Applied linear operators and spectral methods/Weak convergence

< Applied linear operators and spectral methods

Convergence of distributions

Definition:

A sequence of distributions $\{t_n\}$ is said to converge to the distribution $t$ if their actions converge in $\mathbb{R}$ , i.e.,

\left\langle t_n, \phi \right\rangle \rightarrow \left\langle t, \phi \right\rangle \qquad \forall~\phi \in D \quad \text{as}~b \rightarrow \infty

This is called convergence in the sense of distributions or weak convergence.

For example,

t_n := \left\langle \sin(nx), \phi \right\rangle = \int_{-\infty}^{\infty} \sin(nx)~\phi(x)~dx \rightarrow 0 \qquad \text{as}~~ n \rightarrow \infty

Therefore, $\{\sin(nx)\}$ converges to $\sin(0)$ as $n \rightarrow \infty$ , in the weak sense of distributions.

If $\{t_n\} \rightarrow t$ if follows that the derivatives $\{t'_n\}$ will converge to $t'$ since

\left\langle t'_n, \phi \right\rangle = -\left\langle t_n, \phi' \right\rangle \rightarrow -\left\langle t, \phi' \right\rangle = \left\langle t', \phi \right\rangle

For example, $\{t_n\} = \left\{\cfrac{\cos(nx)}{n}\right\}$ is both a sequence of functions and a sequence of distributions which, as $n \rightarrow \infty$ , converge to 0 both as a function (i.e., pointwise or in $L^2$ ) or as a distribution.

Also, $\{t'_n\} = \{-\sin(nx)\}$ converges to the zero distribution even though its pointwise limit is not defined.

Template:Lectures

This article is issued from Wikiversity - version of the Wednesday, June 04, 2008. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.