Applied linear operators and spectral methods/Weak convergence

Convergence of distributions

Definition:

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

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

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

For example,

${\displaystyle t_{n}:=\left\langle \sin(nx),\phi \right\rangle =\int _{-\infty }^{\infty }\sin(nx)~\phi (x)~dx\rightarrow 0\qquad {\text{as}}~~n\rightarrow \infty }$

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

If ${\displaystyle \{t_{n}\}\rightarrow t}$ if follows that the derivatives ${\displaystyle \{t'_{n}\}}$ will converge to ${\displaystyle t'}$ since

${\displaystyle \left\langle t'_{n},\phi \right\rangle =-\left\langle t_{n},\phi '\right\rangle \rightarrow -\left\langle t,\phi '\right\rangle =\left\langle t',\phi \right\rangle }$

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

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