Does Weak Convergence Hold for Sequences Approaching Infinity?

tom_rylex
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Homework Statement


Show that if {x_k} is any sequence of points in space R^n with |{x_k}| \rightarrow \infty, then \delta(x-x_k) \rightarrow 0 weakly


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The Attempt at a Solution


I'm still trying to grasp the concept of weak convergence for distributions. It would appear that this function doesn't converge pointwise. The distribution on a test function is
\int \delta(x-x_k)\theta(x)dx = \theta(x_k) Does the function converge weakly to zero because x_k approaches infinity, and therefore would be outside of the region of support of any locally integrable test function?
 
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I think that's correct. But what exactly is your set of test functions?
 
My set of test functions meet the following criteria:
* function has a finite region of support, inside of which \theta(x) \neq 0, outside of which \theta(x)=0
* \theta(x) has derivatives of all orders.
 
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Then you are right. Finite support is enough.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...

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