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Fordirect differentiation givesandThus ${\theta }_t=D{\theta }_{xx}$.

Its total mass is found from the Gaussian integral:The unit-mass solution converging to the Dirac delta function as $t\downarrow 0$ is therefore the heat kernel

For the second initial condition, the heat-kernel solution is the convolutionWith $u=(x-y)/(2\sqrt{Dt})$ and the definition of the error function, this becomesThis is the heat equation with interval-indicator initial data for $a=1$.

At $t=0$ the graph is the rectangle of height one on $[-1,1]$. For every $t>0$ it is smooth, positive and even, with its maximum at $x=0$. As time increases the graph broadens and its maximum falls, while its total area remains $2$; pointwise it tends to zero as $t\to \infty$.

By the Fourier inversion theorem and Fubini's theorem,Thus multiplication of Fourier transforms corresponds to the convolution $h=f\ast g$, which is the convolution theorem in the convention used here.

Only reaction 1 changes $X_1$. Starting from five molecules, its successive states are $5\to 3\to 1$, with rates $5^2=25$ and $3^2=9$; state one is then absorbing for this channel. ThusThe probability of still being at three is the convolution of the first waiting time with an exponential distribution and survival of the second:Applying the law of total probability to the three possible states givesIt has $g(0)=0$ and tends to one as both reaction waiting times elapse.

Hospital admissions are a delayed thinning of infections. If $f(s)$ is the delay density and $\lambda (t)$ denotes the population infection incidence rate, then the convolutionis the admission incidence rate, with an additional term for infections before time zero if the observation window starts during an epidemic.

A kernel for density estimation is an integrable function $K$ with $\int K=1$, usually also bounded and nonnegative, and its scaled version is $K_h(u)=h^{-1}K(u/h)$. For suitable $g_1,g_2$, their convolution is

Because the observations have length-biased density $g(u)=uf(u)/\mu$,Thus the exact bias is $(K_h\ast f)(x)-f(x)$, exactly the same as for the ordinary kernel density estimator based directly on observations from $f$.

Write $R(K)=\int K(u{)}^{2}du$. The estimator is the average of $Y_i(x)=\mu X_i^{-1}{K}_h(x-X_i)$, soIts mean is $K_h\ast f$. Integrating the pointwise variance therefore givesFinally, Cauchy-Schwarz inequality under density $f$ givesEquality would require $X$ to be constant almost surely, impossible for a density, so $\mu \bar{\mu }>1$. The ordinary KDE has the same negative term but leading integrated variance $R(K)/(nh)$; length-biased sampling strictly inflates it.

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.

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.

For any distribution $E\in {\mathcal{D}}^{\prime }$ and any test function $f$, the convolution $(E\ast f)(x)=⟨E_y,f(x-y)⟩$ is a smooth function. On a compact set of $x$ values, the translated test functions and all their derivatives have supports in one fixed compact set, so distributional continuity permits every derivative:For constant-coefficient operators, $P(D)(E\ast f)=(P(D)E)\ast f$. Hence a fundamental solution of a linear differential operator supplies a smooth particular solution for every compactly supported smooth datum. No temperedness of $E$ is needed.