On the Kaczmarz algorithm of approximation in infinite-dimensional spaces

Tom 148 / 2001

Stanis/law Kwapie/n, Jan Mycielski Studia Mathematica 148 (2001), 75-86 MSC: 41A65, 60G25, 60H25. DOI: 10.4064/sm148-1-7

Streszczenie

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.

Autorzy

  • Stanis/law Kwapie/nInstitute of Mathematics
    Warsaw University
    Banacha 2
    02-097 Warszawa, Poland
    e-mail
  • Jan MycielskiDepartment of Mathematics
    University of Colorado
    Boulder, CO 80309-0395, U.S.A.
    e-mail

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