14.2 Distributions and generalized functions

Distributions are powerful mathematical tools that extend the concept of functions. They allow us to work with objects like the Dirac delta, which aren't traditional functions but are crucial in physics and engineering.

Operations on distributions, like derivatives and convolutions, open up new ways to solve complex problems. These techniques are especially useful in differential equations, helping us tackle issues that stumped classical methods.

Distributions and Their Properties

Properties of distributions

• Distributions are continuous linear functionals that map test functions (smooth, compactly supported functions) to real or complex numbers
• A distribution $T$ is continuous if for every sequence of test functions $\phi_n$ converging to $\phi$, $T(\phi_n)$ converges to $T(\phi)$
• The derivative of a distribution $T$ is defined by $\langle T', \phi \rangle = -\langle T, \phi' \rangle$ for every test function $\phi$, with higher-order derivatives defined iteratively
• A sequence of distributions $T_n$ converges to a distribution $T$ if for every test function $\phi$, $\langle T_n, \phi \rangle$ converges to $\langle T, \phi \rangle$

Distributions vs classical functions

• Locally integrable functions $f$ define distributions $T_f$ by $\langle T_f, \phi \rangle = \int f(x) \phi(x) dx$
• Regular distributions are defined by locally integrable functions
• Singular distributions are not defined by locally integrable functions (Dirac delta function and its derivatives)

Operations on Distributions and Applications

Operations on distributions

• The derivative of a distribution is defined by duality: $\langle T', \phi \rangle = -\langle T, \phi' \rangle$, which is a continuous operation on the space of distributions
• The convolution of two distributions $S$ and $T$ is defined by $\langle S * T, \phi \rangle = \langle S(x), \langle T(y), \phi(x+y) \rangle \rangle$, which is associative and commutative
• The Fourier transform of a distribution $T$ is defined by $\langle \mathcal{F}(T), \phi \rangle = \langle T, \mathcal{F}(\phi) \rangle$, which is a continuous, linear, and bijective operation on the space of tempered distributions

Applications in differential equations

• Distributions can be used to find fundamental solutions to linear partial differential equations, which solve the PDE with a Dirac delta function as the source term
• Green's functions are fundamental solutions to linear PDEs with specific boundary conditions and can be used to solve inhomogeneous boundary value problems
• Distributions provide a rigorous framework for dealing with discontinuities and singularities in physical systems, such as modeling point masses and charge distributions