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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.

For $1\le p<\infty$, an integer $k\ge 0$, and a normalized nonnegative mollifier $\rho \in C_c^{\infty }({\mathbb{R}}^n)$, put ${\rho }_{\epsilon }(x)={\epsilon }^{-n}\rho (x/\epsilon )$. If $u\in W^{k,p}({\mathbb{R}}^n)$, its mollification is smooth and converges to $u$ in the Sobolev norm. The definition of a weak derivative and Fubini's theorem give ${\partial }^{\alpha }({\rho }_{\epsilon }\ast u)={\rho }_{\epsilon }\ast {\partial }^{\alpha }u$ for every multi-index with $|\alpha |\le k$. The approximate identity theorem in the Lp space $L^p$ applied to these finitely many weak derivatives proves the claim. On a domain with boundary this direct convolution argument applies only away from the boundary unless an extension is first supplied.

A static but spatially inhomogeneous linear operator has time translation symmetry without spatial translation symmetry. At fixed source position ${\mathbf{r}}^{\prime }$, one may still Fourier transform its Green function in $\mathbf{x}=\mathbf{r}-{\mathbf{r}}^{\prime }$. The coefficient retains ${\mathbf{r}}^{\prime }$ as a parameter. A multiplication by $V(\mathbf{r})$ becomes a convolution with $V_{{\mathbf{r}}^{\prime }}(\mathbf{q})=e^{i\mathbf{q}\cdot {\mathbf{r}}^{\prime }}\tilde{V}(\mathbf{q})$, not a spatially diagonal multiplier unless $V$ is constant. Suppressing the source-position subscript is only a notational convention.

A sequence of continuous periodic functions satisfying the displayed conditions, with normalized Haar measure $m$ and every $\delta >0$, is a summability kernel. For $f$ in a homogeneous Banach space on the circle, subtract $f\int k_{n}dm$ from its convolution with $k_n$. Translation continuity controls the resulting integral near zero, while boundedness by $2\Vert f{\Vert }_B$ and the vanishing tails control its complement. Thus $\Vert f\ast k_n-f{\Vert }_B\to 0$. The nonnegative Fejér kernel is an example with $L^1$ norm one.

If at least one of $u,v\in {\mathcal{D}}^{\prime }({\mathbb{R}}^n)$ is a compactly supported distribution, thenThe sums are Minkowski sums. The first rule follows from the pairing $⟨u_x\otimes v_y,\varphi (x+y)⟩$: a test function away from the support sum gives zero. For the second rule, localize the output to a small compact set avoiding the sum of the singular supports. The relevant pairs of input points form a compact set because one input support is compact. Cover it by finitely many product neighborhoods on each of which at least one input distribution is smooth; otherwise the output would meet the sum of the singular supports. Smooth cutoffs in these neighborhoods decompose the localized pairing into finitely many convolutions with a smooth compactly supported factor. Each is smooth by smoothing convolution with a test function. This proves the second rule locally.

The same argument works when addition is a proper map on the product of the two supports. Some condition defining the convolution is necessary; arbitrary pairs of distributions cannot always be convolved.

Target: Cauchy kernel fundamental solution. Parent: Fundamental solution of a linear differential operator. Intended file: distribution-theory.bigb.

For a taxi starting at any vertex of a rectangle with side lengths $s,t$, a uniform pickup vertex gives approach-distance law uniform on $0,s,t,s+t$. Conditional on that pickup, a uniform distinct destination gives passenger-distance law uniform on $s,t,s+t$, independent of the pickup vertex. Hence the two distance variables are independent despite sharing that vertex. Their convolution has twelve equally likely pairs. Its mean is $7(s+t)/6$ and its variance is $(s^2+t^2)/4+2(s^2+t^2-st)/9$. Repeated numerical sums combine their multiplicities when some distances coincide.

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 diffusion constant $D>0$, Gaussian increments have variance $2D\Delta t$. Their convolution gives a path weight proportional to $\exp [-\sum (\Delta x{)}^2/(4D\Delta t)]$ with measure factor $(4\pi D\Delta t{)}^{-M/2}$ for $M$ slices. The continuum kinetic action is notation for this limiting measure; typical Brownian paths have no ordinary derivative.

The triangular functionhas angular-frequency Fourier transform $4{\sin }^2(\xi /2)/{\xi }^2$.

Equivalently, this is the convolution of two unit-width rectangular functions: the overlap length is $1-|x|$ when $|x|<1$, and zero otherwise. Time rescaling and amplitude changes give triangular pulses with different widths and heights. The convolution theorem also obtains the displayed transform by squaring $\sin (\xi /2)/(\xi /2)$, with continuous value one at zero.

Truncated convolution makes $L^1(0,1)$ a commutative nonunital Banach algebra. The Tonelli theorem gives $\Vert f\ast g{\Vert }_1\le \Vert f{\Vert }_1\Vert g{\Vert }_1$, and the Fubini theorem gives associativity over the integration simplex. Its unitization is $\mathbb{C}\oplus L^1(0,1)$ with norm $|\alpha |+\Vert f{\Vert }_1$ and product $(\alpha ,f)(\beta ,g)=(\alpha \beta ,\alpha g+\beta f+f\ast g)$. The constant function one has convolution powers $t^{n-1}/(n-1)!$, of norm $1/n!$. Thus it is a nonnilpotent quasinilpotent element in this unitization.

The Weierstrass transform is Gaussian convolution, $Wf(x)=(4\pi {)}^{-1/2}{\int }_{\mathbb{R}}{e}^{-(x-y{)}^2/4}f(y)dy$. It smooths appropriate functions and is the heat semigroup at unit time. The Brownian transition semigroup uses variance-$t$ Gaussian kernels and therefore supplies the scaled time family.