Fourier extension operator

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

Definition

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

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

See also

• Fourier transform § Restriction problems
• Mizohata–Takeuchi conjecture