Informally, the Fourier extension operator is an operator that takes a function defined on the surface of the unit sphere in and applies the inverse Fourier transform to produce a function on the entirety of .
Formally, it is an operator such that where denotes surface measure on the unit sphere , , and for some . Here, the notation denotes the fourier transform of . In this Lebesgue integral, is a point on the unit sphere and is the Lebesgue measure on the sphere, or in other words the Lebesgue analog of .
The Fourier extension operator is the (formal) adjoint of the Fourier restriction operator , where the notation represents restriction to the set .