Applied linear operators and spectral methods/Weak convergence

Convergence of distributions

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Definition:

A sequence of distributions   is said to converge to the distribution   if their actions converge in  , i.e.,

 

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

For example,

 

Therefore,   converges to   as  , in the weak sense of distributions.

If   if follows that the derivatives   will converge to   since

 

For example,   is both a sequence of functions and a sequence of distributions which, as  , converge to 0 both as a function (i.e., pointwise or in  ) or as a distribution.

Also,   converges to the zero distribution even though its pointwise limit is not defined. Template:Lectures