All Questions
13 questions
0
votes
1
answer
106
views
Convergence of mollified functions in weighted $L^p$ norm
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\supp}{\operatorname{supp}}
$
Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a ...
- 5,005
6
votes
2
answers
503
views
Computing a limit on the unit sphere: Riemann Lebesgue?
Let $u\in L^1(\mathbb{S}^{d-1})$. I want to show that
\begin{align*}
\lim_{|\xi|\to \infty}
\int_{\mathbb{S}^{d-1}}(1-\cos(\xi\cdot w))u(w)d \sigma_{d-1}(w)
= \int_{\mathbb{S}^{d-1}}u(w)d \sigma_{d-1}(...
- 271
7
votes
1
answer
259
views
Normal distribution by successive approximation?
$\newcommand\R{\mathbb R}\newcommand\la\lambda$It is well known and easy to see that the rotationally invariant
product of two probability measures on $\R$ has to be a Gaussian (or Dirac) measure; see ...
- 61.9k
6
votes
1
answer
135
views
Small shifts of weakly converging sequences in $L^1$
$\newcommand\R{\mathbb R}$Let $(f_n)$ be a sequence in $L^1(\R)$ converging weakly to some $f\in L^1(\R)$. Let $(a_n)$ be sequence in $\R$ converging to $0$. For each natural $n$, let $g_n$ be the $...
- 128k
0
votes
0
answers
46
views
Taming families of rate functions
$\newcommand\R{\mathbb R}$Let us say that a function $r\colon\R_+\to\R_+$ is a rate function if $r$ is nondecreasing and $r(x)\to\infty$ as $x\to\infty$.
Let us say that a family $(r_j)_{j\in J}$ of ...
- 128k
3
votes
0
answers
205
views
Uniform limit of pointwise limits of continuous functions
Let $X$ be topological spaces, $Y$ a metric space and $(f_n)_{n\in\mathbb{N}}$ a sequence of functions, with $f_n:X\rightarrow Y$ pointwise limit of continuous functions for each $n\in\mathbb{N}$. ...
- 2,286
4
votes
1
answer
273
views
How bad can pointwise convergence in $C$ be?
$\newcommand{\R}{\mathbb R}$Consider the following construction. For real $u$, let
\begin{equation}
f(u):=\frac{2u^2}{1+u^4},
\end{equation}
so that the function $f\colon\R\to\R$ is continuous, $0\...
5
votes
1
answer
654
views
Fréchet L-Spaces
According to the paper The emergence of open sets, closed sets, and limit points
in analysis and topology famous mathematician Maurice Fréchet who introduced the concept of metric spaces has also ...
- 12.7k
4
votes
2
answers
197
views
Symmetry of one-sided partial derivatives
Consider some $f: [0,1)\times [0,1)\to \mathbb{R}$. I'm interested in conditions that guarantee that the following one-sided second partial derivatives at $(x,y)=(0,0)$ are symmetric:
$$
\partial_x^+ ...
- 3,312
4
votes
1
answer
1k
views
The cotangent sum $\sum_{k=0}^{n-1}(-1)^k\cot\Big(\frac{\pi}{4n}(2k+1)\Big)=n$
On the Wolfram Research Reference page for the cotangent function (https://functions.wolfram.com/ElementaryFunctions/Cot/23/01/), I saw the following partial sum formula
$$\sum_{k=0}^{n-1}(-1)^k\cot\...
- 128k
2
votes
2
answers
152
views
Divergence rate of geometric sum of random variables
Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that
$$
0<\lim_{\beta\rightarrow 1}(1-\...
- 128k
1
vote
1
answer
262
views
Relationship between $f(t,x)$ as $t \to \infty$ and $f(t/\epsilon, x/\epsilon^2)$ as $\epsilon \to 0$ (periodic functions)
Let $f: (0,\infty)\times \mathbb {R} \to \mathbb{R}$ be $1$-periodic in the second variable and in $L^\infty((0,\infty)\times \mathbb{R}).$ If it is necessary, we can also assume $f$ to be continuous. ...
- 16.4k
5
votes
1
answer
680
views
When does this interesting sum diverge?
For $x \gt 0,$ what is the greatest $y$ such that $$\sum_ {1\le h^x \le k^y} \frac{1}{h^x k^y}= \infty ?$$
I don't know of any references or methods for this -- not even for $x=1$, for which the ...
- 148k