quality:corruption
Extension of the definition

For 1 < p < 2 {\displaystyle 1<p<2}, the Fourier transform can be defined on L p (R) {\displaystyle L^{p}(\mathbb {R})} by Marcinkiewicz interpolation.

The Fourier transform can be defined on domains other than the real line. The Fourier transform on Euclidean space and the Fourier transform on locally abelian groups are discussed later in the article.

The Fourier transform can also be defined for tempered distributions, dual to the space of rapidly decreasing functions (Schwartz functions). A Schwartz function is a smooth function that decays at infinity, along with all of its derivatives. The space of Schwartz functions is denoted by S (R) {\displaystyle {\mathcal {S}}(\mathbb {R})}, and its dual S ′ (R) {\displaystyle {\mathcal {S}}'(\mathbb {R})} is the space of tempered distributions. It is easy to see, by differentiating under the integral and applying the Riemann-Lebesgue lemma, that the Fourier transform of a Schwartz function (defined by the formula Eq.1) is again a Schwartz function. The Fourier transform of a tempered distribution T ∈ S ′ (R) {\displaystyle T\in {\mathcal {S}}'(\mathbb {R})} is defined by duality: ⟨ T ^, ϕ ⟩ = ⟨ T, ϕ ^ ⟩; ∀ ϕ ∈ S (R). {\displaystyle \langle {\widehat {T}},\phi \rangle =\langle T,{\widehat {\phi }}\rangle ;\quad \forall \phi \in {\mathcal {S}}(\mathbb {R}).}

Many other characterizations of the Fourier transform exist. For example, one uses the Stone–von Neumann theorem: the Fourier transform is the unique unitary intertwiner for the symplectic and Euclidean Schrödinger representations of the Heisenberg group.