Fourier extension operator

Informally, the Fourier extension operator is an operator that takes a function defined on the surface of the unit sphere in $\mathbb {R} ^{n}$ and applies the inverse Fourier transform to produce a function on the entirety of $\mathbb {R} ^{n}$.

Definition

Formally, it is an operator ${\textstyle g\mapsto {\widehat {g\,d\sigma }}}$ such that $${\widehat {g\,d\sigma }}(x)=\int _{S^{n-1}}e^{ix\cdot \xi }g(\xi )\,d\sigma (\xi )$$where $d\sigma$ denotes surface measure on the unit sphere ${\textstyle S^{n-1}}$, ${\textstyle x\in \mathbb {R} ^{n}}$, and ${\textstyle g\in L^{p}(\mathbb {R} ^{n})}$ for some ${\textstyle p\geq 1}$. Here, the notation ${\textstyle {\widehat {f}}}$ denotes the fourier transform of ${\textstyle f}$. In this Lebesgue integral, ${\textstyle \xi }$ is a point on the unit sphere and ${\textstyle d\sigma (\xi )}$ is the Lebesgue measure on the sphere, or in other words the Lebesgue analog of ${\textstyle dx}$.

The Fourier extension operator is the (formal) adjoint of the Fourier restriction operator $g\mapsto {\widehat {f}}\vline _{S^{n-1}}$, where the $\vline _{X}$ notation represents restriction to the set ${\textstyle X}$.

See also

• Fourier transform § Restriction problems
• Mizohata–Takeuchi conjecture