A note on extreme magnitudes of characteristic functions

For T[set membership, variant]R,T[not equal to]0, let [Phi]T be the collection of characteristic functions [phi] such that [phi](T)=0. For t[set membership, variant]R define MT(t)=sup[phi][set membership, variant][Phi]T[phi](t). Obviously, MT(-t)=MT(t). Luo and Zhang [2005. An extremal problem for Fourier transforms of probabilities. C. R. Math. Acad. Sci. Paris 341, 293-296] found, explicitly, MT(t) for t[less-than-or-equals, slant]T. In this note it is shown that MT(t)=1 if t>T, completing the work of Luo and Zhang.