# Rank one density property for a class of $M$-bases

## TL;DR

This paper investigates conditions for a class of $M$-bases in Hilbert spaces to admit linear summation methods, extending previous examples and employing graph theory techniques to analyze their properties.

## Contribution

It generalizes the Larson--Wogen system of $M$-bases and determines when such bases allow linear summation methods using graph theory.

## Key findings

- Identifies conditions for $M$-bases to admit linear summation methods
- Extends Larson--Wogen system to a broader class of $M$-bases
- Uses graph theory techniques in the analysis

## Abstract

In the early 1990s the works of Larson, Wogen and Argyros, Lambrou, Longstaff disclosed an example of a strong $M$-basis that did not admit a linear summation method. We study a class of $M$-bases $\mathfrak{F}=\{f_n\}_{n=1}^\infty$ in the Hilbert space that generalizes the Larson--Wogen system. We determine the conditions under which $\mathfrak{F}$ admits a linear summation method. In order to do that we employ some of the graph theory techniques.