Convergence in distribution (weak convergence)

[christoff]

Homework Statement

Consider the function f_n(x)=n\cdot I\left[|x|<\frac{1}{2n}\right] considered as a distribution in D'(\mathbb{R}), where I denotes the indicator function. Recall that f_n converges to \delta_0, the delta distribution, in D'(\mathbb{R}). Show that f_n^2-n\delta_0 converges in D'(\mathbb{R}).

The Attempt at a Solution

By a few calculations, we can show that for any test function \phi and natural n>0, that
(f_n^2,\phi)=n(f_n-\delta_0,\phi).
At this point, I would be tempted to say that this converges to zero (as a sequence of real numbers) because (f_n-\delta_0,\phi) "must" dominate the convergence of n to infinity. However, this isn't rigorous, and is more of a hunch than anything. However, any attempts I've made to prove it have not gone anywhere. I feel as though it must have something to do with the smoothness of the test functions, but I can't see where to incorporate this.

At this point, I would appreciate a nudge in the right direction.
Thanks in advance!

 

[Ray Vickson]

Homework Statement

Consider the function f_n(x)=n\cdot I\left[|x|<\frac{1}{2n}\right] considered as a distribution in D'(\mathbb{R}), where I denotes the indicator function. Recall that f_n converges to \delta_0, the delta distribution, in D'(\mathbb{R}). Show that f_n^2-n\delta_0 converges in D'(\mathbb{R}).

The Attempt at a Solution

By a few calculations, we can show that for any test function \phi and natural n>0, that
(f_n^2,\phi)=n(f_n-\delta_0,\phi).
At this point, I would be tempted to say that this converges to zero (as a sequence of real numbers) because (f_n-\delta_0,\phi) "must" dominate the convergence of n to infinity. However, this isn't rigorous, and is more of a hunch than anything. However, any attempts I've made to prove it have not gone anywhere. I feel as though it must have something to do with the smoothness of the test functions, but I can't see where to incorporate this.

At this point, I would appreciate a nudge in the right direction.
Thanks in advance!

Try using the second-order Taylor expansion of $\phi(x)$ in the evaluation of $(f_n^2,\phi)$.

 

[christoff]

Yep, that worked. Thanks! It's actually very easy once you apply the Taylor expansion :)