1Institute of Mathematics Warsaw University Banacha 2 02-097 Warszawa, Poland 2Department of Mathematics University of Colorado Boulder, CO 80309-0395, U.S.A.
Studia Mathematica, Tome 148 (2001) no. 1, pp. 75-86
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.
Keywords:
kaczmarz algorithm successive projections suggests following concept sequence unit vectors hilbert space said effective each vector space sequence converges where defined inductively n alpha where alpha langle x x n rangle prove effectivity sequences hilbert spaces generalize concept effectivity sequences vectors banach spaces prove results general concept
Affiliations des auteurs :
Stanis/law Kwapie/n
1
;
Jan Mycielski
2
1
Institute of Mathematics Warsaw University Banacha 2 02-097 Warszawa, Poland
2
Department of Mathematics University of Colorado Boulder, CO 80309-0395, U.S.A.
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author = {Stanis/law Kwapie/n and Jan Mycielski},
title = {On the {Kaczmarz} algorithm of approximation
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Stanis/law Kwapie/n; Jan Mycielski. On the Kaczmarz algorithm of approximation
in infinite-dimensional spaces. Studia Mathematica, Tome 148 (2001) no. 1, pp. 75-86. doi: 10.4064/sm148-1-7
