  A Fourier transform from momentum space to twistor space is introduced in
twistor string theory, for the first time, for the case where the twistor space
is a three-dimensional real projective space, corresponding to ultra-hyperbolic
spacetime. In this case, the Fourier transform is the same as in the standard
analysis. However, when the twistor space is a three-dimensional complex
projective space, corresponding to the complexified Minkowski space, some
aspects of the Fourier transform have yet to be clarified. For example, no
concrete method is known for calculating the complex integral, and nor is it
known which functions it should be applied to or what results are obtained. In
this paper, we define a Fourier transform for momentum-space functions in terms
of a certain complex integral, assuming that the functions can be expanded as
power series. We also define the inverse Fourier transform for twistor-space
functions in terms of a certain complex integral. Then, we show that the
momentum-space functions to which the Fourier transform can be applied and the
twistor-space functions that are obtained can be represented in terms of
cohomology groups. Furthermore, we show that the twistor operator
representations popular in the twistor theory literature are valid for the
Fourier transform and its inverse transform. We also show that the functions
over which the application of the operators is closed can be represented in
terms of cohomology groups.
