8.1 Test functions and distributions

Test functions and distributions form the backbone of generalized function theory. These concepts extend classical functions to include objects like the Dirac delta, crucial in physics and engineering. They provide a rigorous framework for dealing with discontinuous or singular phenomena.

Distributions are defined as continuous linear functionals on test functions, which are smooth and compactly supported. This approach allows for a unified treatment of functions, measures, and more exotic objects, enabling powerful techniques in analysis and differential equations.

Test Functions and Distributions

Defining Test Functions and Distributions

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• Test functions are smooth, compactly supported functions used to define distributions
  • Smooth functions have continuous derivatives of all orders
  • Compactly supported functions vanish outside a bounded set
• Distributions are continuous linear functionals on the space of test functions
  • Linearity property: $\langle T, \alpha f + \beta g \rangle = \alpha \langle T, f \rangle + \beta \langle T, g \rangle$
  • Continuity property: if $f_n \to f$ in the space of test functions, then $\langle T, f_n \rangle \to \langle T, f \rangle$
• Generalized functions extend the notion of functions to include objects like the Dirac delta function
  • Generalized functions are defined by their action on test functions
  • Example: the Dirac delta function $\delta$ is defined by $\langle \delta, f \rangle = f(0)$

Spaces of Test Functions and Distributions

• The space of test functions is denoted by $\mathcal{D}(\mathbb{R}^n)$ or $C_c^\infty(\mathbb{R}^n)$
  • Equipped with a topology that makes it a complete, metrizable, locally convex space
  • Example: $\mathcal{D}(\mathbb{R})$ consists of smooth functions with compact support on the real line
• The space of distributions is the dual space of $\mathcal{D}(\mathbb{R}^n)$, denoted by $\mathcal{D}'(\mathbb{R}^n)$
  • Distributions are continuous linear functionals on $\mathcal{D}(\mathbb{R}^n)$
  • The action of a distribution $T$ on a test function $f$ is denoted by $\langle T, f \rangle$

Properties of Distributions

Support and Regularity of Distributions

• The support of a distribution $T$ is the smallest closed set $\operatorname{supp}(T)$ such that $\langle T, f \rangle = 0$ for all test functions $f$ with $\operatorname{supp}(f) \cap \operatorname{supp}(T) = \emptyset$
  • Intuitively, the support is the set where the distribution "lives"
  • Example: the support of the Dirac delta function $\delta$ is $\{0\}$
• Regular distributions are distributions that can be represented by locally integrable functions
  • A distribution $T$ is regular if there exists a locally integrable function $f$ such that $\langle T, \varphi \rangle = \int_{\mathbb{R}^n} f(x) \varphi(x) dx$ for all test functions $\varphi$
  • Example: any continuous function defines a regular distribution

Singular Distributions and Operations on Distributions

• Singular distributions are distributions that cannot be represented by locally integrable functions
  • Example: the Dirac delta function $\delta$ is a singular distribution
• Operations on distributions are defined by duality
  • Differentiation: $\langle \partial^\alpha T, \varphi \rangle = (-1)^{|\alpha|} \langle T, \partial^\alpha \varphi \rangle$
  • Multiplication by a smooth function $\psi$: $\langle \psi T, \varphi \rangle = \langle T, \psi \varphi \rangle$
  • Convolution: $\langle T * \varphi, \psi \rangle = \langle T, \varphi * \psi \rangle$, where $(\varphi * \psi)(x) = \int_{\mathbb{R}^n} \varphi(x-y) \psi(y) dy$

Special Distributions

The Dirac Delta Function

• The Dirac delta function $\delta$ is a singular distribution defined by $\langle \delta, f \rangle = f(0)$ for all test functions $f$
  • $\delta$ is not a function in the classical sense, but a generalized function
  • It can be thought of as a unit mass concentrated at the origin
• Properties of the Dirac delta function:
  • $\int_{\mathbb{R}} \delta(x) f(x) dx = f(0)$ for any continuous function $f$
  • $\delta(ax) = \frac{1}{|a|} \delta(x)$ for any non-zero constant $a$
  • $\delta(x-a)$ is the Dirac delta function shifted by $a$, defined by $\langle \delta(x-a), f(x) \rangle = f(a)$
• The Dirac delta function has numerous applications in physics and engineering
  • Modeling point masses, point charges, or impulse forces
  • Sampling signals or functions at specific points
  • Representing Green's functions for differential equations