On the Kaczmarz algorithm of approximation in infinite-dimensional spaces

Stanisław Kwapień, and Jan Mycielski. "On the Kaczmarz algorithm of approximation in infinite-dimensional spaces." Studia Mathematica 148.1 (2001): 75-86. <http://eudml.org/doc/284583>.

@article{StanisławKwapień2001,
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 (xₙ) converges to x where (xₙ) is defined inductively: x₀ = 0 and $xₙ = x_\{n-1\} + αₙeₙ$, where $αₙ = ⟨x - x_\{n-1\},eₙ⟩$. 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.},
author = {Stanisław Kwapień, Jan Mycielski},
journal = {Studia Mathematica},
language = {eng},
number = {1},
pages = {75-86},
title = {On the Kaczmarz algorithm of approximation in infinite-dimensional spaces},
url = {http://eudml.org/doc/284583},
volume = {148},
year = {2001},
}

TY - JOUR
AU - Stanisław Kwapień
AU - Jan Mycielski
TI - On the Kaczmarz algorithm of approximation in infinite-dimensional spaces
JO - Studia Mathematica
PY - 2001
VL - 148
IS - 1
SP - 75
EP - 86
AB - 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 (xₙ) converges to x where (xₙ) is defined inductively: x₀ = 0 and $xₙ = x_{n-1} + αₙeₙ$, where $αₙ = ⟨x - x_{n-1},eₙ⟩$. 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.
LA - eng
UR - http://eudml.org/doc/284583
ER -