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The Airy equation is the linear dispersive partial differential equation $q_t+q_{xxx}=0$. Under the spatial Fourier transform, each mode evolves by multiplication by $e^{i{k}^{3}t}$.

For a full Euclidean lattice and a positive definite real symmetric matrix $A$, the Gaussian theta sum obeysUse the Fourier transform kernel $e^{-2\pi ix\cdot z}$, the identity $\widehat{e^{-\pi x^{T}Ax}}(z)=(\det A{)}^{-1/2}{e}^{-\pi z^T{A}^{-1}z}$, and the Poisson summation formula for a Euclidean lattice. No integrality or self-duality of the Euclidean lattice is assumed.

For a zero-mean stationary Gaussian random field $V(\mathbf{r})$, the autocorrelation function iswhich is independent of $\mathbf{r}$. Its Fourier transform is the spatial power spectral density.

A bandlimited function has Fourier transform supported in a bounded frequency interval. An integrable bandlimited function has a bounded continuous representative when its Fourier transform is integrable, and that representative extends analytically using the compact-frequency integral. Scaling arguments scales its frequency support inversely, while multiplying two such functions adds their support intervals through convolution.

For the Maxwell molecule collision operator and the Fourier transform with exponent $-i\xi \cdot v$, put ${\xi }^{\pm }=(\xi \pm |\xi |\sigma )/2$. The angular exchange for elastic collisions, rotation invariance of spherical surface area and factorization of the two velocity integrals give the displayed formula. This identity converts the collision integral into a spherical average of products at two related frequencies.

Using $V=k_0^2(n^2-1)$, the scalar Lippmann-Schwinger equation is $\psi ={\psi }_i+{\int }_D{G}_{k_0}V\psi$. The Born approximation replaces the total field inside the integral by the known incident field:For a unit plane wave, its far-field pattern is $(4\pi {)}^{-1}{\int }_{D}V({\mathbf{r}}^{\prime })e^{i{k}_0({\hat{\mathbf{r}}}_0-\hat{\mathbf{r}})\cdot {\mathbf{r}}^{\prime }}d{\mathbf{r}}^{\prime }$. Thus each incident direction samples a shifted sphere in the Fourier transform of the potential.

For normalized circle measure and a nonnegative cutoff near $(1,0)$, equal to one on $|\theta |\le \delta /C$ and supported on $|\theta |\le 2\delta /C$, remove the constant phase $e^{-i{\xi }_1}$. On $|{\xi }_1|\le {\delta }^{-2}$, $|{\xi }_2|\le {\delta }^{-1}$, the remaining phase has magnitude at most $2/C^2+2/C$. Taking $C$ sufficiently large keeps its real part positive and comparable to one, so the Fourier transform has magnitude at least a constant times the cap mass, which is at least $\delta /(\pi C)$. No upper bound on the cutoff is required for this lower bound.

With kernel $e^{-2\pi i⟨u,x⟩}$ and $\mathrm{Im}\tau >0$, the Fourier transform of $e^{\pi i\tau |x{|}^2}$ is $(-i\tau {)}^{-n/2}{e}^{-\pi i|u{|}^2/\tau }$. Choose the logarithm of $-i\tau$ on the right half-plane. The ordinary real Gaussian Fourier transform proves the identity at $\tau =it$; holomorphic continuation proves the complex formula.

A complex-coordinate parametrization of the two-dimensional Fourier transform. With $z=x+iy$, $k=k_1+i{k}_2$, and $({\xi }_1,{\xi }_2)=(2k_2,-2k_1)$, its exponent is $-i\xi \cdot (x,y)$. Since $d\xi =4dA(k)$, its inverse is $q(z)={\pi }^{-2}\int e^{k\bar{z}-\bar{k}z}Q(k)dA(k)$. The spectral equation $({\partial }_{\bar{z}}-k)F=q$ gives ${\partial }_{\bar{k}}F=-e^{k\bar{z}-\bar{k}z}Q(k)/\pi$, linking Fourier inversion to a spectral dbar equation.

A contact term is a distribution supported where insertion points coincide. Its Fourier transform is local in momentum, typically polynomial. Renormalizing an operator product can require additive contact counterterms even if each operator has zero anomalous dimension.

For a constant-coefficient elliptic differential operator $P(D)$ of order $N\ge 1$, localizing the equation gives $P(D)(\chi u)=\chi P(D)u+[P(D),\chi ]u$. The commutator has order at most $N-1$, so a global Fourier transform estimate raises the known Local Sobolev space index by one until the forcing index plus $N$ is reached.

With inverse Fourier transform normalization $(2\pi {)}^{-1}$, $\int e^{is\theta }{\theta }^{j}d\theta =2\pi i^{-j}{\delta }^{(j)}(s)$ as an oscillatory integral. The factor $i^{-j}$ and the sign convention $⟨{\delta }^{\prime },g⟩=-g^{\prime }(0)$ are essential. For instance, amplitude $-i{x}_2\theta /(2\pi )$ with phase $x_1\theta$ gives $-x_2{\delta }^{\prime }(x_1)$.

Write $e^u\varphi (u)=\int \psi (t)e^{-iut}dt$. The double integral $\iint \psi (t)\psi (t^{\prime })(1+it)(1+i{t}^{\prime })/(2+i(t+t^{\prime }))dtd{t}^{\prime }$ equals the displayed constant. Express the denominator as a Laplace integral and each differentiated Fourier factor as $-{\varphi }^{\prime }(u)$, then use Fubini's theorem. For a real cutoff the resulting square is positive even though the Fourier transform factors have no complex conjugation. The Cauchy-Schwarz inequality and $\varphi (0)=1$, $\varphi (1/3)=0$ give $c_{\varphi }\ge 3$.

At fixed spatial momentum with mode energy $E>0$, $g_E(t)=e^{-iE|t|}/(2E)$ is continuous, with derivatives $-i/2$ and $+i/2$ at zero from opposite sides. Hence its second distributional derivative has a $-i\delta (t)$ term, while away from zero $g_E^{\prime \prime }+E^2{g}_E=0$. Therefore $({\partial }_t^2+E^2)g_E=-i\delta (t)$ and, after the spatial Fourier transform, $(\square +m^2){\Delta }_F=-i{\delta }^4$. This checks the propagator numerator, contour signs and equal-time canonical commutation relations independently of residue calculation.

The dynamic structure factor is the space-time Fourier transform of an equilibrium two-point correlation function. It resolves fluctuations by angular frequency and wavevector and is related to the dissipative part of a retarded Green function by the Fluctuation-dissipation theorem.

For real wave speed $c$ and $t\ge 0$, define the complex-frequency cosine through the entire series $C_t(z)={\sum }_{j\ge 0}(-1{)}^j(ct{)}^{2j}(z\cdot z{)}^j/(2j)!$. The expression is independent of the square-root branch even though the square root itself need not be entire. Writing $z=a+ib$ gives $|\mathrm{Im}\sqrt{z\cdot z}|\le |b|$, and hence $|C_t(z)|\le e^{|c|t|\mathrm{Im}z|}$. The Paley–Wiener–Schwartz theorem therefore makes its inverse Fourier transform a distribution supported in the radius-$|c|t$ ball. Multiplication by $C_t$ evolves initial displacement with zero initial velocity in the wave equation, giving finite propagation speed for compactly supported initial distributions.

If a fast inhibitor satisfies $0=v_{xx}+u-v$, then its spatial Fourier transform obeysSubstitution produces a nonlocal scalar equation for the activator.

A source is nonradiating at frequency $k$ if its outgoing field has zero far-field pattern. For every $\chi \in C_c^{\infty }(A)$, the source $Q=(\Delta +k^2)\chi$ has outgoing field $\psi =\chi$ and is invisible outside $A$. Integration by parts shows that its Fourier transform vanishes on $|\mathbf{\xi }|=k$, so it belongs to the kernel of a linear map defined by the source-to-far-field operator at fixed frequency. This family proves nonuniqueness independently of the instability of the inverse.

For a piecewise smooth decaying function, a jump in its $j$th derivative at $a$ contributes the displayed algebraic term to its high-frequency Fourier transform. Distributionally, $D^{j+1}f$ contains $[f^{(j)}{]}_a\delta (x-a)$, and integration by parts or the derivative-transform identity supplies the factor $(ik{)}^{-j-1}$. Adequate integrability of the subsequent derivatives controls the remainder. Smooth localized polynomial terms give no algebraic cusp contribution.