Kaczmarz algorithm and frames (Q2855822)

\textit{S. Kwapień} and \textit{J. Mycielski} [Stud. Math. 148, No. 1, 75--86 (2001; Zbl 0987.41020)] proposed a Hilbert space version of the Kaczmarz algorithm defined as follows. Given a sequence \(\{e_n\}\) in Hilbert space \(H\), construct \(x_0=\langle x,e_0\rangle e_0\), \(x_n=x_{n-1}+\langle x-x_{n-1},e_n\rangle e_n\), \(n\geq 1\). The sequence \(\{e_n\}\) is called almost effective if \(\forall x\in H\): \(\lim_{n\to\infty}\|x-x_n\|\leq B\|x\|\), \(0\leq B<1\). It is effective if \(B=0\). Kwapień and Mycielski proved that a sequence of unit vectors \(\{e_n\}\) is effective if and only if \(\{g_n\}\) is a tight frame with constant 1 where \(g_n=e_n-\sum_{i=0}^{n-1}\langle e_n,e_i\rangle g_i\), i.e. \([e_0,e_1,\ldots]^t=M[g_0,g_1,\ldots]^t\) with \(M\) unit lower triangular.NEWLINENEWLINEThis paper is about the case where \(\{g_n\}\) is not a tight frame and the Bessel sequence \(\{e_n\}\) is almost effective. Several theorems relate conditions on \(C=M^{-1}\) to a Bessel sequence \(\{e_n\}\) being a frame and being almost effective. Generalizations of the previous result by Kwapień and Mycielski are that a Bessel sequence \(\{e_n\}\) will be a frame or Riesz basis if and only if \(\{g_n\}\) is a frame or a Riesz basis with \(M\) bounded, and \(\{e_n\}\) is orthonormal if and only if \(\{g_n\}\) is orthonormal. As an application, a convergence proof is given for the McCormick version of the Kaczmarz algorithm to solve \(Ax=b\) in \(\ell^2(\mathbb{N})\) (cf. [\textit{S. F. McCormick}, Indiana Univ. Math. J. 26, 1137--1150 (1977; Zbl 0341.65046)]). It suffices that \(A\) is bounded, surjective and that its rows form an orthogonal basis for \(\ell^2(\mathbb{N})\).