SCHWEITZER 1977 Problem 9

Suppose that the components of he vector $\textbf{u}=(u_0,\ldots,u_n)$ are real functions defined on the closed interval $[a,b]$ with the property that every nontrivial linear combination of them has at most $n$ zeros in $[a,b]$ . Prove that if $\sigma$ is an increasing function on $[a,b]$ and the rank of the operator $$A(f)= \int_{a}^b \textbf{u}(x)f(x)d\sigma(x), f \in C[a,b]\ ,$$ is $r \leq n$ , then $\sigma$ has exactly $r$ points of increase.

E. Gesztelyi