A conjecture on the density of a system of weighted polynomial functions (Q1340596)

The density of a system of weighted polynomial functions \(\{w^ n\}^ \infty_{n = 0}\) in \(L^ 2[-1,0]\) arising in the discussion of linear differential delay equations is studied. Characteristics of differential delay equations and their small solutions are given. Conditions are found under which the set \(\{w_ n\}^ \infty_{n = 0}\) forms a dense system in \(L^ 2[-1,0]\) and under which the related differential delay equation \(\dot x(t) = b(t) x(t-1)\) has no small solutions (and thus is dense in \(L^ 2[-1,0])\). A conjecture is proposed on the value of \(b(t)\) for the system \(\{w_ n\}^ \infty_{n = 0}\) to be dense on \(L^ 2[-1,0]\).