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 $ℝ$, i.e.,

$⟨t_n,\varphi ⟩\to ⟨t,\varphi ⟩\mathrm{\forall }\varphi \in D\text{as}b\to \mathrm{\infty }$

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

For example,

$t_n:=⟨\sin (nx),\varphi ⟩=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\sin (nx)\varphi (x)dx\to 0\text{as}n\to \mathrm{\infty }$

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

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

$⟨t{\text{'}}_n,\varphi ⟩=-⟨t_n,{\varphi }^{\prime }⟩\to -⟨t,{\varphi }^{\prime }⟩=⟨t^{\prime },\varphi ⟩$

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

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