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Temporal optical coherence compares a field at one position with a delayed copy. In a Michelson interferometer, unequal round-trip lengths supply this delay. Even a source with rapidly changing wave phase can give high interferometric visibility close to zero delay, because both arms then deliver nearly the same source history. A source narrowed in optical spectrum generally retains correlation over larger delays; a single width does not describe sources with several separated lines. ^[ref].

For a stationary scalar field, choose the carrier convention $e^{-i2\pi \nu t}$, with frequency $\nu$ in cycles per second. Let $S(\nu )\ge 0$ be its spectral density per unit frequency, with $0<\int S(\nu )d\nu <\infty$. Its normalized field correlation is the Fourier transformThe optical spectrum weights the rotating contributions; a broad spread of frequencies allows them to cancel at short delays. This is the stationary field correlation–spectrum relation. The relevant spectrum is that of the optical field in the detected mode, rather than the low-frequency fluctuations of a detector's light intensity. ^[ref].

Here are two stipulated line profiles, each centered at ${\nu }_0$ with full width at half maximum $\Delta \nu >0$. Write $f=\nu -{\nu }_0$ and use normalized profiles on the whole detuning line:The first is a Lorentzian, mathematically a Cauchy distribution; the second is a Gaussian function. Taking their Fourier transforms gives, with the carrier factor $e^{-i2\pi {\nu }_0\tau }$ omitted from the magnitudes,These are calculated model results, not measured line profiles. The detuning approximation treats the narrow line as extending to arbitrarily large positive and negative $f$; for optical applications require $\Delta \nu \ll {\nu }_0$. Exact spectra of the complex positive-frequency field have no negative absolute frequencies. Changing to angular frequency replaces $\Delta \nu$ by $\Delta \omega /(2\pi )$; it must not silently preserve the same numerical constant.

Equal detected single-arm light intensities give interferometric visibility equal to the correlation magnitude. Doubling the Gaussian full width at half maximum from $1$ to $2\mathrm{GHz}$ halves every fixed-contrast delay. At $\tau =0.5\mathrm{ns}$ the model gives Gaussian interferometric visibility approximately $0.410686$ at $1\mathrm{GHz}$ and $0.0284471$ at $2\mathrm{GHz}$; the $1\mathrm{GHz}$ Lorentzian gives $0.207880$. The dashed curve instead keeps the $1\mathrm{GHz}$ Gaussian correlation and changes $I_2/I_1$ to $1/4$, multiplying all interferometric visibilities by $0.8$. Its lower contrast at zero delay therefore comes entirely from intensity imbalance.

A density of states $g(ϵ)=g_0[1+\alpha \cos (2\pi ϵ/\Delta )]$ has zero-temperature oscillatory grand potential $g_0\alpha {\Delta }^2\cos (2\pi \mu /\Delta )/(4{\pi }^2)$. Its Fermi-window convolution multiplies that oscillation by $R_T$ when the lower band edge is far below the chemical potential. The Fourier transform of the even window removes its sine component and gives the title factor. At high $X$, $R_T\sim 2X{e}^{-X}$; at small $X$, $R_T=1-X^2/6+O(X^4)$. The corresponding thermal change relative to zero temperature has amplitude proportional to $1-R_T$, rather than $R_T$.

For an isolated razor-thin disk, define the two-dimensional Fourier transform by $\tilde{g}(\mathbf{k})=\int e^{-i\mathbf{k}\cdot \mathbf{x}}g(\mathbf{x})d^{2}x$. At $k>0$, the razor-thin disk Poisson kernel gives the displayed inversion. Since the transform of $1/|\mathbf{x}|$ is $2\pi /k$, inversion also gives $\Sigma =(4{\pi }^{2}G{)}^{-1}(1/|\mathbf{x}|)\ast {\Delta }_2\Phi$. The forward relation is $\Phi =-G(1/|\mathbf{x}|)\ast \Sigma$. An additive potential constant is irrelevant; a uniform infinite sheet and scale-free divergent potentials require separate zero-mode or potential-difference treatment rather than an arbitrary decaying boundary condition.

Translation shifts the argument without changing the shape of a function. On the real line, the displayed convention shifts a graph to the right by $h$. Lebesgue measure invariance preserves every finite Lp norm; its Fourier transform is multiplied by $e^{-ih\xi }$.

For an integrable function and the angular-frequency convention,so a translation multiplies each frequency component by a factor of modulus one. The amplitude of every component stays the same, while its wave phase changes by $-ka$. For example, delaying $\cos (2\pi \nu t)$ by $a$ produces $\cos (2\pi \nu t-2\pi \nu a)$: faster oscillations accumulate a larger wave phase shift for the same delay. Substituting $y=x-a$ in the defining Fourier transform is the entire proof. Conversely, this multiplier in the Fourier transform produces a translation after Fourier inversion.

The bulk tangent covariance is twice the inverse Fourier transform of $k_{B}T/(A{q}^2+f)$, one term for each transverse direction. Equivalently the inverse of $-A{\partial }_r^2+f$ is $e^{-|r|/\xi }/(2\sqrt{Af})$. This gives the displayed correlation and identifies a force-dependent correlation length, different from the zero-force persistence length. Finite chains require endpoint conditions or a periodic convention.

For the triangular function $f(x)=(1-|x|{)}_{+}$ released with zero initial velocity in the whole-line wave equation, the D'Alembert formula gives the displayed solution. Here $r_{+}=\max (r,0)$. Two equal half-amplitude triangles translate in opposite directions, overlapping at early times and separating after $|c|t>1$. This defines a weak solution, because the initial slope has jumps; away from the translated corner lines it is a classical solution. Its Fourier transform is $2(1-\cos k)\cos (ckt)/k^2$, with removable value one at $k=0$.

The sine and cosine integrals accumulate $\sin t/t$ and $\cos t/t$, with different normalizations at zero and infinity. They arise in oscillatory integrals, Fourier transforms of singular kernels and Fourier series of logarithmic profiles.

The trispectrum is the Fourier transform of a fourth-order cumulant, a fourth-order counterpart of the bispectrum. It describes connected four-point correlations; the primordial trispectrum applies it to cosmological random fields.

An original two-amplitude model makes the timing inference explicit. Use the spectral phase convention of the coherent harmonic pulse-train model, and define positive delay $\tau$ to retard the reference electric field. Let the amplitudes leading to the same final photoelectron state bewith real $b_{\pm }\ge 0$. Squaring their sum givesThe $2\omega$ oscillation comes from opposite signs of the reference photon's phase. For $\Delta {\varphi }_q=2\omega t_0$ and ${\delta }_q=0$, the signal peaks at $\tau =t_0$ modulo $T/2$. A fitted phase contains the target contribution ${\delta }_q$; setting it to zero is an assumption, not an observation. In an experiment, unobserved final channels can add as probabilities, so the displayed formula is one effective-channel idealization.

For a chosen example, impose $\Delta {\varphi }_q=0.80$ radians and ${\delta }_q=0.15$ radians at $800\mathrm{nm}$. The observed oscillation has argument $2\omega \tau -0.65$. Its peak shift is $0.1380\mathrm{fs}$, while the corrected finite-difference group delay is $\Delta {\varphi }_q/(2\omega )=0.1699\mathrm{fs}$. Ignoring the target contribution would bias the inferred delay by $-31.9$ attoseconds. These stipulated phases demonstrate a reconstruction error; they are not atomic calculations or measured delays.

Recovering all neighboring phase differences determines the component spectral phases up to an overall phase and the chosen timing origin, provided the oscillation phases are consistently unwrapped. Known component amplitudes then give a temporal field by a finite inverse Fourier transform. This is why measuring the optical spectrum alone is insufficient: it omits the very information that decides when constructive wave interference occurs.

Choose two point-like astronomical sources with $({\theta }_1,I_1)=(-0.00625,1)$ and $({\theta }_2,I_2)=(0.009375,0.65)$, where the angles are in radians and strengths use arbitrary common units. Their separation is $\Delta \theta =0.015625$ radians. These are invented model parameters, not historical measurements. Their complex visibility follows immediately from the preceding derivation, and its squared magnitude isAt $u=0$, $V=1.65$; at $u=400$, $V=-1+0.65i$; at $u=800$, $V=0.35$. The unequal strengths prevent the complex visibility magnitude from reaching zero. Its modulation contains the source separation, while a common displacement of both sources changes wave phase without changing this magnitude. Amplitude alone loses that location information.

The sparse case samples $u=\pm 12.5k$ for $k\in \{2,5,9,14,20,28,36,44,52,60,64\}$; the denser case samples every $k=1,\dots ,64$, also with both signs. All weights are one. Both reach $|u|=800$, so their shortest fringe period is the same $0.00125$ radians. Their dirty beams and false subsidiary features differ because the intermediate spatial frequencies differ. Neither case includes $u=0$ or frequencies above that maximum. The figure uses a periodic angular interval of length $0.08$ radians; this artificial boundary makes a discrete Fourier transform calculation exact for the chosen grid, rather than modelling an all-sky observation.

The plotted dirty image is the sum of two shifted dirty beams, not a fitted or deconvolved estimate of the source. Adding samples in this example reduces prominent spurious peaks; it does not establish a universal monotonic improvement for every weighting or sky. A missing zero baseline leaves the periodic dirty image with zero mean, so it cannot by itself recover the total positive brightness. Missing short spacings also weaken constraints on extended smooth emission.

Reconstruction is an inverse problem. A perturbation whose Fourier transform vanishes at every sampled coordinate can be added without changing the measurements: incomplete coverage leaves a null space. Nonnegative brightness, a restricted source region, or a model with a small number of components can reduce ambiguity, but each is additional information. Removing dirty beam artefacts means estimating a sky consistent with both data and such assumptions; a narrow central peak alone does not certify a unique image. Measurement error, uncertain calibration and source variability add further limitations.

Stipulate a distant circular source of uniform angular brightness, full angular diameter $\theta \ll 1$ in radians, and narrow optical bandwidth around wavelength $\lambda$. Distinct source patches emit independently. Assume each collector is small enough not to resolve the disk individually. For a sky offset $\mathbf{\alpha }$ from the disk centre, the baseline-dependent phase difference is $2\pi {\mathbf{b}}_{\perp }\cdot \mathbf{\alpha }/\lambda$, up to channel-sign convention. Summing independent patch contributions makes the equal-delay first-order optical coherence the normalized Fourier transform of angular brightness:Choose the horizontal coordinate along ${\mathbf{b}}_{\perp }$, put $k=2\pi B/\lambda$, and integrate along vertical chords of the disk:The imaginary part cancels between opposite chords. Here $J_1$ is the order-one Bessel function of the first kind. One way to obtain the last equality is to integrate the exponential's power series term by term on the finite disk. Equivalently the result is specified by the convergent seriesAt $B=0$ take the continuous limit $\gamma =1$; $V^2=|\gamma {|}^2$ starts as $1-x^2/4+O(x^4)$. This is why very short interferometric baselines give little leverage on angular diameter. Larger disks lose spatial optical coherence at shorter baselines.

The first positive zero of $J_1$ is $x_1\approx 3.831705970$, soThis is a full diameter convention. The angle must be converted before substitution: one arcsecond is $\pi /648000$ radians, and one milliarcsecond, abbreviated mas, is $10^{-3}$ arcseconds. For the stipulated $\theta =5\mathrm{mas}$ and $\lambda =500\mathrm{nm}$, $\theta =2.4240684\times 10^{-8}\mathrm{rad}$ and $B_0\approx 25.1575\mathrm{m}$. At $B=10\mathrm{m}$, the model gives $x\approx 1.52309$ and $V^2\approx 0.542783$.

Beyond $B_0$, $\gamma$ changes sign but $V^2$ stays nonnegative and develops weak further lobes. A single measured squared visibility can therefore match several angular diameters; sampling several interferometric baselines constrains the fit. Limb darkening, spectral averaging, or resolved collector apertures change the uniform-disk prediction. A uniform-disk diameter is a parameter of that brightness model; converting it to a physical stellar radius also requires distance and an appropriate brightness profile.

If a periodic integrable function has all Fourier coefficients zero, it vanishes almost everywhere. Its Fejér sums vanish, whereas the Fejér kernel is an approximate identity and converges to that function in the integrable norm. Applied to a periodized squared Fourier transform, this detects orthogonality of integer translates.

For integrable functions on the real line, equality of their Fourier transforms implies equality almost everywhere. To prove it, subtract and assume $\hat{h}=0$. Convolve $h$ with a Gaussian heat kernel $q_t$. The convolution theorem gives $\widehat{h\ast q_t}=0$, which is integrable, so the Fourier inversion theorem gives $h\ast q_t=0$ almost everywhere. Finally $q_t$ is an approximate identity: $\Vert h\ast q_t-h{\Vert }_1\le \int q_t(y)\Vert h(\cdot -y)-h{\Vert }_{1}dy\to 0$. This follows from translation continuity in Lp with exponent one and the vanishing Gaussian tails. Hence $h=0$ in $L^1$. If the two original functions are continuous, equality almost everywhere strengthens to equality everywhere, since a nonzero continuous difference has magnitude bounded below by a positive constant on some interval.

For a distant spatially incoherent quasimonochromatic source in the small-angle regime, the equal-delay mutual coherence across projected baseline $\mathbf{b}$ is the Fourier transform of its angular intensity. With matched collector response, the normalized degree is $g(\mathbf{b})=\int I(\mathbf{\theta })e^{-i2\pi \mathbf{b}\cdot \mathbf{\theta }/\lambda }{d}^2\theta /\int I(\mathbf{\theta })d^2\theta$. Equal collector intensities and matched polarizations make fringe visibility $|g|$. Finite bandwidth and unmatched optical paths introduce additional temporal-coherence averaging.

Use the Fourier transform convention $\hat{\psi }(\omega )=\int \psi (x)e^{-i\omega x}dx$. If $\psi$ and $x^r\psi$ for $r<q$ are integrable functions, moment differentiation of the Fourier transform yields a $C^{q-1}$ Fourier transform andThe dominated convergence theorem justifies these derivatives. Under these assumptions, $q$ vanishing moments are equivalent to vanishing derivatives through order $q-1$ at zero. For $q>1$ this gives a Peano zero $\hat{\psi }(\omega )=o(|\omega {|}^{q-1})$; for $q=1$ it gives $\hat{\psi }(\omega )=o(1)$. It does not by itself justify a $q$th derivative or an $O(|\omega {|}^q)$ estimate.

If also $\int |x{|}^q|\psi (x)|dx<\infty$, subtract the degree-$(q-1)$ Taylor polynomial of $e^{-i\omega x}$ inside the integral. Its Taylor remainder gives the stronger boundFor an integrable function with compact support, every weighted integral exists and its Fourier transform extends to an entire function. The language “zero of order at least $q$” is then literal analytic function terminology. If the next integral is nonzero, the order is exactly $q$. The smooth-mask vanishing-moment criterion relates these cancellations to a refinement mask under its additional hypotheses.^[ref]

The velocity correlation tensor is $Q_{ij}(\mathbf{r})=⟨u_i(\mathbf{x})u_j(\mathbf{x}+\mathbf{r})⟩$. For isotropic turbulence it splits into longitudinal and transverse parts. Incompressibility constrains them, while the Fourier transform gives the turbulent energy spectrum. Its trace is the scalar velocity-dot-product correlation.

Let a linear Earth response $h$ be recorded using a known source sweep $s$, giving $d=s\ast h+n$. Cross-correlating the record with the sweep yieldsThus a compact sweep autocorrelation $R_{ss}$ produces pulse-compressed arrival wavelets, while unrelated noise has a smaller coherent response. The Fourier transform of the result is $H(\omega )|S(\omega ){|}^2+N(\omega )\overline{S(\omega )}$. Correlation is a matched-filter operation, not an exact inversion of missing source frequencies. Spectral shaping or stabilized deconvolution can further adjust the wavelet within the measured bandwidth.^[ref]

With constant surface brightness over a centred angular rectangle of widths ${\Theta }_x,{\Theta }_y$, and a nearly uniform primary beam over that rectangle, the Fourier transform separates into two one-dimensional integrals. The result is $V(u,v)=S_{\nu }\mathrm{sinc}(\pi {\Theta }_{x}u)\mathrm{sinc}(\pi {\Theta }_{y}v)$, where $\mathrm{sinc}x=\sin x/x$. The zero-spacing value is the integrated spectral flux density, while the first axial null is at $|u|=1/{\Theta }_x$. Long baselines therefore do not measure the total flux of a resolved rectangle.

For two coherent square pulses of equal duration $\tau$, pulse-start separation $T\ge \tau$, resonant Rabi angular frequency $\Omega$, and Rabi detuning $\delta$, the initially lower-state transition probability has the displayed leading weak-excitation form when $|\Omega |\tau \ll 1$. Integrating the first-order probability amplitude over both windows gives a sum of two equal Fourier transforms, with relative quantum phase $\delta T$. The factor four from coherent addition makes the central value $(\Omega \tau {)}^2$. The rotating-wave approximation additionally requires many carrier cycles per pulse and near-resonant weak driving.