Distribution theory

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.

Restriction of a distribution ( $u ∣_{V}$ )

For open sets

V \subseteq U

and

u \in D^{'} (U)

, define

⟨ u ∣_{V}, ϕ ⟩ = ⟨ u, ϕ ⟩

, where

ϕ

extends the test function

ϕ \in 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.

Tensor product of distributions ( $u \otimes v$ )

For distributions

u

R^{m}

and

v

R^{n}

, their tensor product acts on a joint test function by

⟨ u \otimes v, Φ ⟩ = ⟨ u_{x}, ⟨ v_{y}, Φ (x, y)⟩⟩

. The inner function is smooth with compact support in the projection of

supp Φ

, 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.

Regular distribution ( $⟨ u_{f}, φ ⟩ = \int f φ$ )

A locally integrable function

f

defines a distribution by

⟨ u_{f}, φ ⟩ = \int f φ

. 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.

Order of a distribution

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

Distribution of unbounded order

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 \geq 1} \partial^{j} δ_{j}

on the real line has orders growing without bound near the integers.

Order-zero distribution

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.

Hadamard finite-part integral

A Hadamard finite-part integral subtracts the divergent terms of a truncated integral and retains its finite limit. It defines distributions such as the extension of

1/ x

obtained by differentiating

lo g x_{+}

Finite-part inverse-absolute-value distribution ( $Pf (1/∣ k ∣)$ )

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

sgn k

Hadamard finite-part reciprocal-power distribution ( $Λ_{m} = Fp (x^{- m})$ )

For

m \geq 2

, the symmetric Hadamard finite-part integral defining

Λ_{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 satisfies

Λ_{m} = \frac{( - 1 ) ^{m - 1}}{( m - 1 )!} (\frac{d}{d x})^{m} lo g ∣ x ∣.

The lower-order subtraction terms are integrable at infinity, while the remaining odd

1/ x

singularity cancels under symmetric truncation.

Weak convergence of distributions

A sequence of distributions

u_{j}

converges weakly to

u

when

⟨ u_{j}, φ ⟩ \to ⟨ u, φ ⟩

for every test function

φ

Rapid distributional decay of periodic oscillations ( $\int_{0}^{1} \overline{f} (j x) ϕ (x) d x \leq j^{- N} ∥ f ∥_{1} ∥ ϕ^{(N)} ∥_{\infty}$ )

f \in L^{1} (0, 1)

has mean zero,

j

is a positive integer, and

ϕ \in D (0, 1)

, the displayed estimate holds for every integer

N \geq 1

. Let

h_{N} = P^{N} f

use periodic primitives with zero mean. Repeated integration by parts gives

\int_{0}^{1} \overline{f} (j x) ϕ (x) d x = j^{- N} \int_{0}^{1} \overline{h}_{N} (j x) ϕ^{(N)} (x) d x .

There are no boundary terms, and integration over exactly

j

periods gives

∥ \overline{h}_{N} (j \cdot) ∥_{L^{1} (0, 1)} = ∥ h_{N} ∥_{L^{1}} \leq ∥ f ∥_{L^{1}}

. This proves the estimate. In particular, if zero-mean

g_{j}

satisfy

∥ g_{j} ∥_{L^{1}} = O (j^{m})

, their periodic extensions satisfy

\overline{g}_{j} (j \cdot) \to 0

in weak convergence of distributions by choosing

N = m + 1

Target: Essential closure. Parent: Lebesgue measure. Intended file: measure-theory.bigb.

Oscillating point-mass defect ( $u_{j} = v + c_{j} δ_{p} + o_{D^{'}} (1)$ )

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} δ_{p} + o_{D^{'}} (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.

Radial quadratic oscillation defect in two dimensions ( $2 δ (∣ x ∣^{2} - a) - π cos (ma) δ_{0}$ )

For

a > 0

, polar-coordinate reduction and integration by parts give

m sin (m ∣∣ x ∣^{2} - a ∣) = 2 δ (∣ x ∣^{2} - a) - π cos (ma) δ_{0} + o_{D^{'}} (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.

Translation of a distribution ( $τ_{h} u$ )

τ_{h} f (x) = f (x - h)

for functions, its extension to distributions is defined by

⟨ τ_{h} u, φ ⟩ = ⟨ u, φ (\cdot + h)⟩ .

Differentiability of distribution translations ( $\frac{d}{d h} τ_{h} u = - τ_{h} u^{'}$ )

With translation of a distribution defined by

⟨ τ_{h} u, φ ⟩ = ⟨ u, φ (\cdot + h)⟩

, differentiation of the test function gives

\frac{d}{d h} τ_{h} u = - τ_{h} u^{'}

in weak convergence of distributions. The Taylor theorem shows convergence in the space of test functions, including a common compact support. Thus

(τ_{- h} u - u) / h \to u^{'}

; fixing the translation convention prevents a sign error.

Translation invariance and vanishing distributional derivative

A distribution 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.

Space of smooth functions ( $E (X)$ )

For an open set

X \subseteq R^{n}

, the space

E (X) = C^{\infty} (X)

carries the Fréchet topology in which

f_{j} \to f

exactly when every derivative converges uniformly on every compact set contained in

X

Compactly supported distribution space ( $E^{'} (X)$ )

The continuous dual

E^{'} (X)

E (X)

is the space of compactly supported distributions on

X

. Its weak topology is pointwise convergence on

E (X)

Test function ( $C_{c}^{\infty}$ )

A test function is a smooth function with compact support. Weak and distributional equations are identities obtained by integrating against every test function.

Cutoff function

A cutoff function is a smooth function that equals one on a chosen inner set and vanishes outside a slightly larger set. On concentric balls

B_{r} \subset B_{R}

, it can be chosen with

∣ Dη ∣ \leq C / (R - r)

Space of test functions ( $D (U)$ )

The space of test functions is

D (U) = C_{c}^{\infty} (U)

. A sequence converges in

D (U)

when all its supports eventually lie in one compact subset of

U

and every derivative converges uniformly there.

Test-function inductive limit topology ( $D (X) = lim_{K} D_{K} (X)$ )

The space of test functions has the locally convex inductive-limit topology of the spaces of smooth functions supported in a fixed compact set

K

. Each

D_{K}

uses seminorms

p_{m} (φ) = max_{∣ α ∣ \leq m} ∥ \partial^{α} φ ∥_{\infty}

. Sequential convergence means a common compact support and uniform convergence of every derivative. A linear functional on this space is a distribution exactly when its restriction to each

D_{K}

satisfies a finite order of a distribution estimate; the order and constant may depend on

K

Density of test functions in distributions

Every distribution is a limit of test functions in weak convergence of distributions. With a mollifier

ρ_{ε}

and an expanding smooth cutoff function

χ (x / j)

, the functions

χ (x / j) (u * ρ_{1/ j}) (x)

have compact support and converge to

u

. For each fixed test, the cutoff eventually equals one on its support, while the reflected mollifier approximates that test with all derivatives in a common compact set.

Locally integrable function ( $L_{loc}^{1} (U)$ )

A function is locally integrable when its absolute value has finite integral over every compact subset of its domain. It defines a regular distribution by integration against test functions.

Distributional identity

A distributional identity is an equality that holds after both sides act on every smooth compactly supported test function.

Dirac delta function

 2  0

The Dirac delta is the distribution defined by

⟨ δ, φ ⟩ = φ (0)

Long-time resonance kernel ( $sin (t x) / x \to π δ (x)$ )

t \to + \infty

, the displayed limit holds in distribution theory. Indeed

sin (t x) / x = \frac{1}{2} \int_{- t}^{t} e^{i ux} d u

and the Fourier representation of the Dirac delta function gives the limit on smooth rapidly decaying test functions. For a regular detuning

Δ (J)

, the coarea formula reduces its use under an integral over action variables to this one-dimensional identity. The kernel grows like

t

at exact resonance, so this is not pointwise convergence.

Lorentzian resonance approximation ( $γ / (x^{2} + γ^{2}) \to π δ (x)$ )

For

γ > 0

, the kernels

γ / [π (x^{2} + γ^{2})]

have unit integral and concentrate at zero as

γ \to 0^{+}

. Splitting the integral into a neighborhood of zero and its complement proves convergence against bounded continuous test functions. Thus

[(k v - ω)^{2} + γ^{2}]^{- 1} ≃ π δ (k v - ω) /∣ γ ∣

under an integral, with

δ (k v - ω) = δ (v - ω / k) /∣ k ∣

. This is the narrow-width limit of a Cauchy distribution, not a pointwise replacement away from the peak.

Dirac comb ( $\sum_{j \in Z} δ (x - j D)$ )

A Dirac comb is a periodic train of Dirac delta distributions. With Fourier transform convention

f (q) = \int f (x) e^{- i q x} d x

and period

D > 0

F [\sum_{j \in Z} δ (x - j D)] (q) = \frac{2 π}{D} \sum_{m \in Z} δ (q - \frac{2 πm}{D}) .

The identity holds as a tempered distribution. Truncating the real-space train by a finite window convolves the reciprocal train with the transform of that window, broadening its peaks. Convolving the train with one slit's transmission instead multiplies the reciprocal peaks by the slit's field transform, the basic construction of a diffraction grating pattern.

Folded sine approximation to a Dirac delta ( $m sin (m ∣ t ∣) \to 2 δ_{0}$ )

As distributions on the line,

m sin (m ∣ t ∣) \to 2 δ_{0}

. Pairing with a test function

A

folds the integral to

\int_{0}^{\infty} m sin (m s) [A (s) + A (- s)] d s

. One integration by parts yields

2 A (0)

plus a cosine integral tending to zero by the Riemann-Lebesgue lemma. Pointwise convergence is unnecessary.

Surface delta distribution ( $δ_{S}$ )

For a smooth hypersurface

S

, its surface delta is the distribution

⟨ δ_{S}, φ ⟩ = \int_{S} φ d S

. Its normalization uses geometric surface measure rather than a defining function. On the unit circle,

δ_{S^{1}}

pairs by integrating

φ (cos t, sin t)

over

0 \leq t \leq 2 π

Spherical surface measure convolution ( $u_{a} * u_{b}$ )

For the geometric surface delta distribution on the radius-

a

sphere in

R^{3}

⟨ u_{a}, ϕ ⟩ = a^{2} \int_{S^{2}} ϕ (aω) d σ (ω)

. Angular integration gives the Fourier transform

u_{a} (ξ) = 4 πa sin (a ∣ ξ ∣) /∣ ξ ∣

, with removable value

4 π a^{2}

at zero. The convolution of distributions with a compactly supported factor gives the regular distribution

u_{a} * u_{b} = \frac{2 πab}{∣ x ∣} 1_{{∣ a - b ∣ < ∣ x ∣ < a + b}} .

This density has total mass

16 π^{2} a^{2} b^{2}

. Values on endpoint spheres do not affect the distribution. When

a = b

, the inverse-distance singularity is locally integrable in three dimensions and is not an additional point mass. The annulus expresses the triangle inequality for the sum of two vectors of fixed lengths.

Level-set normalization of a surface delta ( $δ_{S} = ∣\nabla g ∣ δ (g)$ )

g

is smooth and

\nabla g \neq = 0

S = {g = 0}

, transverse local coordinates give

⟨ δ (g), φ ⟩ = \int_{S} φ /∣\nabla g ∣ d S

. Thus

δ_{S} = ∣\nabla g ∣ δ (g)

as distributions. In particular

δ_{S^{1}} = 2 δ (∣ x ∣^{2} - 1)

. Omitting this Jacobian confuses the delta of a defining function with a geometric surface delta distribution.

Fourier representation of the Dirac delta function ( $δ (x) = \frac{1}{2 π} \int_{- \infty}^{\infty} e^{ik x} d k$ )

The identity

δ (x) = \frac{1}{2 π} \int_{- \infty}^{\infty} e^{ik x} d k

is interpreted distributionally: symmetric frequency cutoffs give

sin (n x) / (π x)

, which converge to the Dirac delta function.

Delta derivatives from polynomial oscillatory amplitudes ( $\int e^{i s θ} θ^{j} d θ = 2 π i^{- j} δ^{(j)} (s)$ )

With inverse Fourier transform normalization

(2 π)^{- 1}

\int e^{i s θ} θ^{j} d θ = 2 π i^{- j} δ^{(j)} (s)

as an oscillatory integral. The factor

i^{- j}

and the sign convention

⟨ δ^{'}, g ⟩ = - g^{'} (0)

are essential. For instance, amplitude

- i x_{2} θ / (2 π)

with phase

x_{1} θ

gives

- x_{2} δ^{'} (x_{1})

Distributional derivative of the Heaviside step function

For every test function

φ

⟨ H^{'}, φ ⟩ = - \int_{0}^{\infty} φ^{'} (t) d t = φ (0),

H^{'} = δ

as distributions.

Distributional jump formula for a Heaviside product ( $D^{k} (H u) = H u^{(k)} + \sum_{s = 0}^{k - 1} u^{(k - 1 - s)} (0) δ_{0}^{(s)}$ )

For a smooth function

u

, the distributional derivative of the product of

u

with the Heaviside function is

D (H u) = H u^{'} + u (0) δ_{0}

. Induction gives the displayed formula. It records all boundary sources, including derivatives of the Dirac delta distribution, and determines which initial derivatives must vanish in a retarded fundamental solution.

Dirac delta scaling

For nonzero

a

δ (a x) = δ (x) /∣ a ∣

in the distributional sense.

Derivative of the Dirac delta ( $δ^{'}$ )

The derivative of the Dirac delta is the distributional derivative defined by

⟨ δ^{'}, φ ⟩ = - φ^{'} (0)

for every test function

φ

Multi-index notation ( $α = (α_{1}, \dots, α_{n})$ )

A multi-index is a tuple of nonnegative integers used to abbreviate products and partial derivatives:

∣ α ∣ = \sum_{j} α_{j}

x^{α} = \prod_{j} x_{j}^{α_{j}}

, and

D^{α} = \prod_{j} D_{j}^{α_{j}}

Distributional derivative

For a distribution

u

and a multi-index

α

, its distributional derivative is defined by

⟨ D^{α} u, φ ⟩ = (- 1)^{∣ α ∣} ⟨ u, D^{α} φ ⟩ .

Jump contribution to a piecewise weak derivative ( $D u = u_{pieces}^{'} + [u]_{a} δ_{a}$ )

For a function smooth on either side of a joining point

a

, integration by parts on the two intervals adds a Dirac mass whose coefficient is the right trace minus the left trace. If the one-sided derivatives are locally integrable, a first weak derivative can be represented by a locally integrable function across the join only when the jump vanishes. Derivative jumps themselves do not obstruct first-order Sobolev membership: a continuous piecewise Lipschitz function still has a bounded weak first derivative.

Distributional second derivative across a jump ( $D^{2} y = y_{reg}^{''} + [y^{'}]_{c} δ_{c} + [y]_{c} δ_{c}^{'}$ )

Let

y

be piecewise twice continuously differentiable, with finite one-sided values and derivatives at

c

, and write

[y]_{c} = y (c +) - y (c -)

. Its distributional derivative is

Dy = y_{reg}^{'} + [y]_{c} δ_{c}

, and differentiating again gives

D^{2} y = y_{reg}^{''} + [y^{'}]_{c} δ_{c} + [y]_{c} δ_{c}^{'} .

Here

δ_{c}

is the Dirac delta at

c

, and the subscript reg means the ordinary derivative on the two open intervals, interpreted as a regular distribution. To prove the first formula, split

- \int y φ^{'}

c

and use integration by parts on each piece for a test function

φ

. The two boundary contributions add to

[y]_{c} φ (c)

. Applying the same argument to the regular first derivative proves the second formula. Thus a derivative jump produces a delta source, whereas a value jump produces the derivative of a delta source.

Continuity of distributional differentiation ( $u_{j} \to u in D^{'} \Rightarrow D^{α} u_{j} \to D^{α} u$ )

u_{j} \to u

in weak convergence of distributions, then every fixed distributional derivative converges too. For each test function

ϕ

, its derivative

D^{α} ϕ

is another test function, and

⟨ D^{α} u_{j}, ϕ ⟩ = (- 1)^{∣ α ∣} ⟨ u_{j}, D^{α} ϕ ⟩ ⟶ (- 1)^{∣ α ∣} ⟨ u, D^{α} ϕ ⟩ .

Thus a series converging as distributions can be differentiated term by term in that topology. For example, uniform convergence on compact sets implies convergence of the associated regular distributions, even when the ordinary series of derivatives does not converge pointwise.

Target: Even-weight principal-value distribution. Parent: Principal-value reciprocal distribution. Intended file: distribution-theory.bigb.

Distributional derivative of the logarithmic modulus ( $\frac{d}{d x} lo g ∣ x ∣ = pv (1/ x)$ )

The locally integrable function

lo g ∣ x ∣

has distributional derivative equal to the principal-value reciprocal distribution. Symmetric integration by parts makes the boundary term

lo g ε [φ (ε) - φ (- ε)]

tend to zero.

A distribution with zero derivative is constant

v^{'} = 0

R

, then

v

is a constant distribution. Every test function of integral zero is the derivative of its compactly supported primitive, so

v

vanishes on those test functions. Choosing one test function

η

of integral one gives

⟨ v, φ ⟩ = ⟨ v, η ⟩ \int φ

. Iteration shows that a distribution with

m

th derivative zero is a polynomial of degree at most

m - 1

Weak derivative

A locally integrable function

g

is the

i

th weak derivative of

u

when

\int_{U} u D_{i} φ = - \int_{U} g φ

for every test function

φ \in C_{c}^{\infty} (U)

. Thus a weak derivative is a distributional derivative that is represented by a locally integrable function.

Continuous weak gradient gives a C1 representative ( $D_{j} u \in C ⟹ u = U a.e. with U \in C^{1}$ )

If a locally integrable function has weak derivatives represented by a continuous vector field

g

, it has a representative

U \in C^{1}

with

\nabla U = g

. The assertion is local, so work on a ball compactly contained in the domain and regularize using a mollifier. The regularizations satisfy the segment identity

u_{ε} (x) - u_{ε} (y) = \int_{0}^{1} \nabla u_{ε} (y + t (x - y)) \cdot (x - y) d t

. Averaging over a smaller ball in

y

fixes the additive constant through local

L^{1}

convergence. The gradients converge uniformly on compact sets, so the averaged identity gives local uniform convergence to

U

. Passing to the limit on each segment then gives

U (x + h) - U (x) - g (x) \cdot h = o (∣ h ∣)

. Continuity of

g

proves that

U

is continuously differentiable.

Bounded isolated-point removability for weak first derivatives ( $n \geq 2$ )

A bounded function which is

C^{1}

away from one point in dimension

n \geq 2

, and whose classical gradient is locally integrable across that point, has that gradient as its weak derivative. Cut a test function off within distance

ε

of the point. The error from differentiating the cutoff is bounded by

C ∥ u ∥_{\infty} ε^{n - 1}

and vanishes. The other terms converge by local integrability. In dimension one the Heaviside step function is a counterexample: its classical derivative off the point is zero, but its distributional derivative is a Dirac delta.

Locality of weak differentiability

A locally integrable function has weak derivatives on an open set exactly when it has them near every point. Local derivatives agree almost everywhere on overlaps because they define the same distributional derivative. A countable local cover therefore gives globally defined, locally integrable derivative functions. A smooth partition of unity subordinate to the cover splits each compactly supported test function; summing the local integration identities yields the global weak-derivative identity.

Weak gradient ( $\nabla u$ )

The weak gradient is the vector of first-order weak derivatives of a function. Its

i

th component is specified by

\int u \partial_{i} φ = - \int (\partial_{i} u) φ

for compactly supported test functions. For a differentiable function with appropriate local integrability it agrees with the classical gradient.

Sobolev chain rule ( $D (Φ (u)) = Φ^{'} (u) D u$ )

u \in W_{loc}^{1, 2}

and

Φ

is continuously differentiable with bounded derivative on the range in use, then

D (Φ (u)) = Φ^{'} (u) D u

almost everywhere. One proves this first for smooth functions and then uses density of smooth functions in a Sobolev space and boundedness of

Φ^{'}

. On compact sets one may truncate

Φ

away from the range of a bounded

u

Gradient of a Sobolev function vanishes on a level set ( $D u = 0 a.e. on {u = t}$ )

u \in W_{loc}^{1, 2}

, then

D u = 0

almost everywhere on every level set

{u = t}

. Choose smooth truncations

Φ_{ε} (s)

with

∣ Φ_{ε} ∣ \leq 2 ε

, derivative one near zero, bounded derivative, and derivative zero outside

(- 2 ε, 2 ε)

. The Sobolev chain rule gives

D Φ_{ε} (u - t) \to 1_{{u = t}} D u

in local

L^{2}

by dominated convergence theorem, while

Φ_{ε} (u - t) \to 0

. The limiting weak derivative is therefore zero.

Multiplication of a distribution by a smooth function

For a distribution

u

and a smooth function

a

, the product

a u

is defined by

⟨ a u, φ ⟩ = ⟨ u, a φ ⟩

. It obeys the Leibniz rule; in one dimension,

(a u)^{'} = a^{'} u + a u^{'}

Distribution annihilated by a function with simple zeros

If a smooth function

a

R

has finitely many zeros

r_{j}

, all simple, then

a w = 0

exactly when

w = \sum_{j} c_{j} δ_{r_{j}}

. Away from the zeros, divide a test function by the nonvanishing

a

to show

w

vanishes. Near

r_{j}

, write

a (x) = (x - r_{j}) b_{j} (x)

with smooth nonzero

b_{j}

. The kernel of multiplication by a coordinate, after translation and multiplication by

b_{j}

, says that the local distribution is a multiple of

δ_{r_{j}}

. A partition of unity gives the global sum. Simplicity of the zeros excludes derivatives of deltas.

Kernel of multiplication by a coordinate

In one dimension,

x w = 0

if and only if

w = c δ_{0}

. Choose a cutoff function

η = 1

near zero and write every test function as

φ = φ (0) η + x ψ

. The pairing with

x ψ

vanishes, leaving

⟨ w, φ ⟩ = φ (0) ⟨ w, η ⟩

Dirac delta multiplication identity

For smooth

a

and a point

r

a δ_{r} = a (r) δ_{r}, a δ_{r}^{'} = a (r) δ_{r}^{'} - a^{'} (r) δ_{r} .

These follow by applying both sides to a test function and using the Leibniz rule. In particular

(x - r) δ_{r}^{'} = - δ_{r}

Principal-value reciprocal distribution ( $pv (1/ x)$ )

The Cauchy principal value

⟨ pv \frac{1}{x}, φ ⟩ = lim_{ε ↓ 0} \int_{∣ x ∣ > ε} \frac{φ ( x )}{x} d x

defines a distribution. Near zero the odd constant contribution cancels, and the remaining numerator is

O (x)

. It satisfies

x pv (1/ x) = 1

. Hence all solutions of

xv = 1

are

v = pv (1/ x) + c δ_{0}

by the kernel of multiplication by a coordinate.

Even-weight principal-value distribution ( $pv (f (x) / x)$ )

For a continuous even function

f

R

, the Cauchy principal value

⟨ pv \frac{f ( x )}{x}, ϕ ⟩ = \int_{0}^{\infty} f (x) \frac{ϕ ( x ) - ϕ ( - x )}{x} d x

defines a distribution of order of a distribution at most one. For test functions supported in

[- R, R]

, the absolute value is at most

2 R max_{[0, R]} ∣ f ∣ ∥ ϕ^{'} ∥_{\infty}

, since the numerator is bounded by

2 x ∥ ϕ^{'} ∥_{\infty}

. When

ϕ (0) = 0

, the expression equals the absolutely convergent ordinary integral

\int f ϕ / x

It is a regular distribution exactly when

\int_{0}^{r} ∣ f (x) ∣/ x d x < \infty

for some

r > 0

. Necessity follows by testing away from zero: any representing locally integrable function must equal

f / x

almost everywhere off zero. Sufficiency follows from absolute convergence. Continuity with

f (0) = 0

alone does not suffice:

f (x) = 1/ lo g (1/∣ x ∣)

near zero gives the divergent integral

\int_{1}^{\infty} d s / s

Target: L1-normalized dilation of a distribution. Parent: Dilation of a distribution. Intended file: distribution-theory.bigb.

Dilation of a distribution ( $δ_{t} u$ )

For

t > 0

, the dilation

δ_{t} u

of a distribution on

R^{n}

is defined by

⟨ δ_{t} u, φ ⟩ = t^{- n} ⟨ u, φ (\cdot / t)⟩ .

For a regular distribution this agrees with the function

x \mapsto u (t x)

L1-normalized dilation of a distribution ( $⟨ u_{ε}, ϕ ⟩ = ⟨ u, ϕ (ε \cdot)⟩$ )

For

ε > 0

and

u \in D^{'} (R^{n})

, define

u_{ε}

by the displayed pairing. If

u

is a regular distribution, this agrees with

u_{ε} (x) = ε^{- n} u (x / ε)

and preserves its L1 space norm. With the convention

δ_{t} u (x) = u (t x)

for dilation of a distribution, it equals

ε^{- n} δ_{1/ ε} u

u \in L^{1} (R^{n})

, the dominated convergence theorem gives

⟨ u_{ε}, ϕ ⟩ = \int u (y) ϕ (ε y) d y ⟶ ϕ (0) \int u, u_{ε} \to (\int u) δ_{0} .

The constant distribution

u = 1

instead gives

u_{ε} = ε^{- n}

and has no limit in weak convergence of distributions.

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

Support law for distributional dilation

For

t > 0

, the support of a distribution satisfies

supp (δ_{t} u) = t^{- 1} supp u .

The duality formula

⟨ δ_{t} u, φ ⟩ = t^{- n} ⟨ u, φ (\cdot / t)⟩

proves one inclusion by transporting test functions; applying the inverse dilation proves the other. For a compact support, the Paley–Wiener–Schwartz theorem gives another proof:

δ_{t} u (ζ) = t^{- n} u (ζ / t)

rescales the exponential growth indicator by

1/ t

Homogeneous distribution

A distribution

u

is homogeneous of degree

σ

when

δ_{t} u = t^{σ} u

for every

t > 0

. With the unnormalized angular-frequency convention, its Fourier transform is homogeneous of degree

- n - σ

Fourier transform of a homogeneous distribution

The scaling property of the Fourier transform extends by duality to

δ_{t} u = t^{- n} δ_{1/ t} u .

Consequently a homogeneous distribution of degree

σ

transforms into one of degree

- n - σ

Fourier transform of a reciprocal positive quadratic form ( $F [(x^{T} G x)^{- 1}] (ξ) = \frac{2 π ^{2}}{det G ξ ^{T} G ^{- 1} ξ}$ )

For a real symmetric positive-definite three-dimensional matrix, the reciprocal quadratic form is locally integrable and defines a regular tempered distribution. With the Fourier kernel

e^{- i x \cdot ξ}

, radial Abel regularization gives

F (∣ x ∣^{- 2}) = 2 π^{2} /∣ ξ ∣

. The linear change of variables

y = G^{1/2} x

gives the displayed formula. No principal value or origin-supported correction is required. The result is homogeneous of degree

- 1

on the frequency side.

Accretive complex quadratic reciprocal Fourier transform ( $F [(x^{T} A x)^{- 1}] (ξ) = \frac{2 π ^{2}}{d ( A ) ξ ^{T} A ^{- 1} ξ}$ )

For a complex symmetric three-dimensional

A = G + i B

with real symmetric

B

and positive-definite

G

, the reciprocal is a regular tempered distribution since

∣ x^{T} A x ∣ \geq x^{T} G x

. Holomorphic continuation of the real quadratic transform proves the displayed expression. The determinant branch is the analytic determinant square root for accretive symmetric matrices. The other root has positive real part because

Re (ξ^{T} A^{- 1} ξ) = ξ^{T} G^{- 1/2} (1 + C^{2})^{- 1} G^{- 1/2} ξ > 0

for nonzero real

ξ

. Both the original and transformed singularities are locally integrable, so the continued identity holds as distributions at the origin too.

Riesz potential ( $I_{α}$ )

For

0 < α < n

, the Riesz potential is convolution with a constant multiple of

∣ x ∣^{α - n}

. Its Fourier multiplier is a constant multiple of

∣ ξ ∣^{- α}

Riesz kernel

The locally integrable homogeneous function

x \mapsto ∣ x ∣^{- α}

0 < α < n

, is a Riesz kernel. For

f (ξ) = \int e^{- i x \cdot ξ} f (x) d x

∣ x ∣^{- α} (ξ) = 2^{n - α} π^{n /2} \frac{Γ (( n - α ) /2 )}{Γ ( α /2 )} ∣ ξ ∣^{α - n} .

Hedberg inequality

1 < p < n / α

, the pointwise Hedberg inequality is

I_{α} ∣ f ∣ (x) \leq C_{n, p, α} ∥ f ∥_{p}^{α p / n} (M f (x))^{1 - α p / n} .

Split the integral at radius

r

. Dyadic annuli bound the near part by

C r^{α} M f (x)

, while Hölder's inequality bounds the far part by

C ∥ f ∥_{p} r^{α - n / p}

. Choosing

r^{n / p} = ∥ f ∥_{p} / M f (x)

balances the terms.

Mollifier

A standard mollifier is a nonnegative test function

ρ

with integral one. The rescaling

ρ_{ε} (x) = ε^{- n} ρ (x / ε)

forms an approximation to the identity, and

u * ρ_{ε}

is smooth wherever the convolution stays inside the domain.

Mollification ( $u_{δ} = u * ρ_{δ}$ )

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.

Mollification of a divergence-forcing equation ( $Δ u_{σ} = div (b u)_{σ} + (c u)_{σ} + f_{σ}$ )

The distributional equation

Δ u = div (b u) + c u + f

becomes the smooth interior equation

Δ u_{σ} = div (b u)_{σ} + (c u)_{σ} + f_{σ}

. The products are convolved as wholes. Only coefficient bounds on compact interior sets are needed, and the divergence-forcing interior H1 estimate provides a uniform first derivative bound.

Support of a distribution ( $supp u$ )

The support of a distribution is the complement of the largest open set on which its pairing vanishes for every supported test function. A partition of unity shows that the union of such open sets still has this vanishing property. For example, the Dirac delta distribution and all its derivatives have support

{0}

Compactly supported distribution

A distribution has compact support when it vanishes on every test function supported outside some compact set. Every compactly supported distribution has finite order.

Negative Sobolev regularity of a compactly supported distribution ( $u \in H^{s}, s < - M - n /2$ )

A compactly supported distribution of order of a distribution at most

M

has a smooth function as its Fourier transform, bounded by

C ⟨ ξ ⟩^{M}

. It therefore belongs to every Sobolev space

H^{s}

with

s < - M - n /2

Fourier transform of a compactly supported distribution

For

u \in E^{'} (R^{n})

, its Fourier transform is the smooth function

u (λ) = ⟨ u (x), e^{- iλ \cdot x} ⟩ .

It has at most polynomial growth, and its extension to complex frequency is controlled more precisely by the Paley–Wiener–Schwartz theorem.

Shrinking-cutoff exponential-type estimate ( $∣ u (z) ∣ \leq C (1 + ∣ z ∣)^{M} e^{H_{K} (Im z)}$ )

Let a distribution have support in a compact convex set

K

. Fixed neighborhood cutoffs initially give an exponential bound larger than the support function

H_{K}

. Cutoffs of width

ε

cost powers of

ε^{- 1}

in derivative estimates but enlarge the exponential by only

e^{Cε ∣ Im z ∣}

. Choosing

ε = (1 + ∣ z ∣)^{- 1}

absorbs the cutoff cost into a polynomial and retains the exact exponential

e^{H_{K} (Im z)}

Fourier decay in a bounded complex strip

For a test function

φ

supported in a fixed ball of radius

R

and a fixed height bound

H

, repeated integration by parts gives

∣ φ (ξ + i η) ∣ \leq C_{R, H, L} max_{∣ α ∣ \leq 2 L} ∥ \partial^{α} φ ∥_{\infty} (1 + ∣ ξ ∣^{2})^{- L}, ∣ η ∣ \leq H .

Apply

(1 - Δ)^{L}

to the compactly supported smooth function

e^{x \cdot η} φ (x)

inside the real-frequency Fourier integral. The bounded

η

absorbs its derivative and exponential factors into the constant. This estimate makes the numerator of a Hörmander staircase integral absolutely integrable and makes truncated contour endpoints vanish.

Structure theorem for compactly supported distributions

Every compactly supported distribution is a finite sum of distributional derivatives of bounded continuous functions. One proof convolves it with a sufficiently high-order Bessel potential and then applies a power of

1 - Δ

Compact continuous-derivative representation of a distribution ( $u = \sum_{α} \partial^{α} f_{α}$ )

Every compactly supported distribution on an open set

X

is a finite sum

u = \sum_{α} \partial^{α} f_{α}

with continuous

f_{α}

compactly supported inside

X

. One can first write

u = \partial^{γ} F

for a continuous global convolution primitive, then multiply by a cutoff function equal to one near the support. The identity

χ \partial^{γ} F = \sum_{β \leq γ} (- 1)^{∣ β ∣} (β γ) \partial^{γ - β} [(\partial^{β} χ) F]

localizes every coefficient. Finite global representation can fail for a distribution of unbounded order.

Bessel potential

The Bessel potential of order

s

is the Fourier multiplier

(1 + ∣ ξ ∣^{2})^{- s /2}

. Sufficiently high order turns a compactly supported finite-order distribution into a bounded continuous function.

Fundamental solution of a linear differential operator

A fundamental solution of a linear differential operator

P (D)

is a distribution

E

satisfying

P (D) E = δ_{0}

Cauchy kernel fundamental solution ( $\partial_{\overset{z}{ˉ}} (1/ (π z)) = δ_{0}$ )

The locally integrable function

E (z) = 1/ (π z)

on the plane is a fundamental solution for the Wirtinger derivative

\partial_{\overset{z}{ˉ}} = (\partial_{x} + i \partial_{y}) /2

. Let

L = (4 π)^{- 1} lo g (x^{2} + y^{2})

be the fundamental solution of the Laplace equation. Its first distributional derivatives agree with its ordinary ones: deleting a radius-

ε

disk in integration by parts gives an error

O (ε ∣ lo g ε ∣)

, tending to zero. Hence

E = 4 \partial_{z} L

and

\partial_{\overset{z}{ˉ}} E = 4 \partial_{\overset{z}{ˉ}} \partial_{z} L = Δ L = δ_{0} .

The analogous fundamental solution for

\partial_{z}

1/ (π \overset{z}{ˉ})

. The singular support of each kernel is exactly

{0}

: they are smooth elsewhere, and smoothness near zero would make their Wirtinger derivatives smooth there, contradicting the Dirac delta distribution identity.

Target: Distributional Cauchy-Riemann lemma. Parent: Cauchy-Riemann equations. Intended file: analysis.bigb.

Malgrange–Ehrenpreis theorem

Every nonzero constant-coefficient linear partial differential operator has a distributional fundamental solution.

Hörmander staircase

Let

P (ξ^{'}, z)

be a polynomial that is monic in the complex variable

z

. A Hörmander staircase partitions the real

ξ^{'}

-space into measurable sets

Δ_{j}

and assigns bounded heights

c_{j}

so that

P (ξ^{'}, s + i c_{j})

stays uniformly away from zero for

ξ^{'} \in Δ_{j}

and

s \in R

. Integrating

1/ P

over the resulting union of horizontal contours constructs a fundamental solution of a linear differential operator.

Finite-height polynomial root avoidance

Suppose

P (ξ^{'}, z)

has degree

m

z

and a nonzero constant leading coefficient

a

. Among the

m + 1

heights

h_{ℓ} = 3 ℓ

, at least one satisfies

∣ P (ξ^{'}, s + i h_{ℓ}) ∣ \geq ∣ a ∣ for all s \in R

at each real

ξ^{'}

. Factor into

m

linear factors with multiplicity. Each root's imaginary part can be within distance less than one of at most one candidate height. The pigeonhole principle leaves a height whose distance from every root is at least one, proving the product bound. The height can be chosen measurably: the set where a candidate succeeds is the closed intersection of

∣ P (ξ^{'}, q + i h_{ℓ}) ∣ \geq ∣ a ∣

over rational

q

, and selecting the first successful candidate gives a Borel set partition. This avoids any need for continuous global root labels.

Retarded fundamental solution

A retarded fundamental solution is supported in a chosen forward region, such as

[0, \infty)

for an ordinary differential operator.

Retarded Fourier pole prescription ( $e^{- iω t} : ω \mapsto ω + i 0$ )

For inverse time Fourier transform

\int G (ω) e^{- iω t} d ω / (2 π)

, a causal stable oscillator has its poles below the real frequency axis. Undamped real poles are therefore displaced by the prescription

ω \mapsto ω + i 0

in its frequency polynomial. For

t > 0

, closing the contour below encloses poles with clockwise orientation; for

t < 0

, closing above gives zero. Changing the transform's sign reverses the corresponding prescription. For a positive oscillator operator

\partial_{t}^{2} + ω_{0}^{2}

, the inverse is

[ω_{0}^{2} - (ω + i 0)^{2}]^{- 1}

and its time response is

Θ (t) sin (ω_{0} t) / ω_{0}

Retarded fundamental solution of a constant-coefficient ordinary differential operator ( $E = H u, P (D) E = δ_{0}$ )

For

P (z) = a_{N} z^{N} + \dots + a_{0}

with

N \geq 1

and

a_{N} \neq = 0

, the unique retarded fundamental solution is

E = H u

, where

P (D) u = 0

u^{(j)} (0) = 0

for

j < N - 1

, and

u^{(N - 1)} (0) = 1/ a_{N}

. The distributional jump formula for a Heaviside product gives

P (D) E = δ_{0}

. Its explicit expression is

u (x) = \sum_{P (λ) = 0} Res_{z = λ} \frac{e ^{z x}}{P ( z )} .

The residue theorem gives the initial derivatives from the coefficients at infinity. Equivalently, the inverse Fourier transform is integrated below all its poles, producing support of a distribution in

[0, \infty)

. Uniqueness follows because the difference of two retarded solutions is an analytic homogeneous solution vanishing on the negative half-line.

Elliptic differential operator

A constant-coefficient operator of order

N

is elliptic when its homogeneous principal symbol

P_{N} (ξ)

is nonzero for every real

ξ \neq = 0

Strict ellipticity ( $ξ^{T} A (x) ξ > 0 (ξ \neq = 0)$ )

For a real symmetric matrix field

A

, strict ellipticity means that

A (x)

is a positive-definite matrix at every point. Continuity makes its smallest eigenvalue positive uniformly on each compact subset of the domain. A uniform positive lower bound on the entire domain is the stronger uniform ellipticity condition. Some conventions include a finite uniform upper bound in the latter term, while others state coefficient boundedness separately.

Elliptic system of differential equations ( $det A_{N} (ξ) \neq = 0 (ξ \neq = 0)$ )

A square system whose entries have common maximal order

N

is elliptic when its matrix principal symbol is invertible at every nonzero real frequency. Unlike scalar operators, a system can be first-order elliptic in three dimensions: the Pauli matrices give

A (ξ) = \sum_{j} σ_{j} ξ_{j}

with

A (ξ)^{2} = ∣ ξ ∣^{2} 1

Odd-order obstruction for scalar elliptic operators in at least three dimensions ( $n \geq 3 \Rightarrow N even$ )

The principal symbol of an odd-order scalar differential operator is an odd map from the frequency sphere to the complex plane. The Borsuk-Ulam theorem forces a zero on a two-dimensional sphere, contradicting ellipticity. Restriction to three variables proves the same obstruction in every greater dimension.

High-frequency lower bound for an elliptic polynomial ( $∣ P (ξ) ∣ ≳ ⟨ ξ ⟩^{N}$ )

A scalar elliptic differential operator of order

N

has a principal symbol nonzero on the unit sphere. Compactness gives a positive minimum there, and homogeneity makes that term dominate all lower-degree terms at sufficiently large real frequencies.

Elliptic regularity

Elliptic regularity says that solutions of an elliptic equation gain derivatives wherever the coefficients and forcing are regular. For the Laplacian, local estimates typically control two derivatives of

u

Δ u

together with a lower-order norm of

u

Reaction-diffusion regularity from Laplacian regularity ( $∥ z ∥_{H^{2}} \leq 2 C ∥ e ∥_{L^{2}}$ )

Suppose the mixed boundary value problem

Δ w = g

, with zero Dirichlet boundary condition on a nonempty boundary portion and zero Neumann boundary condition on the remainder, has an

H^{2}

solution with

∥ w ∥_{H^{2}} \leq C ∥ g ∥_{L^{2}}

. Assume the domain is connected and that zero Sobolev trace on the specified portion excludes nonzero constants. Then the same boundary conditions for

- Δ z + z = e

give

∥ z ∥_{H^{2}} \leq 2 C ∥ e ∥_{L^{2}}

Indeed, the Lax-Milgram theorem first constructs

z

in the Sobolev space with a partial Dirichlet condition; testing with

z

shows

∥ z ∥_{H^{1}} \leq ∥ e ∥_{L^{2}}

. Apply the assumed elliptic regularity result to

g = z - e

. Its solution

w

and

z

satisfy the same weak Laplace equation, so testing their difference shows

\nabla (w - z) = 0

. The Sobolev function with zero weak gradient property and zero Sobolev trace give

w = z

. Finally

∥ z - e ∥_{L^{2}} \leq 2∥ e ∥_{L^{2}}

. This argument supplies the dual regularity needed by the Aubin–Nitsche duality argument for the equation with a positive reaction term.

Interior second-derivative estimate for the Poisson equation ( $∥ u ∥_{H^{2} (U)} \leq C (∥ u ∥_{H^{1} (V)} + ∥Δ u ∥_{L^{2} (V)})$ )