  In the early 1990s the works of Larson, Wogen and Argyros, Lambrou, Longstaff
disclosed an example of a strong tridiagonal $M$-basis that was not rank one
dense. Later Katavolos, Lambrou and Papadakis studied $k$ point density
property of this example. In this paper we present new methods for the analysis
of $k$ point density and rank one density properties for band-diagonal
$M$-bases.
