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Miklos Schweitzer 1977_9

Source: rank of integral operator

January 25, 2009
vectorfunctionintegrationreal analysisreal analysis unsolved

Problem Statement

Suppose that the components of he vector <spanclass=′latex−bold′>u</span>=(u0,…,un) <span class='latex-bold'>u</span>=(u_0,\ldots,u_n) are real functions defined on the closed interval [a,b] [a,b] with the property that every nontrivial linear combination of them has at most n n zeros in [a,b] [a,b]. Prove that if σ \sigma is an increasing function on [a,b] [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] , A(f)= \int_{a}^b <span class='latex-bold'>u</span>(x)f(x)d\sigma(x), \;f \in C[a,b]\ , is r≤n r \leq n, then σ \sigma has exactly r r points of increase. E. Gesztelyi
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