On the completeness of the functions e−np(x) cos nx,e−np(x) sin nx for n ≥ 0 and p(x) a 2π periodic function

It is shown how a conjecture of G. B. Whitham on the completeness of the functions { ψ _n } = { e^?np(x) cos nx, e^?np(x) sin nx } in the interval 0, 2π] is indeed true provided p has a Hölder continuous first derivative, i.e., p ∈ C^{1, α}0, 2π]. Also an elementary proof of Whitham's conjecture is given for p ∈ C^{2, α}0, 2π].