Applied linear operators and spectral methods/Weak convergence

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