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We investigate connections between the geometry of linear subspaces and the convergence of the alternating projection method for linear projections. The aim of this article is twofold: in the first part, we show that even in Euclidean spaces the convergence of the alternating method is not determined by the principal angles between the subspaces involved. In the second part, we investigate the properties of the Oppenheim angle between two linear projections. We discuss, in particular, the question of existence and uniqueness of consistency projections in this context.
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We study infinite products of reproducing kernels with view to their use in dynamics (of iterated function systems), in harmonic analysis, and in stochastic processes. On the way, we construct a new family of representations of the Cuntz relations. Then, using these representations we associate a fixed filled Julia set with a Hilbert space. This is based on analysis and conformal geometry of a fixed rational mapping $R$ in one complex variable, and its iterations.
Let $mathcal{H}$ be a complex separable Hilbert space. We prove that if ${f_{n}}_{n=1}^{infty}$ is a Schauder basis of the Hilbert space $mathcal{H}$, then the angles between any two vectors in this basis must have a positive lower bound. Furthermore, we investigate that ${z^{sigma^{-1}(n)}}_{n=1}^{infty}$ can never be a Schauder basis of $L^{2}(mathbb{T}, u)$, where $mathbb{T}$ is the unit circle, $ u$ is a finite positive discrete measure, and $sigma: mathbb{Z} rightarrow mathbb{N}$ is an arbitrary surjective and injective map.
We study the alternating algorithm for the computation of the metric projection onto the closed sum of two closed subspaces in uniformly convex and uniformly smooth Banach spaces. For Banach spaces which are convex and smooth of power type, we exhibit a condition which implies linear convergence of this method. We show these convergence results for iterates of Bregman projections onto closed linear subspaces. Using an intimate connection between the metric projection onto a closed linear subspace and the Bregman projection onto its annihilator, we deduce the convergence rate results for the alternating algorithm from the corresponding results for the iterated Bregman projection method.
We prove that iterating projections onto convex subsets of Hadamard spaces can behave in a more complicated way than in Hilbert spaces, resolving a problem formulated by Miroslav Bav{c}ak.
Let $G $ be a noncompact semisimple Lie group with finite centre. Let $X=G/K$ be the associated Riemannian symmetric space and assume that $X$ is of rank one. The spectral projections associated to the Laplace-Beltrami operator are given by $P_{lambda}f =fast Phi_{lambda}$, where $Phi_{lambda}$ are the elementary spherical functions on $X$. In this paper, we prove an Ingham type uncertainty principle for $P_{lambda}f$. Moreover, similar results are obtained in the case of spectral projections associated to Dunkl Laplacian.
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