## New answers tagged limits-and-convergence

0
votes

### A limit involving the largest prime factor of a prime gap

I'm sorry in advance for potential mistakes. I don't have an answer to the question, but I might have some basic result that is not contradictory with the simulation you carried out.
Let $n$ be a ...

5
votes

### Does weak $H^1$ convergence imply $L^2$ convergence when multiplied with an exponentially decaying function?

Yes. Indeed, we have $R^2f_n\to R^2f_*$ in $L^2$. I take the integration domain to be $\mathbb R^d$, i.e., $L^2=L^2(\mathbb R^d)$ and $H^1=H^1(\mathbb R^d)$. We assume that $R(x)^2\leq Ce^{-\alpha\|x\|...

0
votes

Accepted

### A complex question related to a certain convergence of Lévy measures

At first, we consider an example.
Let
\begin{gather*}
f(x)=\frac{I_{\{x>0\}}(x)}{2x^2(1\vee x^2)}
=\frac{I_{\{(0,1)\}}(x)}{2x^2} + \frac{I_{\{[1,\infty)\}}(x)}{2x^4},\\
\nu(\mathrm{d}x)=...

2
votes

### Nature of $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $

You might try to regularize the sum,
$$S(\alpha)=\sum_{n=1}^\infty \frac{ \cos(\alpha n) \sin(n+1) }{n}=
-\tfrac{1}{4} i \left[e^{-i} \ln \left(1-e^{i (\alpha-1)}\right)+e^{-i} \ln \left(1-e^{-i (\...

0
votes

### Does weak-* convergence in $W^{1,\infty}$ imply weak-* convergence in $L^\infty$?

I was wondering about the same question and was surprised that I could not find an answer to this anywhere. The answer to your question is yes.
Let $(X, \|\cdot\|)$ be a normed vector space and $Y \...

2
votes

### Weak convergence of measures on continuous function spaces

$\newcommand{\sgn}{\operatorname{sgn}}\newcommand{\ep}{\varepsilon}$Here is an elementary proof that $\mu_r$ converges weakly (as $r\to\infty$) the measure $\mu$ that is the distribution of the ...

2
votes

### Weak convergence of measures on continuous function spaces

These measures do converge weakly to a measure with two atoms (of equal probabilities $1/2$) at the two paths $\phi_{\pm}:t\mapsto \pm t$. One way to see it is to consider $\frac{1}{r}B_t$, so that ...

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