Fourier extension operator
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.
Definition
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 adjoint of the Fourier restriction operator, where the notation represents restriction to the set.