Applied linear operators and spectral methods/Weak convergence

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Convergence of distributions[edit | edit source]


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