Distributions

Distributions, also known as Schwartz Distributions are

• The product of a distribution by a smooth function cannot be extended to an associative product on the space of distributions.

  Example:
  If \(p.v.\frac{1}{x}\) is the distribution obtained by the Cauchy Principal Value,

  \((p.v.\frac{1}{x}(\phi)=\lim_{\epsilon\to0^+}\int_{|x|\geq3}{\frac{\phi(x)}{x}dx}\) for all \(\phi \in S(R)\)

  If \(\delta\) is the Dirac delta distribution, then
  \((\delta\times x)\times p.v.\frac{1}{x}=0\)
  but,
  \(\delta\times (x\times p.v.\frac{1}{x})=\delta\)