On the Kaczmarz algorithm of approximation in infinite-dimensional spaces
Volume 148 / 2001
Studia Mathematica 148 (2001), 75-86
MSC: 41A65, 60G25, 60H25.
DOI: 10.4064/sm148-1-7
Abstract
The Kaczmarz algorithm of successive projections suggests the following concept. A sequence $(e_k)$ of unit vectors in a Hilbert space is said to be effective if for each vector $x$ in the space the sequence $(x_n)$ converges to $x$ where $(x_n)$ is defined inductively: $ x_0 =0$ and $x_n = x_{n-1} +\alpha _n e_n$, where $\alpha _n = \langle x-x_{n-1},e_n\rangle $. We prove the effectivity of some sequences in Hilbert spaces. We generalize the concept of effectivity to sequences of vectors in Banach spaces and we prove some results for this more general concept.
