# How Fourier transform relates to interpolation space.

This refers to the link :

http://en.wikipedia.org/wiki/Interpolation_space

where in the History section it mentions that:

"Many methods were designed to generate such spaces of functions, including the Fourier transform, complex interpolation,[1] real interpolation,[2] as well as other tools (see e.g. fractional derivative)."

May I ask how how Fourier transform can be used to generate such space? As I understand Fourier transform; it is a transformation from one domain to another; how can it be used as a interpolation space? Few comments would be highly appreciated.

The word interpolation generally used to mean "interpolation of a function". However the link refers to interpolation of "spaces" not "functions". It is known from Riesz–Thorin theorem that Fourier Transform transform element of one $L^p$ space to another $L^q$ space.