.png){kind=link}
 - Vertical.png){kind=link}
.png){kind=link}
 - Vertical.png){kind=link}
.svg){kind=link}
.png){kind=link}
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,
\(\delta\times (x\times p.v.\frac{1}{x})=\delta\)