Distributions

Posted December 16th, 2007 by Structure

Distributions are continuous linear applications on different test functions.
Usually test functions are in the space $\mathcal{C}_{0}^{\infty}(R^n)$ with topology $\mathcal{T}$ defined by convergent sequence, as follows
A sequence $(\phi_n)_{n\in N},\phi\in\mathcal{C}_{0}^{\infty}(R^n)$ is convergent to $\phi$ if and only if there is a compact $K\subset R^n$ such as for all $n\in N,supp \phi_n\subset K$ and for all $\alpha=(\alpha_1,\alpha_2,...\alpha_n)$ ,

$\partial^{\alpha}\phi_n=\frac{\partial^{|\alpha|}\phi_n}{\partial x_1^\alpha_1\partial x_2^\alpha_2...\partial x_n^\alpha_n} \:is \:uniform \:convergent \:to \:\:\partial^{\alpha}\phi$

We write $\mathcal{D}(R^n)$ for the space $(\mathcal{C}_{0}^{\infty}(R^n),\mathcal{T})$
There is a useful equivalent condition for a linear map on $\mathcal{C}_{0}^{\infty}(R^n)$ to be a distribution:
Theorem. A linear map u on $\mathcal{C}_{0}^{\infty}(R^n)$ is a distribution if and only if
$\forall K\subset R^n$ , there are constants $M>0 \:and \: k\in N$ such as $\forall \phi\in\mathcal{D}_K(R^n)$ we have

$|<u,\phi>|\leq M \sum_{|\alpha|\leq k}sup_{x\in K}|\partial^{\alpha}\phi(x)|$

The minimal integer k in this definition is called the order of distribution on K.
An important example of distribution is Dirac $\delta_{a}$ defined by

$<\delta_{a},\phi>=\phi(a)$

Dirac $\delta_{a}$ is a distribution of order zero.

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