖x_n−P_Mx‖=o(n^−α)$\Vert x_n-P_{M}x\Vert =o\left(n^{-\alpha }\right)$ as n→∞$n\to \infty$ for all  α>0$\alpha >0$. 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_Mx$P_{M}x$ is in fact the limit of a series which converges unconditionally.