Distribution (mathematical analysis, ${\mathcal{D}}^{\prime }(U)$)

A distribution on an open set $U$ is a continuous linear functional on the space of test functions $\mathcal{D}(U)=C_c^{\infty }(U)$.

A distribution has period $T>0$ if the displayed equality holds for every test function $\varphi$. This agrees with $f(x+T)=f(x)$ almost everywhere for a regular distribution. If $u$ is a tempered distribution, density of test functions in the Schwartz space extends the identity to all Schwartz functions.

For a test function $\Psi$ with ${\sum }_k\Psi (x+kT)=1$, one hasIndeed, expand $\varphi ={\sum }_k\Psi (x+kT)\varphi (x)$ in the Schwartz space and move each translation to $\varphi$ by periodicity. All Schwartz seminorms of the localized terms decay faster than any power of $|k|$, justifying the sum of pairings. This localization permits rigorous Fourier coefficients without multiplying a distribution by a discontinuous window.

A distribution which vanishes on a neighborhood of each point vanishes globally. To prove this, choose a smooth partition of unity subordinate to the neighborhoods. Every test function is a finite sum of its products with the partition members, since its support is compact. The distribution vanishes on each summand. Applying the result to a difference proves that two distributions agreeing locally agree globally. This justifies defining the support of a distribution by removing the union of all open regions on which it is zero.

Target: Periodic primitive with zero mean. Parent: Periodic extension. Intended file: function.bigb.

For open sets $V\subseteq U$ and $u\in {\mathcal{D}}^{\prime }(U)$, define $⟨u{|}_V,\varphi ⟩=⟨u,\tilde{\varphi }⟩$, where $\tilde{\varphi }$ extends the test function $\varphi \in \mathcal{D}(V)$ by zero. Because its compact support lies inside $V$, this extension is smooth. Extension preserves convergence in the space of test functions, proving continuity of the restricted distribution. Successive restrictions agree with direct restriction.

Target: Locality of distributions. Parent: Distribution (mathematical analysis). Intended file: distribution-theory.bigb.

For distributions $u$ on ${\mathbb{R}}^m$ and $v$ on ${\mathbb{R}}^n$, their tensor product acts on a joint test function by $⟨u\otimes v,\Phi ⟩=⟨u_x,⟨v_y,\Phi (x,y)⟩⟩$. The inner function is smooth with compact support in the projection of $\mathrm{supp}\Phi$, and finite order of a distribution estimates prove continuity. Reversing the pairings gives the same value: finite sums of product kernels are dense in the required smooth seminorms, and the two orders agree on product kernels. Its support is contained in the Cartesian product of the two supports. This provides a rigorous replacement for formal integration in separate distribution variables.

A locally integrable function $f$ defines a distribution by $⟨u_f,\phi ⟩=\int f\phi$. The pairing is bilinear if distributions are taken complex-linear. This identifies ordinary functions with a subset of distributions without claiming that every distribution is a function.

The order of a distribution on a compact set is the least nonnegative integer $m$ for which its action is bounded by a constant times the sup norms of test-function derivatives through order $m$.

A distribution can have finite order of a distribution on every compact set without having one order bound valid on the whole domain. For example, the locally finite sum ${\sum }_{j\ge 1}{\partial }^j{\delta }_j$ on the real line has orders growing without bound near the integers.

An order-zero distribution is locally bounded by the sup norm of its test function. Such distributions are precisely the distributions represented locally by signed or complex Radon measures.

A Hadamard finite-part integral removes explicit divergent terms from a truncated integral and retains the finite limit. For example, the one-sided finite part of $H(x)/x$ can be defined byfor each test function $\varphi$. The logarithmic subtraction cancels the divergence at zero. The locally integrable function $L(x)=H(x)\log x$, interpreted as zero for $x<0$, has exactly this distributional derivative: integrate $-{\int }_0^{\infty }\log x{\varphi }^{\prime }(x)dx$ by parts with lower cutoff $ϵ$.

This is distinct from the symmetric principal-value reciprocal distribution, which is the derivative of $\log |x|$ on the whole line. More singular reciprocal powers require additional Taylor polynomial subtractions. The subtraction and normalization must be specified, because distributions supported at the singular point can distinguish different extensions.

Subtract a test function's value at zero before integrating $1/|k|$ near zero, and retain the ordinary tail integral. This defines an even tempered distribution. Changing the subtraction cutoff adds a multiple of the Dirac delta distribution. Multiplication by $k$ gives $\mathrm{sgn}k$.

For $m\ge 2$, the symmetric Hadamard finite-part integral defining ${\Lambda }_m$ subtracts the Taylor polynomial through degree $m-2$ from a test function before division by $x^m$. It has finite order of a distribution and satisfiesThe lower-order subtraction terms are integrable at infinity, while the remaining odd $1/x$ singularity cancels under symmetric truncation.

A sequence of distributions $u_j$ converges weakly to $u$ when $⟨u_j,\phi ⟩\to ⟨u,\phi ⟩$ for every test function $\phi$.

If $f\in L^1(0,1)$ has mean zero, $j$ is a positive integer and $\varphi \in \mathcal{D}(0,1)$, then for every integer $N\ge 1$,Here the overline denotes the periodic extension, not complex conjugation. Let $h_N=P^{N}f$ use periodic primitives with zero mean. Repeated integration by parts gives the signed identityThere are no boundary terms, and integration over exactly $j$ periods gives $\Vert {\overline{h}}_N(j\cdot ){\Vert }_1=\Vert h_N{\Vert }_1\le \Vert f{\Vert }_1$. This proves the estimate. If zero-mean $g_j$ satisfy $\Vert g_j{\Vert }_1=O(j^m)$ for fixed real $m\ge 0$, choose an integer $N>m$ to obtain ${\overline{g}}_j(j\cdot )\to 0$ in weak convergence of distributions.

A sequence can converge as distributions off one point while retaining a nonconvergent multiple of a Dirac delta function at that point. If $u_j=v+c_j{\delta }_p+o_{{\mathcal{D}}^{\prime }}(1)$ with $c_j$ nonconvergent, restriction to the punctured domain tends to $v$, but a test nonzero at $p$ detects the obstruction to global convergence.

For $a>0$, polar-coordinate reduction and integration by parts give $m\sin (m||x{|}^2-a|)=2\delta (|x{|}^2-a)-\pi \cos (ma){\delta }_0+o_{{\mathcal{D}}^{\prime }}(1)$ in two dimensions. The circle term comes from the cusp in the folded phase, while the origin term is a radial endpoint contribution. A punctured-domain limit therefore need not extend across the quadratic critical point.

If ${\tau }_{h}f(x)=f(x-h)$ for functions, its extension to distributions is defined by

With translation of a distribution defined by $⟨{\tau }_{h}u,\phi ⟩=⟨u,\phi (\cdot +h)⟩$, differentiation of the test function gives $\frac{d}{dh}{\tau }_{h}u=-{\tau }_h{u}^{\prime }$ in weak convergence of distributions. The Taylor theorem shows convergence in the space of test functions, including a common compact support. Thus $({\tau }_{-h}u-u)/h\to u^{\prime }$; fixing the translation convention prevents a sign error.

A distribution on ${\mathbb{R}}^n$ is invariant under every translation parallel to the $i$th coordinate axis exactly when its $i$th distributional derivative vanishes. Differentiate the translated pairing for one direction; integrate that derivative in the translation parameter for the converse.