  We make a relativistic extension of the one-dimensional J-matrix method of
scattering. The relativistic potential matrix is a combination of vector,
scalar, and pseudo-scalar components. These are non-singular short-range
potential functions (not necessarily analytic) such that they are well
represented by their matrix elements in a finite subset of a square integrable
basis set that supports a tridiagonal symmetric matrix representation for the
free Dirac operator. Transmission and reflection coefficients are calculated
for different potential coupling modes. This is the first of a two-paper
sequence where we develop the theory in this part then follow it with
applications in the second.
