# On $k$ point density problem for band-diagonal $M$-bases

## TL;DR

This paper introduces new analytical methods to study the $k$ point density and rank one density properties of band-diagonal $M$-bases, building on historical examples and extending understanding of their density characteristics.

## Contribution

It develops novel techniques for analyzing $k$ point density and rank one density in band-diagonal $M$-bases, advancing the theoretical framework in this area.

## Key findings

- New methods for analyzing $k$ point density
- Insights into rank one density properties
- Extension of previous examples and results

## Abstract

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.