The Dual Kaczmarz Algorithm

The Kaczmarz algorithm is an iterative method for solving a system of linear equations. It can be extended so as to reconstruct a vector $x$ in a (separable) Hilbert space from the inner-products ${\langle x, \phi_{n} \rangle}$. The Kaczmarz algorithms defines a sequence of approximations from the sequence $\{⟨x,{\varphi }_n⟩\}$; these approximations only converge to $x$ when $\{{\varphi }_n\}$ is $effective$. We dualize the Kaczmarz algorithm so that $x$ can be obtained from $\{⟨x,{\varphi }_n⟩\}$ by using a second sequence $\{{\psi }_n\}$ in the reconstruction. This allows for the recovery of $x$ even when the sequence $\{{\varphi }_n\}$ is not effective; in particular, our dualization yields a reconstruction when the sequence $\{{\varphi }_n\}$ is $almost$ $effective$. We also obtain some partial results characterizing when the sequence of approximations from ${\langle x, \phi_{n} \rangle}$using${\psi_{n}}$convergesto$x$,inwhichcase${(\phi_n, \psi_n)}$iscalledan$effective$ $pair$.