Distribution

1. Definition of distribution

Without explanation, we always assume that

Ω \subset R^{n}

is a nonempty open set, where

n ⩾ 1

. Given the compact set

K \subset R^{n}

, we use

C_{K}^{\infty} (Ω)

to represent the set of smooth functions supported by

K \subset Ω

, i.e

C_{K}^{\infty} (Ω) = {f \in C^{\infty} (Ω) ∣ supp (f) \subset K} .

Given a multiple indicator

α = (α_{1}, \dots, α_{n})

, where each component is a nonnegative integer, and the symbol

\partial^{α} φ

represents the following multiple partial derivatives

\partial_{x_{1}}^{α_{1}} \partial_{x_{2}}^{α_{2}} \dots \partial_{x_{n}}^{α_{n}} φ,

where

\partial_{x_{i}}^{α_{i}} = \frac{\partial^{α_{i}}}{\partial x_{i}^{α_{i}}}

. In addition, we order

| α | = \sum_{i = 1}^{n} α_{i} .

We use

D (Ω) = C_{0}^{\infty} (Ω)

represents a set of smooth functions defined on

Ω

and with compact support. We call it {\bf test function space}. By definition, for any

φ \in D (Ω)

, there is a compact set

K \subset Ω

(this is the compact set in

R^{n}

), so that

{f |}_{Ω - K} \equiv 0

, that is, for any

X \in Ω - K, f (x) = 0

Definition 1(Test function space): On the space

D (Ω)

, we specify the following convergence (topology): given the function sequence

{φ_{p}}_{p ⩾ 1} \subset D (Ω)

, the so-called sequence converges to

0 \in D (Ω)

(is recorded as φ_{p} \overset{D (Ω)}{⟶} 0)

, which refers to

There is a compact set of $K \subset Ω$ , so that for each $p ⩾ 1$ , there is $\sup (φ_{p}) \subset K$ ;
For each multiple indicator $α$ , function sequence ${\partial^{α} φ_{p}}_{p ⩾ 1}$ converge to $0$ uniformly on $K$ , i.e., $lim_{p \to \infty} {∥ \partial^{α} φ_{p} ∥}_{L^{\infty} (K)} = 0.$

Definition 2(distribution) A distribution (also known as generalized function) on

Ω

refers to a linear functional (linear mapping) on

D (Ω)

u : D (Ω) \to C, φ \mapsto ⟨ u, φ ⟩,

meet the following two conditions

For any $φ, ψ \in D (Ω)$ and $α, β \in C$ , we have $⟨ u, α φ + β ψ ⟩ = α ⟨ u, φ ⟩ + β ⟨ u, ψ ⟩$
For any compact set $K \subset Ω$ , there are nonnegative integers $p$ and normal numbers $C$ ( $p$ and $C$ depend on $K$ ), so that for any $φ \in C_{K}^{\infty} (Ω)$ , both $| ⟨ u, φ ⟩ | ⩽ C sup_{| α | ⩽ p} {∥ \partial^{α} φ ∥}_{L^{\infty} (K)}$

If the above selection of

p

does not depend on the selection of compact set

k

, then we call the smallest such nonnegative integer

p

the order of distribution

u

We use

D^{'} (Ω)

to represent the whole distribution defined on

Ω

, and specify the following convergence (topology): the so-called distribution sequence

{u_{p}}_{p ⩾ 1} \subset D^{'} (Ω)

converges to

0 \in D^{'} (Ω)

(this is a linear mapping that maps all functions to 0), which is recorded as

u_{p} \overset{D^{'} (Ω)}{⟶} 0

, which means that we have any test function

φ \in D (Ω)

lim_{p \to \infty} ⟨ u_{p}, φ ⟩ = 0

We usually say that a distribution can be paired with a (compactly supported) smooth function to obtain a number, that is

D^{'} (Ω) \times D (Ω), (u, φ) \mapsto ⟨ u, φ ⟩

Remark: Any given distribution of

u \in D^{'} (Ω)

, which is actually a {\bf continuous} linear functional on

D (Ω)

. We have not discussed the theory of topological linear space related to distribution in detail, so we do not intend to expand this point too much (which has no impact on the understanding of distribution theory). The so-called continuity can be described in the following sequence language: for any test function sequence

φ_{p} \overset{D}{⟶} 0

, we all have

lim_{p \to \infty} ⟨ u, φ_{p} ⟩ = 0

The proof of this proposition only needs to use the concept of distribution.

2. Examples of distribution

As an example, let's first learn some important distributions:

Example 1 (Dirac function)

For any

a \in Ω

, we can define the distribution

Δ_{a} \in D^{'} (Ω)

. Where, for any

φ \in D (Ω)

, we define

⟨ δ_{a}, φ ⟩ = φ (a) .

Let's verify

Δ_{a}

is actually a distribution:

For any compact set

K \subset Ω

, if

a \notin K

, then for any

φ \in C_{K}^{\infty} (Ω)

, we all have

⟨ δ_{a}, φ ⟩ = 0 .

a \in K

, then it makes any

φ \in C_{K}^{\infty} (Ω)

, we have

| ⟨ δ_{a}, φ ⟩ | = | φ (a) | ⩽ 1 \cdot sup_{| α | ⩽ 0} {∥ \partial^{α} φ ∥}_{L^{\infty} (K)} .

Therefore, we can take

q = 0

and

C = 1

in the definition of distribution.

In particular, we also know The order of

δ_{a}

0

Example 2 (locally integrable function)

Given the open set

Ω

(always equipped with Borel algebra and Lebesgue measure), the so-called locally integrable function refers to the function integrable on each compact local, that is, the function

f

(the equivalent class of the corresponding almost everywhere equal function). For any compact set

K \subset Ω

, the function

f \cdot 1_{K} \in L^{1} (Ω)

. We use

L_{l o c}^{1} (Ω)

represents a locally integrable function on

Ω

For any

f \in L_{l o c}^{1} (Ω)

, we define the linear functional on

D (Ω)

T_{f} : D (Ω) \to C, φ \mapsto ⟨ T_{f}, φ ⟩ = \int_{Ω} f (x) φ (x) d x .

Proposition 1 For given

χ (x) \in D (R^{n})

, it is assumed that

\int_{R^{n}} χ (x) d x = 1.

For any

ε > 0

, we define

χ_{ε} (x) = \frac{1}{ε^{n}} χ (\frac{x}{ε}) .

Then we have, in distrubution sense,

χ_{ε} \overset{D^{'}}{⟶} δ_{0}

ε \to 0

Distribution

1. Definition of distribution

2. Examples of distribution

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