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For finite nonnegative measures with equal mass and first moment and finite second moments, the constant and linear terms of their Fourier transforms cancel. Taylor's remainder bounds their difference by a constant times $|\xi {|}^2$, making this distance finite. Fourier uniqueness gives definiteness. The origin is excluded from the supremum; the quotient need not have a direction-independent limit there.

A Fourier extension operator forms an oscillatory integral from a density on a frequency surface with measure $\sigma$. The opposite sign in the phase merely reflects the output variable, so it gives identical Lp norms. In the usual dual formulation it is the adjoint of restricting the Fourier transform to that surface.

Nonzero compactly supported distributions with pairwise disjoint supports are linearly independent: a smooth cutoff function isolating one support isolates its coefficient. Injectivity transfers this independence to their Fourier transforms. Exponential-type bounds and the Translation property of the Fourier transform can prove the needed support separation before the distributions themselves are known explicitly.

A Fourier multiplier is a function $m$ on frequency space used to define an operator by multiplication after taking a Fourier transform.

The Fourier sine transform is a half-line transform adapted to a homogeneous Dirichlet boundary condition. With the displayed normalization, inversion is $f(x)=(2/\pi ){\int }_0^{\infty }\sin (kx)S(k)dk$. It is equivalent to the Fourier transform of an odd extension. If $f(0)=0$ and boundary terms vanish, two integrations by parts give the transform of $f^{\prime \prime }$ as $-k^{2}S(k)$.

Fourier space labels a field by the frequency or wavevector variables of its Fourier transform, rather than by its original coordinates. Spatial gradients act by multiplication by $i\mathbf{k}$, and the negative Laplacian acts by $k^2$.

The Fourier transform with negative exponential and inverse factor $(2\pi {)}^{-n}$ is a continuous isomorphism of the Schwartz space. Weighted frequency derivatives are transforms of derivatives of weighted input functions, so their suprema are controlled by finitely many Schwartz seminorms. Fourier inversion gives ${\mathcal{F}}^2=(2\pi {)}^n\mathcal{R}$, where $\mathcal{R}f(x)=f(-x)$. Transposition gives a continuous isomorphism on tempered distributions as well.

For $u\in {\mathcal{E}}^{\prime }({\mathbb{R}}^n)$, its Fourier transform is the smooth functionIt has at most polynomial growth, and its extension to complex frequency is controlled more precisely by the Paley–Wiener–Schwartz theorem.

The angular-frequency Fourier transform of a finite measure, or a complex measure of finite total variation, is defined by the displayed integral. It is bounded by total variation and continuous by dominated convergence. An integrable density recovers the ordinary Fourier transform of a function. A measure supported on a curved surface instead leads to a Fourier extension operator.

With $\hat{\phi }(\xi )=\int e^{-ix\cdot \xi }\phi (x)dx$, the Fourier transform of a tempered distribution is defined byContinuity of the Fourier transform on the Schwartz space makes this well defined. For regular integrable functions, Fubini's theorem shows that it agrees with the ordinary transform. The inverse convention has factor $(2\pi {)}^{-n}$ and the positive exponential.

Fourier transform spectroscopy measures an interferogram as a function of optical path length difference. Its Fourier transform recovers the spectrum. An imaging implementation obtains an interferogram at every detector pixel, but a scanned cube is not simultaneous at all wavelengths.

On the unit-period spatial circle, free streaming has solution $f_t(x,v)=f_0(x-tv,v)$. With Fourier phase $e^{-2\pi i(kx+\xi v)}$, its mixed Fourier transform is ${\hat{f}}_t(k,\xi )={\hat{f}}_0(k,\xi +kt)$. A nonzero spatial mode of the velocity-integrated density therefore samples increasingly large velocity frequencies. The conserved spatial zero mode remains, while extra velocity regularity supplies quantitative decay of the other modes.

For $\varphi >0$ and nonnegative variances, $\gamma (h)={\tau }^2{1}_{\{h\ne 0\}}+{\sigma }^2(1-e^{-{\varphi }^2\Vert h{\Vert }^2})$ admits covariance $C(h)={\sigma }^2{e}^{-{\varphi }^2\Vert h{\Vert }^2}+{\tau }^2{1}_{\{h=0\}}$. The Gaussian kernel is positive definite because it is the Fourier transform of the density of a Gaussian distribution. Adding independent white noise supplies the nugget effect. Its scale is $a=1/\varphi$. For ${\sigma }^2>0$, its exact range of a semivariogram is infinite. If ${\sigma }^2=0$, it is a pure nugget effect with no nontrivial structured range.

For a centered shell Gaussian measure of a real scalar field with positive shell kernel $K(p)=\kappa p^2+r$, its Gaussian functional integral gives $⟨\tilde{\varphi }(p)\tilde{\varphi }(q)⟩=(2\pi {)}^D{\delta }^{(D)}(p+q)/K(p)$ on the shell. Inverse Fourier transforms therefore yield $G_{>}(x-y)={\int }_{\Lambda /b<|p|\le \Lambda }{d}^{D}p(2\pi {)}^{-D}{e}^{ip\cdot (x-y)}/K(p)$. The kernel must be positive throughout the shell. Translation invariance makes the separation essential; a numerator involving $x$ alone is valid only with $y=0$ or if $x$ is redefined as the separation.

For a flat background dominated by one constant-$w$ barotropic equation of state, assume $\delta P=w\delta \rho$ and no scalar anisotropic stress. The Einstein field equations then reduce the common Newtonian-gauge potential to the stated Fourier transform equation, with $\mathcal{H}=2/[(1+3w)\tau ]$. For radiation the regular solution is $3{\Phi }_0(\sin x-x\cos x)/x^3$, $x=k\tau /\sqrt{3}$, and oscillates with a decaying envelope after sound-horizon entry. For pressureless matter the two solutions are a constant and ${\tau }^{-5}$. A non-adiabatic pressure perturbation would supply an additional source, so constant background $w$ alone does not justify the homogeneous equation.

For $t>0$ this Green function is the response to a point impulse on the infinite line, using the continuous square root of $1+i{c}_d$. The Fourier transform multiplies the initial transform by $e^{\mu t-ikUt-(1+i{c}_d)k^{2}t}$; its Gaussian inverse gives the displayed expression. At fixed $x$, its exponential rate is $\mu -U^2/[4(1+c_d^2)]$, whereas along $x=Ut$ it is $\mu$. Thus a temporally growing flow with $0<\mu <U^2/[4(1+c_d^2)]$ is convectively rather than absolutely unstable.

For $\mathcal{H}f(x)={\pi }^{-1}\mathrm{pv}\int f(y)/(x-y)dy$ and $\hat{f}(\xi )=\int e^{-ix\xi }f(x)dx$, the Fourier transform multiplier is $-i\mathrm{sgn}\xi$. The Heaviside function transform and the double-transform reflection identity give the sign. Plancherel theorem proves the $L^2$ isometry, and multiplication by $i\xi$ proves commutation with differentiation. The map need not preserve the Schwartz space.

A distribution $u$ is homogeneous of degree $\sigma$ when ${\delta }_{t}u=t^{\sigma }u$ for every $t>0$. With the unnormalized angular-frequency convention, its Fourier transform is homogeneous of degree $-n-\sigma$.

A homogeneous Sobolev space controls derivatives without an independent low-frequency L2 norm: formally $\Vert f{\Vert }_{{\dot{H}}^s}^2=\int |\xi {|}^{2s}|\hat{f}(\xi ){|}^{2}d\xi$, with the normalization chosen according to the Fourier transform. Completions or quotient conventions must be specified when polynomial ambiguities occur. In three dimensions ${\dot{H}}^1$ is represented by $L^6$ functions with gradient in $L^2$, and $\Vert f{\Vert }_6\le C\Vert \nabla f{\Vert }_2$.