Toeplitz bases in matrix fields

It is the first purpose of this paper to give some of the criteria in order that the unit vectors (e^k) [or more general sequences of points] should be a Toeplitz T-basis in a matrix field EA. We then develop some specific applications to Cesäro-type matrix fields, but using matrices with additional components (including slowly-varying functions) with an eye on later applications of these basis results to summabil ity- factor theorems involving order conditions.