  We give the asymptotics of the Fourier transform of self-similar solutions to
the modified Korteweg-de Vries equation, through a fixed point argument in
weighted W^{1,\infty} around a carefully chosen, two term ansatz. Such
knowledge is crucial in the study of stability properties of the self-similar
solutions for the modified Korteweg-de Vries flow. In the defocusing case, the
self-similar profiles are solutions to the Painlev\'e II equation. Although
they were extensively studied in physical space, no result to our knowledge
describe their behavior in Fourier space. We are able to relate the constants
involved in the description in Fourier space with those involved in the
description in physical space.
