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Observed symptom-onset intensity is the convolution of past infection intensity with the incubation delay density: $\mu (t)={\int }_0^{\infty }h(t-s)f(s)ds$. Back-calculation infers the infection history from that relation. Include pre-observation infections unless their absence is known.

Every compactly supported distribution on an open set $X$ is a finite sum $u={\sum }_{\alpha }{\partial }^{\alpha }{f}_{\alpha }$ with continuous $f_{\alpha }$ compactly supported inside $X$. One can first write $u={\partial }^{\gamma }F$ for a continuous global convolution primitive, then multiply by a cutoff function equal to one near the support. The identity $\chi {\partial }^{\gamma }F={\sum }_{\beta \le \gamma }(-1{)}^{|\beta |}(\genfrac{}{}{0ex}{}{\gamma }{\beta }){\partial }^{\gamma -\beta }[({\partial }^{\beta }\chi )F]$ localizes every coefficient. Finite global representation can fail for a distribution of unbounded order.

If $v$ is a compactly supported distribution and $u$ is any distribution, define $⟨u\ast v,\varphi ⟩=⟨u_x,⟨v_y,\varphi (x+y)⟩⟩$. The inner function is smooth and supported in $\mathrm{supp}\varphi -\mathrm{supp}v$, hence is a test function for $u$. Finite-order estimates prove continuity, while the tensor product of distributions proves commutativity after inserting compact cutoffs. For every test function $\psi$, $(u\ast v)\ast \psi =u\ast (v\ast \psi )$; evaluating at zero against reflected test functions proves uniqueness. If both distributions have compact support, $\mathrm{supp}(u\ast v)\subset \mathrm{supp}u+\mathrm{supp}v$ and the convolution theorem gives $\widehat{u\ast v}=\hat{u}\hat{v}$. Without a compact factor or another support/growth condition, arbitrary distributional convolution is not generally defined.

If independent real random variables $X$ and $Y$ have densities $f_X$ and $f_Y$, their sum has the convolution density

For functions whose Laplace transforms converge on a common right half-plane, the causal convolution $(f\ast g)(t)={\int }_0^{t}f(s)g(t-s)ds$ satisfies $\mathcal{L}(f\ast g)=\mathcal{L}(f)\mathcal{L}(g)$. This converts a transformed boundary forcing into a known time-domain integral representation.

For a set of density $\alpha$ in a finite cyclic group, select frequencies whose normalized Fourier coefficients on a finite abelian group have magnitude at least ${\alpha }^{3/2}/2$. Parseval identity bounds their number by $4{\alpha }^{-2}$. Fourier inversion of the fourfold convolution shows it is positive on the Bohr set in phase-distance convention of radius $1/6$. This is an explicit form of the Bogolyubov lemma.

For a distribution and a compactly supported smooth kernel, differentiation with constant coefficients commutes with convolution wherever the kernel support stays inside the domain. The identity follows by testing against the translated kernel and tracking the sign of its derivative. A variable coefficient must remain inside the product before convolution; in general $(bu{)}_{\sigma }\ne b{u}_{\sigma }$.

For a mixture distribution of exponential distributions, introduce one convolution state $A_j(u)={\int }_0^u\phi (z)e^{-{\beta }_j(u-z)}dz$ for each rate ${\beta }_j$. It satisfies $A_j^{\prime }=\phi -{\beta }_j{A}_j$. Together with the survival integro-differential equation these form a constant-coefficient first-order system; applying the differential operators $D+{\beta }_j$ eliminates the convolution states. For equal mixing weights at rates one and one half, with $r=\lambda /c$, the result is ${\phi }^{\prime \prime \prime }+(3/2-r){\phi }^{\prime \prime }+(1/2-3r/4){\phi }^{\prime }=0$. The original integral equation supplies initial conditions lost in elimination.

Gaussian blur applies convolution with $G_{\sigma }(x)=(2\pi {\sigma }^2{)}^{-n/2}{e}^{-|x{|}^2/(2{\sigma }^2)}$ to an image signal. It averages over a spatial scale $\sigma$, suppressing rapid variation while also blurring image edges. The heat-kernel convolution realizes this filter at time $t={\sigma }^2/2$.

For the heat equation on ${\mathbb{R}}^n$ with diffusion coefficient one, the heat kernel is $K_t(x)=(4\pi t{)}^{-n/2}{e}^{-|x{|}^2/(4t)}$. Its convolution with bounded initial data is smooth for positive time, solves the equation by differentiation under the integral, and returns continuous initial data locally uniformly through the approximate identity property. It is Gaussian filtering with per-coordinate variance $2t$.

The infectious disease renewal equation expresses current incidence as a convolution of past incidence with an infectivity profile, multiplied by a time-varying transmission level.

Mollification is the regularization of a function or distribution by convolution with a rescaled mollifier. It creates smooth functions on interior regions, preserves constant-coefficient distributional equations there, and converges back to the original object in the appropriate local topology as the radius tends to zero. The process is distinct from the kernel used to perform it.