We begin a study of Schur analysis in the setting of the Grassmann algebra,
when the latter is completed with respect to the 1-norm. We focus on the
rational case. We start with a theorem on invertibility in the completed
algebra, and define a notion of positivity in this setting. We present a series
of applications pertaining to Schur analysis, including a counterpart of the
Schur algorithm, extension of matrices and rational functions. Other topics
considered include Wiener algebra, reproducing kernels Banach modules, and
Blaschke factors.Comment: 35 page
