  We present novel reconstruction and stability analysis methodologies for
two-dimensional, multi-coil MRI, based on analytic continuation ideas. We show
that the 2-D, limited-data MRI inverse problem, whereby the missing parts of
$\textbf{k}$-space (Fourier space) are lines parallel to either $k_1$ or $k_2$
(i.e., the $\textbf{k}$-space axis), can be reduced to a set of 1-D Fredholm
type inverse problems. The Fredholm equations are then solved to recover the
2-D image on 1-D line profiles (``slice-by-slice" imaging). The technique is
tested on a range of medical in vivo images (e.g., brain, spine, cardiac), and
phantom data. Our method is shown to offer optimal performance, in terms of
structural similarity, when compared against similar methods from the
literature, and when the $\textbf{k}$-space data is sub-sampled at random so as
to simulate motion corruption. In addition, we present a Singular Value
Decomposition (SVD) and stability analysis of the Fredholm operators, and
compare the stability properties of different $\textbf{k}$-space sub-sampling
schemes (e.g., random vs uniform accelerated sampling).
