Distribution Theory

Table of Contents

1. Test Function

  • The space of test function is \(D(\mathbb{R}^n) = C_{c}^\infty(\mathbb{R}^n)\) equipped with the notion of convergence.
    • \(C_c^\infty(U)\) with \(U\subset \mathbb{R^n}\), is a dense subset of \(C^\infty(U)\), such that for all \(f\in C_c^\infty(U)\) have compact support.

1.1. Convergence

  • \[ \varphi_k \stackrel{\mathcal{D}}{\to} \varphi \]
  • \(\exists \text{ bounded set }M\colon \forall k, x\not\in M\implies \varphi_k(x) = 0.\)
  • \(\varphi_k \to \varphi\text{ unifromly}\) and \(\forall \alpha, D^\alpha \varphi_k \to D^\alpha\varphi\text{ uniformly}\) where \(\alpha\) is the multi-index.

2. Distribution

2.1. Definition

  • A distribution \(T\) on \(U\) is a linear functional on \(C^\infty_c(U)\), that is continuous when \(C^\infty_c(U)\) is given a topology called the canonical LF topology.
  • \(T\colon \mathcal{D}(\mathbb{R}^n) \to \mathbb{R}\) with
    • Linearity
    • Continuity
      • Sequentially continuous with respect to the convergence of the test functions. \[ \varphi_k\stackrel{D}{\to} \varphi \implies T(\varphi_k) \to T(\varphi). \]

2.2. Multiplication

  • Product of two distributions is also a distribution: \[ (f\cdot S)(\varphi) = (T_f\cdot S)(\varphi) := S(f\cdot \varphi) \] where \(S\in \mathcal{D}'(\mathbb{R}^n), f\in C^\infty(\mathbb{R}^n)\).
  • Equivalently: \[ \langle f\cdot S, \varphi\rangle := \langle S, f\cdot\varphi\rangle. \]
  • Motivated by \(T_f\cdot T_g = T_{f\cdot g}\).

2.3. Coordinate Transformation

  • For a invertible linear map \( A\colon \mathbb{R}^n \to \mathbb{R}^n \): \[ \langle T\circ A, \varphi\rangle := \langle T, |\det A|^{-1}\varphi\circ A^{-1}\rangle. \]
  • With different notation: \[ \langle T(Ax), \varphi(x)\rangle := \langle T(x), |\det A|^{-1}\varphi(A^{-1}x)\rangle. \]

2.4. Convolution

  • Extension of convolution.
  • \[ \langle \psi\ast T, \varphi\rangle := \langle T, \check{\psi}\ast \varphi\rangle \] where the check operator simply reverses the argument: \(\check{\psi}(x) = \psi(-x)\).
  • The convolution \(*\colon \mathcal{D}(\mathbb{R}^n)\times \mathcal{D}'(\mathbb{R}^n) \to \mathcal{D}'(\mathbb{R}^n)\) is a bilinear map, which can be thought of as a multiplication.
  • Notably, \[ \psi \ast \delta = \psi. \]

3. Notation

  • \(T(x)\), although not correct, may be used to denote the variable that the distribution is evaluated by.

4. Properties

4.1. Continuity

  • For every compact subset \(K\subseteq U\), there exists constants \(C > 0\) and \(N\in \mathbb{N}\) such that for all \(f\in C_c^\infty(U)\) with support contained in \(K\): \[ |T(f)| \le C\sup\{|\partial^\alpha f(x)| : x\in U, |\alpha| \le N\}. \]
  • For every compact subset \(K\subseteq U\) an every sequence \(\{f_i\}_{i=0}^\infty\) in \(C^\infty_c(U)\) whose supports are contained in \(K\): If \(\{\partial^\alpha f_i\}_{i=1}^\infty\) converges uniformly to zero on \(U\) for every multi-index \(\alpha\), then \(T(f_i) \to 0\).

4.2. Vector Space

  • Distribution can be added and scalar multiplied.
  • Distributions forms a vector space equipped with the bilinear map: \[ \langle \cdot, \cdot \rangle\colon \mathcal{D}'(\mathbb{R}^n)\times \mathcal{D}(\mathbb{R}^n) \to \mathbb{R} (\text{or $\mathbb{C}$}). \]

5. Dirac Delta Distribution

  • Dirac delta funciton, \(\delta\) Distribution, Unit Impulse

A generalized function with the property of: \[ \int_R f(x)\delta(x)\,dx=f(0). \]

5.1. Definition

5.1.1. As a Measure

  • Dirac measure \(\delta\) over \(\mathbb{R}\) satisfies:
    • \(0\in A\subset \mathbb{R} \implies \delta(A) = 1\), and otherwise \(\delta(A) = 0\).
  • \[ \int_\mathbb{R} f(x)\delta(x)\,dx := \int_\mathbb{R}f(x) \delta(dx) \] in terms of Lebesgue integral.

5.1.2. As a Distribution

  • A distribution \(\delta\) such that: \[ \delta[\varphi] = \varphi(0). \]

5.2. Properties

  • Even: \(\delta(x) = \delta(-x)\)
  • Dirac delta \(\delta(x-y)\) is the continuous version of the Kronecker delta \(\delta_{ij}\)

5.3. Nascent Delta Function

  • The family of function \(\eta_\varepsilon (x)\) such that: \[ \lim_{\varepsilon\to 0+}\eta_\varepsilon(x) = \delta(x). \]

6. Regularity

  • A distribution \(T\in \mathcal{D}'(\mathbb{R}^n)\) is regular, if \[ \exists f \in L_{1, \rm loc}(\mathbb{R}^n)\colon T(\varphi) = \int_{\mathbb{R}^n}f(x)\varphi(x)\,dx. \]

7. Distributional Derivative

  • For a distribution \(T\in \mathcal{D}'(\mathbb{R}^n)\), the distributional partial derivative \(D^\alpha T\in \mathcal{D}'(\mathbb{R}^n)\) of \(T\) is: \[ \langle D^\alpha T, \varphi\rangle = (-1)^{|\alpha|}\langle T, D^\alpha \varphi\rangle \] where \(\alpha\) is the multi-index.
  • Motivated by \(\langle T_{f'}, \varphi\rangle = -\langle T_f, \varphi'\rangle.\)
  • The differential operator \(D^\alpha\colon \mathcal{D}'(\mathbb{R}^n)\to \mathcal{D}'(\mathbb{R}^n)\) is linear and continuous.
    • \[ D^\alpha \left(\sum_{k=1}^\infty T_k\right) = \sum_{k=1}^\infty D^\alpha T_k. \]
    • The linearity is enough for finite sum, but continuity is required for the infinite sum.
  • The fundamental solution of a differential equation can be obtained as a distribution.
  • For all multi-indices \(\alpha\): \[ D^\alpha (\psi\ast T) = (D^\alpha \psi)\ast T = \psi\ast (D^\alpha T). \]
  • Given a fundamental solution \(\mathcal{P}(D)E = \delta\), the partial differential equation \(\mathcal{P}(D)u = f\) is solved by \(f\ast E\), as long as \(f\) is one of the test function.

8. Reference

Created: 2025-05-19 Mon 01:30