Some criteria and properties of Chebyshev systems (Q1920872)

Let \(T^n [a, b]\) be the Chebyshev system of \(n\) functions on \([a, b]\). An isolated zero \(t\in [a, b]\) of a continuous function \(x\) is called nodal if either \(t\in \{a, b\}\) or \(t\in ]a, b[\) and the function \(x\) changes sign upon passage across \(t\), and \(t\) is nonnodal otherwise. Some systems of \(n\) continuous functions whose linear spans contain functions with given nodal and non-nodal zeros are introduced. The relation between the introduced systems and the system \(T^n [a, b]\) are studied. The results depend on the value of \(n\in \mathbb{N}\) essentially.