On the Kaczmarz algorithm of approximation in infinite-dimensional spaces

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.

Studia Mathematica, Tome 148 (2001) no. 1, pp. 75-86

Résumé

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.

Détail

DOI : 10.4064/sm148-1-7

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