Fourier transform ($\mathcal{F}$)

For $f\in L^1({\mathbb{R}}^n)$, the angular-frequency Fourier transform is

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.

For $p>-1$ this formula is interpreted away from $k=0$ as a tempered distribution or by exponential regularization. It also supplies local high-frequency cusp terms after multiplying by a smooth cutoff. Even nonnegative integer $p$ give smooth polynomials and only contact terms at $k=0$, so their localized forms have no algebraic tail. For example, the cusp expansion of $(1+|x{|}^3{)}^{-1}$ gives $-12|k{|}^{-4}+29!|k{|}^{-10}+O(|k{|}^{-16})$.

Multiplication by a unit-modulus linear phase shifts the Fourier transform. Substituting $e^{ia\cdot x}$ into the defining integral gives the displayed identity. For a surface measure this permits independent translations of cap transforms without changing their input norms or angular supports.

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.

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 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.

With the $e^{ikx}$ convention, a right-supported decaying function has a transform analytic in the upper half-plane and a left-supported one in the lower half-plane. Incident modes continued with small imaginary frequency can have poles outside the initial convergence strip; these poles must be tracked separately from outgoing scattered waves.

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.

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 $-k^{2}S(k)$.

For sufficiently regular integrable functions on a half-line, Fourier sine transform inversion recovers the odd extension at continuity points. A zero endpoint value is required when demanding continuity at the endpoint.

The unnormalized sinc is $\mathrm{sinc}(x)=\sin x/x$, extended by value one at zero. The normalized form is $\sin (\pi x)/(\pi x)$. It is the inverse Fourier transform of a frequency interval indicator, so it describes ideal low-pass filtering and the Shannon scaling function.

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 $k^2$.

If $f\in L^1(\mathbb{R})$ and $x^{k}f\in L^1(\mathbb{R})$ for $0\le k\le r$, then its Fourier transform is $r$ times continuously differentiable and the displayed formula holds. The dominated convergence theorem permits each derivative under the integral. Thus $r+1$ vanishing moments give a zero Taylor polynomial through degree $r$ at the origin. Unweighted $L^1$ membership only guarantees a continuous function as the Fourier transform.

The Riemann-Lebesgue lemma and Fourier inversion give a bounded injective map $L^1(\mathbb{R})\to C_0(\mathbb{R})$ with dense image. It is not onto: inverse transforms of ${\chi }_{[-n,n]}\ast {\chi }_{[-1,1]}$ have unbounded $L^1$ norms while the targets have uniform norm $2$, contradicting the bounded inverse that the open mapping theorem would give.

The Wigner distribution of two fields is the Fourier transform of a field's two-point product in its separation variable:Its inverse reconstructs $u(\mathbf{z})\overline{v({\mathbf{z}}^{\prime })}$. For vector fields, specify whether the product is an outer product or a scalar contraction; the latter can encode an average over receiver channels.

For receiver-dependent Green functions $g_{\mathbf{\zeta }}(\mathbf{z})$ and aperture weight $a(\mathbf{\zeta })$, define ${\mathcal{W}}_A(\mathbf{Z},\mathbf{p})=\int a(\mathbf{\zeta })W[g_{\mathbf{\zeta }},g_{\mathbf{\zeta }}](\mathbf{Z},\mathbf{p})d\mathbf{\zeta }$. Its inverse separation Fourier transform is the kernel of the time reversal operator on the source plane.

The Japanese bracket is a smooth positive frequency weight, comparable to $1+|\xi |$. It defines Sobolev space norms and the growth estimates of a symbol class without singularities at zero frequency.

For real $s$ and vectors $\xi ,\eta$, the displayed bound follows from $⟨a+b⟩\le \sqrt{2}⟨a⟩⟨b⟩$, applied in the opposite direction when $s<0$. One can take $C_s=2^{|s|/2}$. Combined with Young's convolution inequality, it proves Sobolev multiplication by a smooth cutoff for all real orders: the rapidly decreasing transform of the cutoff absorbs the extra weight. The negative-order case is particularly useful when a distribution is not initially a function.

For the angular-frequency convention,

The Fresnel integral is an oscillatory Gaussian integral. In the angular-frequency convention, its distributional form isfor nonzero real $\omega$.

For $\alpha >0$, polar coordinates reduce the integral to $2\pi {\int }_0^{\infty }r{e}^{-(\alpha -i\lambda )r^2}dr$. The result is $\pi /(\alpha -i\lambda )$. Its damped limit for real $\lambda >0$ is $i\pi /\lambda$, which fixes the sign of the oscillatory complex Gaussian integral. The Fresnel integral convention can be compared with NIST definitions.

For $t>0$, contour rotation of the Gaussian integral gives

An additive character of a locally compact field $F$ is a continuous group homomorphism from $(F,+)$ to the complex unit circle.

For a nontrivial additive character $\psi$ on a local field, the character annihilator of an additive subgroup $H$ is $H^{\perp }=\{y:\psi (xy)=1\text{ for all }x\in H\}$. If ${\mathcal{O}}_F^{\perp }={\pi }^c{\mathcal{O}}_F$, then $({\pi }^n{\mathcal{O}}_F{)}^{\perp }={\pi }^{c-n}{\mathcal{O}}_F$. This is a character pairing construction, distinct from the annihilator of a module.

For a local field $F$, an additive character $\psi$, and a Haar measure $dx$, the Fourier transform isIt maps the Schwartz-Bruhat space to itself.

For a compact open additive subgroup $H$ of a local field, translation and additive character orthogonality give the displayed transform of a coset indicator. With the self-dual Haar measure, transforming again gives $1_{a+H}(-x)$. Every Schwartz-Bruhat function is a finite linear combination of these indicators.

Relative to a nontrivial additive character, the self-dual additive Haar measure makes the twice-applied Fourier transform over a local field equal to reflection. If the annihilator of the valuation ring is ${\pi }^c{\mathcal{O}}_F$ and its residue field has $q$ elements, the normalization is $\mathrm{vol}({\mathcal{O}}_F)=q^{c/2}$. Volumes of a compact open subgroup and its character annihilator then multiply to one.

For an additive character $\psi$ of a non-Archimedean local field $F$,For the character induced from the standard character of ${\mathbb{Q}}_p$, this annihilator is the inverse different.

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

The inverse Fourier transform of $(1-|\xi |{)}_{+}$ in three dimensions, with $a=2\pi |x|$, is $-4\pi \sin a/a^3+8\pi (1-\cos a)/a^4$. Its leading oscillatory $|x{|}^{-3}$ tail is not absolutely integrable against the radial volume element $4\pi |x{|}^{2}d|x|$. Multiplying by a Gaussian Fourier transform retains a nonzero boundary-derivative tail, so the radial cutoff need not map even Schwartz functions into $L^1$. A tensor product of one-dimensional triangular cutoffs is a valid approximate identity, and must not be confused with this Euclidean radial cutoff.

A Fourier multiplier operator $T_m$ is defined by $\widehat{T_{m}f}(\xi )=m(\xi )\hat{f}(\xi )$. On $L^2$, a bounded symbol gives a bounded operator by the Plancherel theorem, while an unbounded symbol defines a closed linear operator on the functions for which $m\hat{f}$ remains square-integrable.

Suppose $w$ is locally integrable andThen $\hat{w}(\lambda )+(c_{+}-c_{-})\log |\lambda |$ has finite limits at zero from both sides. With the convention $\hat{w}(\lambda )=\int e^{-i\lambda x}w(x)dx$, the positive-frequency limit minus the negative-frequency limit isSubtracting suitable reflected copies of $x^{-1}{1}_{(1,\infty )}(x)$ leaves an $L^1$ function, whose Fourier transform is continuous; the jump then follows from the Dirichlet integral.

If $f_{\lambda }(x)=f(\lambda x)$ on ${\mathbb{R}}^n$ with $\lambda >0$, thenand consequently

If a nontrivial uniform estimateholds for all integrable $L^p$ functions on ${\mathbb{R}}^n$, dilation by every $\lambda >0$ forcesThus $q$ must be the conjugate exponent of $p$.

As a tempered distribution on ${\mathbb{R}}^3$,for a nonzero convention-dependent constant $C$. One proof writesand applies the Fourier transform of a Gaussian before substituting $u=|\xi {|}^2/(4t)$.

With one common sign convention,Reversing the denominator reverses the sign.

For a Schwartz function $f$, the Hilbert transform satisfies $\mathcal{H}f(x)=(\int f)/(\pi x)+o(|x{|}^{-1})$ at both infinities. Splitting the inverse Fourier transform at the sign multiplier's jump and integrating each half-line by parts extracts the zero-frequency value $\hat{f}(0)$; the remainder vanishes by the Riemann-Lebesgue lemma. A nonzero zeroth moment therefore obstructs Schwartz decay.

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.

For $f=1_{[-a,a]}$ and the convention $\hat{f}(\xi )=\int f(x)e^{-ix\xi }dx$,with its continuous value $2a$ at zero.

With the standard Fourier convention, Plancherel theorem applied to $1_{[-1,1]}$ gives