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Time independence of $V$ makes the Green function depend on $t,t^{\prime }$ through $\tau =t-t^{\prime }$. A spatially varying potential does not imply spatial translation invariance. Fix ${\mathbf{r}}^{\prime }$ and set $\mathbf{x}=\mathbf{r}-{\mathbf{r}}^{\prime }$; write the Fourier coefficient more fully as $G_{{\mathbf{r}}^{\prime }}(\mathbf{k};\omega )$. The source position is an implicit parameter in the printed notation. DefineThe ordinary derivative terms transform into $(-{\omega }^2+k^2+m^2)G_{{\mathbf{r}}^{\prime }}$. The product $V{\partial }_{t}G$ becomes a convolution, and $2ie(-i\omega )=2e\omega$. More explicitly, insert both inverse spatial Fourier transforms into $VG$: the integral over $\mathbf{x}$ enforces $\mathbf{q}+{\mathbf{k}}^{\prime }=\mathbf{k}$. The transformed delta source is one, leavingThis is a source-position-dependent Fourier Green function. Dropping the explicit ${\mathbf{r}}^{\prime }$ subscripts recovers the requested notation with exactly the potential transform defined in the source. No assumption of a homogeneous potential is needed. The factor $(2\pi {)}^{-3}$ and the source-position phase are necessary for the stated Fourier convention.

The delta function in the specified convolution kernel collapses the ${\mathbf{k}}^{\prime }$ integral to $-i\gamma G$. Set $\lambda =e\gamma$ and ${\omega }_0^2=k^2+m^2$. ThenThere is a change of assumptions here: the specified kernel is the transform of $V=-i\gamma$, which is not real. These later parts define a formal analytic continuation to an effective damped equation, $[{\partial }_{\tau }^2+2\lambda {\partial }_{\tau }+{\omega }_0^2]G=\delta (\tau )$; they cannot simultaneously describe the earlier real external potential. The following pole analysis uses that continued operator.

Completing the square gives $D(\omega )=-(\omega +i\lambda {)}^2+({\omega }_0^2-{\lambda }^2)$, soFor $e>0$, hence $\lambda >0$, the underdamped regime has two poles at $\pm \Omega -i\lambda$, with $\Omega =\sqrt{k^2+m^2-{\lambda }^2}>0$. Both lie below the real axis. In the overdamped regime they lie on the imaginary axis at $-i(\lambda \mp \kappa )$, where $\kappa =\sqrt{{\lambda }^2-k^2-m^2}<\lambda$ for $k^2+m^2>0$; both are again below the real axis. At equality they coalesce into a double pole at $-i\lambda$. If $m<|\lambda |$, the crossover occurs at $k_c=\sqrt{{\lambda }^2-m^2}$; increasing $k$ moves the two imaginary poles together and then apart horizontally. If $m>|\lambda |$, every $k$ is underdamped; if equal, only $k=0$ is critical.

The sign of $e$ is not specified: for $e<0$ all nonzero-frequency poles are instead in the upper half-plane, describing antidamping; for $e=0$ they lie on the real axis and a causal prescription is required. When $m=k=0$, one pole is at zero even for nonzero $\lambda$ and a real-axis prescription is again needed. The figure uses illustrative $\lambda =1$, not supplied numerical data.

Take $\tau =t_f-t_i>0$ and split it into $M$ intervals of width $\Delta t=\tau /M$. The markov property and Gaussian convolution giveThis finite-dimensional identity is exact for every positive $M$. Define the path-integral measure by these normalized time slices:In the time-sliced diffusion path integral, the discrete exponent is ${\sum }_j[(x_{j+1}-x_j)/\Delta t{]}^2\Delta t/(4D)$. Its conventional continuum notation is thereforeThe time-slice normalization is essential, not an arbitrary multiplicative constant. Brownian paths are almost surely nondifferentiable, so the displayed action is shorthand for the Gaussian limiting measure rather than an ordinary integral of the derivative of each typical path.

Let $X_1,X_2$ be the two independent Poisson random variables. For $k\ge 0$, their discrete convolution iswhere the last equality is the binomial theorem. Thus $X_1+X_2\sim \mathrm{Poisson}(\lambda +\mu )$.

By the preceding convolution, $Z=X_1+X_2$ has a Poisson distribution with parameter $\nu =\lambda +\mu$. Directly from its probability mass function, shifting the index in the absolutely convergent sums gives $\mathbb{E}[Z]=\nu$ and $\mathbb{E}[Z(Z-1)]={\nu }^2$. Since $Z^2=Z(Z-1)+Z$, $\mathbb{E}[(X_1+X_2{)}^2]=(\lambda +\mu {)}^2+(\lambda +\mu )$. This is also the identity $\mathbb{E}[Z^2]=\mathrm{Var}(Z)+(\mathbb{E}[Z]{)}^2$.

Let ${\hat{u}}_0$ be the Fourier transform of the initial temperature. Solving the scalar linear ordinary differential equation givesRecover $u$ using the Fourier inversion theorem, with inverse factor $(2\pi {)}^{-2}$ for this two-dimensional convention. Equivalently, take the convolution of the initial data with the two-dimensional heat kernel. On a periodic square, the same exponential applies to each discrete Fourier mode, with $k_x=2\pi m/L_x$ and $k_y=2\pi n/L_y$.

For $0<\epsilon \le 1/2$, this continuous function on $\mathbb{R}/\mathbb{Z}$ has nonnegative Fourier coefficientsIt is ${\epsilon }^{-1}$ times the periodic convolution of two indicators of $[-\epsilon /2,\epsilon /2]$. Its Fourier series is absolutely convergent, ${\sum }_r{\hat{b}}_{\epsilon }(r)=b_{\epsilon }(0)=1$, and ${\sum }_{|r|>R}{\hat{b}}_{\epsilon }(r)\ll 1/(\epsilon R)$. Consequently, if all $N$ points $t_n$ avoid the interval $\Vert t\Vert <\epsilon$, some frequency $1\le r\ll {\epsilon }^{-2}$ satisfies $|{\sum }_{n}e(r{t}_n)|\gg N\epsilon$. This detects missing small neighborhoods by a large exponential sum.

The velocity induced by planar vorticity is a convolution with this kernel, whose sign depends on the stream function and vorticity conventions. Its magnitude is $(2\pi |z|{)}^{-1}$ and its derivative is bounded by $C|z{|}^{-2}$. Splitting the convolution at radius $R$ gives $\Vert K\ast \omega {\Vert }_{\infty }\le R\Vert \omega {\Vert }_{\infty }+\Vert \omega {\Vert }_1/(2\pi R)$. The same singularity yields a log-Lipschitz modulus, enough for a Yudovich characteristic flow.

A point spread function is an imaging system's response to an optical point source. Here it means the distribution of image light intensity: an ideal point object becomes a finite spot, often with faint rings, because of diffraction. An unresolved star and a sufficiently small bead emitting fluorescence are practical approximations to an optical point source. Their image tells us how nearby object details will overlap, and reveals blurring from optical aberrations, defocus, and, for an optical telescope, astronomical seeing.

A useful convention separates shape from throughput. At a fixed image plane, let $p(\mathbf{x})\ge 0$ be the point spread function with ${\int }_{{\mathbb{R}}^2}p(\mathbf{x})d^2\mathbf{x}=1$. Its units are inverse area. A point delivering detected radiant flux $F$ at image position ${\mathbf{x}}_0$ produces light intensity $Fp(\mathbf{x}-{\mathbf{x}}_0)$; this point spread function normalization makes the integral over the image equal to $F$. A profile divided by its own peak is convenient for comparing widths, but is a different point spread function normalization and does not by itself compare total radiant flux.

In incoherent imaging, different object points have negligible mutual spatial optical coherence over the measurement, so their image light intensities add. If the response also obeys translation invariance over the chosen field of view, the expected continuous image is the convolutionHere $O$ is the object brightness distribution expressed in image-scaled coordinates, with magnification already included, and $\tau$ is constant transmission. For $O(\mathbf{u})=F_0\delta (\mathbf{u}-{\mathbf{u}}_0)$, where $\delta$ is the two-dimensional Dirac delta function, this gives $I=\tau F_{0}p(\mathbf{x}-{\mathbf{u}}_0)$. A changing response across the field of view instead requires a position-dependent integral kernel $p(\mathbf{x};\mathbf{u})$; one measured point spread function is then only a local calibration.

In coherent imaging, the superposition principle applies to the complex wave amplitudes. With amplitude spread function $h$, the corresponding ideal model isThus $|h\ast E{|}^2$ generally differs from $|h{|}^2\ast |E{|}^2$: coherent contributions produce wave interference terms that depend on their relative wave phases. Their time average vanishes for mutually incoherent emitters, which explains the intensity convolution. Partial optical coherence requires the correlations between object points as well.^[ref]

The figure is an original diffraction model, not measured data. All panels have the same total radiant flux: each member of a pair contributes half the single-source radiant flux. Dashed curves show the separate light intensity contributions; the solid curve is their sum. The point spread functions have tails, so even sources satisfying the Rayleigh criterion overlap. As separation decreases, the image changes continuously; there is no abrupt disappearance of all information at one separation.

A motivating measured example is Measurement of point-spread function (PSF) for confocal fluorescence microscopy by InCheon Song and others (2005). A $543.8\mathrm{nm}$ laser, delivered through an optical fiber, excited fluorescent beads on a slide through a $100\times$, numerical aperture $1.3$ oil-immersion objective. A moving stage scanned the bead; emitted light returned through the objective, an emission filter and a pinhole to a photodiode. The readout was fluorescence light intensity against scanned position. Under the same setup, nominal bead diameters $0.17$, $0.5$, and $1\mu \mathrm{m}$ produced progressively broader measured spots. The size comparison controls for the finite object: the measured response is the convolution of bead emission with the effective point spread function, rather than a pure point response. Inverting this convolution model gave a response compared with the theoretical one. The smaller bead gave a weaker signal relative to noise; uncertainty in actual bead diameter also affected the restored width. These are limits of the calibration. In confocal microscopy, excitation and pinhole detection jointly determine the effective point spread function; this experiment does not measure the ideal widefield Airy disk plotted above.^[ref]

For a new calibration, useful additional controls are to measure detector background, avoid saturating the photodetector, repeat the scan to check stability, and compare isolated beads at several positions in the field of view. Record wavelength, numerical aperture, pinhole setting, sampling interval and optical focus. These are practical recommendations, not extra measurements claimed for the published experiment.

For $0<r<1$, the geometric series gives $P_r(t)=1+2{\sum }_{n\ge 1}{r}^n\cos nt$. Hence this positive even kernel has normalized integral one, and normalized convolution with it multiplies the $n$th Fourier coefficient by $r^{|n|}$. Its mass outside any fixed neighbourhood of zero tends to zero as $r\uparrow 1$. If an integrable function has finite one-sided limits $A,B$ at zero, evenness gives half the mass to each side and the convolution tends to $(A+B)/2$.

The real Gaussian function heat kernel $(4\pi \epsilon z{)}^{-1/2}\exp [-(\varphi -\theta {)}^2/(4\epsilon z)]$ is integrable only when $\epsilon z>0$. For forward time $z>0$, this requires positive diffusivity. A Cole-Hopf transformation is an algebraic substitution for any nonzero coefficient, but that does not make the real forward convolution valid for negative diffusivity. Constant initial heat data already produce a divergent integral with the wrong sign.

In the two-level saturation population model, $p_e$ and the population-difference reduction $q$ reach half their resonant values at $|\Delta |=(\Gamma /2)\sqrt{1+s}$. Their half width at half maximum is therefore $(\Gamma /2)\sqrt{1+s}$ and their full width at half maximum is $W_{\omega }=\Gamma \sqrt{1+s}$ in angular frequency, or $W_{\nu }=W_{\omega }/(2\pi )$ in hertz. This is power broadening: stronger excitation addresses a wider range of detunings even though the lifetime has not changed.

Increasing $s$ improves the local population-difference reduction $q(0)=s/(1+s)$ but broadens the response. From $s=1$ to $s=9$, this reduction increases from $0.5$ to $0.9$, while the width increases from $\sqrt{2}\Gamma$ to $\sqrt{10}\Gamma$. The contrast approaches a limit and the width keeps growing. This single-atom width is not automatically the width of a velocity-averaged pump–probe feature.

For example, if the distribution of velocities is approximately constant across the narrow resonant classes, the population-only overlap integral contains two Lorentzian line shapes: a probe of half width $\gamma =\Gamma /2$ and a pump hole of half width $a=\gamma \sqrt{1+s}$. Their centres in $kv$ are separated by $2\delta$. The convolution of these profiles has half width $a+\gamma$ in that separation, so its full width at half maximum as a function of $\delta$ is $a+\gamma =(\Gamma /2)(1+\sqrt{1+s})$. Even this simplified pump–probe width differs from the single-atom power broadening formula. Finite Doppler broadening and coherent or multilevel effects require further analysis.

On the torus ${\mathbb{T}}^2$ with normalized Haar measure, choose $c_N$ so that $\int K_N=1$. The kernel is a nonnegative trigonometric polynomial. Its base $2+\cos s+\cos t$ has unique maximum four at the origin. Outside any fixed neighborhood its maximum is $q<4$, while in a smaller neighborhood of positive measure it is at least some $q^{\prime }>q$. Normalization consequently bounds the exterior kernel by $C(q/q^{\prime }{)}^N$, which tends to zero. Convolution with these kernels gives uniform trigonometric polynomial approximation to every continuous function on the torus.

The rectangular function has value one on $|t|<1/2$ and zero on $|t|>1/2$. Endpoint values are convention-dependent and do not affect its ordinary integrals or Fourier transform. Shifts, amplitude changes and time rescaling produce a rectangular pulse of arbitrary height, center and duration. With $\hat{f}(k)=\int f(t)e^{-ikt}dt$, for $a>0$ the width-$2a$ pulse $f(t)=1_{|t|<a}$ has Fourier transform $2\sin (ak)/k$, with value $2a$ at zero by a limit. Its convolution with itself is a rescaled triangular function, $(2a-|t|)$ on $|t|<2a$ and zero outside, because the convolution measures the overlap length of two intervals.

For a twice continuously differentiable interaction potential with $\Vert D^{2}W\Vert \le L$ and a probability measure with finite first absolute moment, this subtraction produces a finite potential even if $W\ast \mu$ is infinite. Taylor's formula and linear growth of the gradient giveDominated differentiation under the integral sign gives $\nabla V_{\mathrm{ren}}=(\nabla W)\ast \mu$ and $D^2{V}_{\mathrm{ren}}=(D^{2}W)\ast \mu$. If the unrenormalized convolution is finite, the two potentials differ only by a constant, so they generate the same velocity field when that field depends on their gradient.

In the classical risk model, condition on the first arrival of the Poisson process. If available capital just before it is $v$, ruin has probability ${\overline{F}}_X(v)+{\int }_0^v{f}_X(x)\psi (v-x)dx$. Integrate this expression at $v=u+ct$ against $\lambda e^{-\lambda t}$ and change variables from $t$ to $v$. Differentiating the lower limit gives the displayed equation. The claim-density convolution is continuous for bounded $\psi$, so the first-arrival representation also supplies the differentiability required by the equation.

For $a>0$ and $x\ge 0$, split the convolution integral at $y=0$ and $y=x$. The three contributions are $e^{-ax}/(2a)$, $x{e}^{-ax}$ and $e^{-ax}/(2a)$. Evenness gives the displayed answer for all real $x$.

With Fourier transform convention $\tilde{f}(k)=\int f(x)e^{-ikx}dx$, one has $\tilde{f}(k)=2a/(a^2+k^2)$. The convolution theorem and Fourier inversion theorem therefore yieldThe value at $x=0$ is $\pi /(2a^3)$. Multiplying $f$ by $a/2$ instead gives a normalized Laplace distribution; its self-convolution is the density $a(1+a|x|)e^{-a|x|}/4$.

Signal processing transforms or analyzes signals to extract information, change a representation, or improve a measurement. Examples include smoothing a sensor record, selecting a band of frequencies, and decomposing a recording through Fourier analysis. A signal is a mathematical representation of a physical record; deciding which variation is meaningful and which is noise is part of the problem.

A linear filter offers a simple entry point. Its output is assembled from weighted copies of its input. A moving average can reduce rapid variation, but it also changes timing and can erase useful detail. The worked example below makes that tradeoff explicit rather than treating a smoother plot as automatic improvement.

Two complementary views make the subject appealing. An impulse response describes the effect of a localized input; a frequency response describes the effect of a sustained oscillation. For a linear time-invariant system, both describe the same operation, connected by convolution and Fourier analysis. ^[ref].

For causal convolution on $t\ge 0$, the Laplace transform convolution theorem converts $x+\sin \ast x=F$ into $\bar{x}=[(p^2+1)/(p^2+2)]\bar{F}$. Since this multiplier is $1-1/(p^2+2)$, inversion gives the displayed resolvent kernel. The formula remains valid for locally integrable forcing on finite intervals: substitution and the Fubini theorem reduce verification to an identity between the two continuous sine kernels. Thus a global Laplace transform is a convenient derivation, rather than an essential growth restriction on every local forcing.

A continuous map from a smooth manifold to ${\mathbb{R}}^d$ can be approximated by a smooth map with any prescribed positive continuous pointwise error bound. In coordinate charts, smoothing by local convolution and then combining with a smooth partition of unity gives the approximation. On a compact subset of an open chart domain, a smooth cutoff allows the resulting straight-line homotopy to be supported in that domain. This permits the Sard theorem to be used after approximation; the original continuous map can be space-filling and need not omit a point.