Suppose that the components of he vector <spanclass=′latex−bold′>u</span>=(u0,…,un) 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 σ is an increasing function on [a,b] and the rank of the operator A(f)=∫ab<spanclass=′latex−bold′>u</span>(x)f(x)dσ(x),f∈C[a,b] , is r≤n, then σ has exactly r points of increase.
E. Gesztelyi vectorfunctionintegrationreal analysisreal analysis unsolved