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Asymptotic behavior of the Hermite functions

I would like to understand the asymptotic behavior of the Hermite function : $$\psi_k(x) = \frac{1}{\sqrt{2^k k!}}H_k(x) e^{-\frac{x^2}{2}},$$ where $H_k(x)$ is the $k-$th Hermite polynomial. For ...
Darius's user avatar
  • 21
-2 votes
0 answers
64 views

A Problem using Limits of Sequences of Functions

Suppose $\{f_n\}$ is a sequence of nonnegative extended real-valued functions on $X$ and $\lim_{n\to\infty}f_n=f$. Take a simple function $0\leq\varphi\leq f$. If $X_{\infty}=\{x\in X: \varphi(x)=a>...
hunter's user avatar
  • 1
1 vote
1 answer
157 views

Is finding the CDF from the Laplace transform well-posed?

In my study of Dynamic Light Scattering, I came across the following inverse problem. Let $F(s):[0,T]\rightarrow[0,T]$ be the Laplace transform of a probability distribution $f(t)$ on the real line ...
Riemann's user avatar
  • 654
1 vote
1 answer
330 views

Does $\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$ converge?

I am trying to study the converge of the series $$\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$$ But $e^{\sin{n}}$ is not monotone, and the Abel's test rule fails here. Can someone help me? ...
pxchg1200's user avatar
  • 287
5 votes
0 answers
285 views

How do you go about making ranges (for integer variables) independent?

Basic question: say you have a sum $$\sum_{n_1 n_2 \dotsb n_k \leq x} f(n_1,\dotsc,n_k),$$ where $f$ decomposes in some sense (say: $f(n_1,\dotsc,n_k) = g(n_1) + \dotsb + g(n_k)$, or $f(n_1,\dotsc,n_k)...
H A Helfgott's user avatar
  • 20.2k
1 vote
1 answer
60 views

Does Monotone (linear) convergence of iterates imply monotone (linear) convergence of function values?

I am considering a proof that would require a certain connection between convergence of iterates and corresponding function values: Consider an algorithm with iterates $\left\{{\mathbf{x}}^k\right\}_{...
AY Wer's user avatar
  • 13
9 votes
1 answer
553 views

Does the sequence formed by Intersecting angle bisector in a pentagon converge?

I asked this question on MSE here. Given a non-regular pentagon $A_1B_1C_1D_1E_1$ with no two adjacent angle having a sum of 360 degrees, from the pentagon $A_nB_nC_nD_nE_n$ construct the pentagon $...
pie's user avatar
  • 541
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 ...
Akira's user avatar
  • 835
9 votes
1 answer
845 views

Convergence of sequences formed by orthocenters, incenters, and centroids in repeated triangle constructions

I asked this question on MSE here. Given a scalene triangle $A_1B_1C_1$ , construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ from the triangle $A_nB_nC_n$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, ...
pie's user avatar
  • 541
0 votes
0 answers
106 views

How to prove that $f(x) := |x|^{\frac{\lambda - n}{p}}(1 - \psi(x))$ satisfies a specific property related to its limit at the origin

Disclaimer. I have asked this question a month ago on MSE (click here to access the original post) and even bountied it. I got an answer on MSE, but unfortunately I don't feel like it has enough ...
Rodrigo's user avatar
  • 51
0 votes
1 answer
53 views

Rate of convergence of the minimum point over a product space

Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that $f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$. $f(\theta, \epsilon) > 0$ for ...
MathLearner's user avatar
0 votes
1 answer
77 views

Decay rate of minimum point over a product space

Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that $f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$. $f(\theta, \epsilon) > 0$ for ...
MathLearner's user avatar
2 votes
0 answers
79 views

Limit of lacunar power series at $1^-$

I've asked this question on MSE but I didn't get a convincive answer so I'm trying here. Here is the question : Let $\sigma:\mathbb{N}\longrightarrow\mathbb{N}$ be strictly increasing, and consider ...
Tuvasbien's user avatar
  • 186
2 votes
1 answer
183 views

Example of a conditionally convergent series $\sum_{n=1}^\infty b_n$ such that $n^2(b_n-b_{n+1})$ is bounded

Let $(b_n)_{n \in \mathbb{N}}$ be a real sequence such that $(nb_n)$ is bounded. I know that if the series $\sum_{n=1}^\infty b_n$ is conditionally convergent, then $(n^2b_n)_n$ is not bounded. But, ...
Kanydo Mat's user avatar
0 votes
1 answer
414 views

Necessary conditions for convergence of convolution

In math.SE, I've asked a question about the convergence of convolution of two functions which have bilateral Laplace transform and also have disjoint Region Of Convergence (ROC) but the question didn'...
S.H.W's user avatar
  • 61
-1 votes
2 answers
87 views

Limits of integral series

Suppose we have the series of functions: \begin{equation} F(x)=\sum_{n=1}^{\infty} f_n(x) \end{equation} where convergence is uniform. Additionally, consider the partial functions of the series: \...
george andrade's user avatar
7 votes
1 answer
355 views

High dimensional Fekete's subadditive lemma: does $|\vec x_{n+m}|\leq |\vec x_n+\vec x_m|$ imply the convergence of $\{\vec x_n/n\}$?

Let $d\geq 1$ be a positive integer. If $\{\vec x_n\}_{n=1}^\infty$ is a sequence of $d$-dimensional vectors satisfying $$\lvert\vec x_{n+m}\rvert\leq \lvert\vec x_n+\vec x_m\rvert\qquad \text{for all ...
Feng's user avatar
  • 517
0 votes
1 answer
270 views

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

I'm trying to determine the nature of this series $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $, but I'm not getting anywhere. I've tried using the Abel and trigonometric formulas, but I can't ...
user avatar
11 votes
2 answers
1k views

Twice continuously differentiable implied by existence of limit

I have the following question. Let $f,g:\mathbb{R}\to\mathbb{R}$ be two continuous functions (vanishing at infinity) and assume that $$ \frac{f(x+t)+f(x-t)-2f(x)}{t^2}\to g(x) $$ for all $x\in X$ when ...
Sonam Idowu's user avatar
22 votes
1 answer
4k views

A challenging (for me) limit calculation

How to calculate the following limit $$ \lim_{n\to\infty}\sqrt{n}\underbrace{{}\sin(\sin(\sin(\sin(\cdots\sin(\frac{1}{\sqrt{n}})\cdots))))}_{n \text{ sin's}} \text{?} $$ ${}{}$
C. WANG's user avatar
  • 549
2 votes
0 answers
70 views

A lemma in the application of Lions's concentration compactness pricnciple in Hardy-Littlewood-Sobolev inequality

I'm encountering some problems when reading Lions' paper "the concentration-compactness principle in the calculus of variations. The limit case, Part 2". The Hardy-Littlewood-Sobolev (HLS) ...
IMOS's user avatar
  • 121
1 vote
0 answers
79 views

Asymptotics of ${2n \choose n+k} {2n \choose n}^{-1}$ when $k$ grows with $n$

The quotient $Q(n,k) := \frac{{2n \choose n+k}}{{2n \choose n}}$ clearly converges to one for $k \in \mathbb{N}$ fixed and $n \rightarrow \infty$. Simultaneously it converges to zero, if $k$ grows ...
Ben Deitmar's user avatar
  • 1,295
3 votes
2 answers
293 views

On convergence of convex-concave functions

Let $(f_n)$ be a sequence of twice differentiable functions on $\mathbb R$ such that for each $n$ there exists some $x_n\in\mathbb{R}$ such that: $f_n$ is strictly convex on $(-\infty,x_n)$, $f_n$ is ...
Iosif Pinelis's user avatar
1 vote
1 answer
300 views

Convergence of concave/convex function

Let assume that you have a sequence of twice differentiable functions $(f_n)_{n\in\mathbb{N}}\in\mathscr{C}^2(\mathbb{R})^{\mathbb{N}}$. Let suppose that for each $f_n$, it exists a $x_n\in\mathbb{R}$ ...
NancyBoy's user avatar
  • 393
2 votes
1 answer
165 views

Continuity of an upper semi-continuous function over periodic points

Let $f: X \to \mathbb{R}$ be an upper semi-continuous function on $X$, which is a compact subspace of a vector space. Let sequence $x_n, n \in \mathbb{N}$, with positive elements - periodic: there ...
Adam's user avatar
  • 1,043
1 vote
1 answer
89 views

Supremum or upper bound of bivariate function involving logarithms and combinatorial coefficients or the gamma function over a region of the integers

This is a repost from MSE because I got no answers there. I have been trying to find the supremum of this bivariate function over a specific region. However, the expressions that I get are horrible. I ...
Hvjurthuk's user avatar
  • 573
0 votes
0 answers
138 views

Under what conditions is $\lim_{x\to a}\left|\varphi\circ f(x)-\tau \circ g(x)\right|=0$ true?

This question is inspired from another much easier problem I was trying to solve which I tried to generalize. The question is essentially as follows (assuming all the limits exist) If $a\in \mathbb R\...
Sayan Dutta's user avatar
1 vote
1 answer
185 views

Does $\sum_{n=1}^\infty e^{-n^2 T} \int_0^T e^{n^2 t} \lvert f(t)\rvert \, dt$ converge for $L^1_\text{loc}$ $f : [0,\infty) \to \mathbb{R}$?

Let $f(t) : [0,\infty) \to \mathbb{R}$ be an $L^1_\text{loc}$ function. Then, I wonder if the following series \begin{equation} \sum_{n=1}^\infty e^{-n^2 T} \int_0^T e^{n^2 t} \lvert f(t)\rvert \, dt ...
Isaac's user avatar
  • 3,477
4 votes
1 answer
254 views

$\limsup_{n\rightarrow \infty, n\in\mathbb{N}} \sin(n)^{n^x}$ for various $x$

Let $$f(x)=\limsup_{n\rightarrow \infty, n\in\mathbb{N}} \sin(n)^{n^x}.$$ Compute $f(1)$ and $f(2)$.
ninepointcircle's user avatar
3 votes
0 answers
454 views

Surprisingly difficult limit of a sequence

Is there an easy way to prove that $|\operatorname{Re}(a_n)| \to \infty$ where $a_n=\left(\frac{1}{2}+i\frac{\sqrt{7}}{2}\right)^n$? Of course $|a_n| \to \infty$, but we have $$ \operatorname{Re}(a_n)=...
J.Mayol's user avatar
  • 489
1 vote
1 answer
139 views

Which kind of convergence can we get from Laplace transform convergence?

This question is a related question see this post Vague convergence VS Laplace transform convergence. But now we assume that \begin{equation} \int_0^\infty e^{-sx}\mu_n(dx)\to \int_0^\infty e^{-sx}...
Fractional analysics's user avatar
1 vote
1 answer
301 views

Vague convergence VS Laplace transform convergence?

If we assume that $\int_0^\infty e^{-sx}\mu_n(dx)\to \int_0^\infty e^{-sx}\mu(dx), \forall s\geq0$, it is possible to show that $\mu_n\to\mu$ vaguely. Where $\mu_n$ is a measure. Please check here for ...
Fractional analysics's user avatar
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}(...
Guy Fsone's user avatar
  • 1,101
4 votes
2 answers
352 views

Solving the functional equation $2f(x)=f(x+a_n)+f(x-a_n)$

Let $a_n$ be a sequence of strictly positive real numbers such that $\lim_{n \to \infty}a_n=0$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ that admit primitives (i.e. there exists a function $F:...
Shthephathord23's user avatar
0 votes
0 answers
341 views

Main ideas behind the proof of the Carleson theorem

I tried to read a few years ago the book "Pointwise Convergence of Fourier Series" (Springer, Juan Arias De Reyna) which is a detailed proof of the Carleson theorem, but I was lost after a ...
Basj's user avatar
  • 587
1 vote
2 answers
169 views

Asymptotic properties of weighted random walks / infinite convolutions of random variables

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d. real-random variables. Let further $0<c<1$. I'm interested in the asymptotic properties of $$ \sum_{k=1}^n c^k X_k. $$ I can prove that this ...
SetofMeasureZero's user avatar
2 votes
2 answers
314 views

Convergence of series related to partial fraction expansion of cotangent function

I am looking at the convergence of the series $$ \cos(t\theta) = \frac{\sin(\pi t)}{\pi} \cdot \Bigg[\frac{1}{t} + 2t \sum_{k=1}^\infty (-1)^k \frac{\cos(k\theta)}{t^2 - k^2}\Bigg].$$ Here $t\in\...
Vincent Granville's user avatar
5 votes
0 answers
158 views

Weaker versions of the Riemann series theorem in constructive mathematics

The classical Riemann series theorem states that given a sequence $(a_n)_{n \in \mathbb{N}}$ of real numbers such that the series $\sum_{n = 0}^\infty a_n$ is conditionally convergent, for all real ...
Madeleine Birchfield's user avatar
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 $...
Iosif Pinelis's user avatar
4 votes
2 answers
548 views

Convergence of a sequence

Let $x_0=1$ and $$x_{k+1} = (1-a_k)\left(\frac{3}{2}-\frac{1}{2}\frac{1}{x_k}\right)$$ where $a_n$ is a known sequence satisfying that $a_k\in(0,1)$ for all $k$ and $a_k\to 0$ as $k\to\infty$. How to ...
Jean Legall's user avatar
1 vote
0 answers
96 views

Limit of alternating sum of factorial moments which diverge

Consider the non-negative, integer valued random variable $X$, and its $i^{\text{th}}$ factorial moment $E_{i}[X]$. Then we have that $$ P(X=0) = \sum _{i=0}^{\infty} \frac{(-1)^i E_{r}[X]}{ i!} $$ ...
apg's user avatar
  • 640
17 votes
2 answers
2k views

"Find $\lim_{n \to \infty}\frac{x_n}{\sqrt{n}}$ where $x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$" -where does this problem come from?

Recently, I encountered this problem: "Given a sequence of positive number $(x_n)$ such that for all $n$, $$x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$$ Find the limit $\lim_{n \rightarrow \infty} \...
Paresseux Nguyen's user avatar
1 vote
1 answer
206 views

Does pointwise convergence yield the convergence under Skorokhod topology?

Let $D_+$ be the set of non-increasing functions $f: [0,T]\to [0,1]$ that are right-continuous. Let $(f_n)_{n\ge 1}\subset D_+$ be a sequence of continuous functions s.t. $\lim_{n\to\infty }f_n(t)$ ...
user avatar
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 ...
Iosif Pinelis's user avatar
9 votes
2 answers
490 views

Rearrangement, conditional convergence, and "placid" permutations

This question came out of a conversation with my students about Riemann's rearrangement theorem, and the general problem of which permutations are "safe" w/r/t summing infinite series. It ...
Noah Schweber's user avatar
2 votes
1 answer
290 views

One series converges iff the other converges

In Show that $\sum\limits_pa_p$ converges iff $\sum\limits_{n}\frac{a_n}{\log n}$ converges it is said that this sequence of partial sums converges $$ \begin{split} \sum_{1<n\leq N}\frac{a_{n}}{\...
tongyang2357's user avatar
3 votes
1 answer
91 views

Boundedness and convergence

If I know that $\Phi_\varepsilon$ is bounded in $L^{\infty}(\mathbb{R}^{2d})$ and that $\nabla \Phi_\varepsilon$ is bounded in $L^{\infty}(\mathbb{R}^{2d})$, is it true that $\nabla \Phi_\varepsilon \...
Markus's user avatar
  • 31
2 votes
1 answer
137 views

Convergence of the average weight of an infinite path through a weighted directed graph

Consider a directed graph $G = (V, E, w)$, where $V$ is the set of vertices, $E \subseteq V \times V$ is the set of directed edges (with self-loops allowed), and $w : E \to \mathbb{R}_+$ is a weight ...
David's user avatar
  • 21
0 votes
0 answers
59 views

Is $g = \sum_{n \in \mathbb{Z}} f(\cdot - n)$ continuous if $f$ is vanishing, continuous, and integrable?

Let $f \in \mathcal{C}_0(\mathbb{R}) \cap L^1(\mathbb{R})$ be a continuous and integrable function such that $f(x) \rightarrow 0$ when $|x|\rightarrow \infty$. The sequence of a functions $f_N = \sum_{...
Goulifet's user avatar
  • 2,306
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 $n\in\mathbb{N}$. ...
Lorenzo's user avatar
  • 2,286