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,

  $(p.v.\frac{1}{x}(\varphi )={\lim }_{ϵ\to 0^{+}}{\int }_{|x|\ge 3}\frac{\varphi (x)}{x}dx$ for all $\varphi \in S(R)$

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