All Questions
Tagged with real-analysis limits-and-convergence
44 questions with no upvoted or accepted answers
8
votes
1
answer
2k
views
Summary of sufficient conditions for convergence of Fourier series
I would like to summarize various sufficient conditions for various modes of convergence of Fourier series. The followings are what I have gathered so far:
$L^p$ convergence:
if $f \in L^p(\mathbb{T}...
- 315
6
votes
0
answers
283
views
Is the arithmetic-geometric mean of 1 and 2 rational?
It is easy to show that, for two fixed real numbers $\alpha, \beta > 0$, the sequences given by $a_ 1 = \frac{\alpha + \beta }{2}$, $ g_1 = \sqrt{\alpha\beta}$, and $a_{n+1} = \frac{a_n + g_n}{2}$, ...
- 161
6
votes
0
answers
267
views
Convergence of $\sum_{n=1}^\infty x_n^k$
I thought that this question is more suitable for MSE, and asked it there. (Link to the MSE question) However, it does not get any answer despite the upvotes. It appears that I might have ...
- 1,755
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)...
- 20.2k
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 ...
- 2,624
5
votes
0
answers
313
views
Uniqueness of a SDE with non-negativity constraint
I am working on the following SDE (but we will dealing only with deterministic object: $\omega\in\Omega$ is fixed):
\begin{equation}\label{sde}%sde
x_t=\underbrace{\xi_0+\int_0^tb(s,x_s)\,ds+\int_0^t\...
- 779
4
votes
0
answers
459
views
Is there any closed form expression for $\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$?
Is there any closed form expression for the following serie?
$$\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$$
Or at least a proof that it is an irrational number. The ...
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)=...
- 489
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
3
votes
0
answers
238
views
Dominated convergence Theorem
I am struggling to understand the proof in the paper, Learning Temporal Evolution of Spatial Dependence with
Generalized Spatiotemporal Gaussian Process Models.
Theorem 2.1 in the page 33 uses ...
- 31
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 ...
- 186
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) ...
- 121
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 ...
- 21
2
votes
0
answers
57
views
Is the lattice of bounded Henstock Kurzweil integrable functions countably complete?
The set of HK integrable functions with an integrable upper bound $f$ forms a lattice, and satisfies the MCT and DCT. Does this mean that the lattice is countably complete?
Indexing any countable set, ...
- 1,947
2
votes
0
answers
126
views
Mixed partial derivatives of planar functions converging to delta distribution
Given a sequence $(f_k)_{k\in\mathbb{N}}\subset C^2(\mathbb{R}^2)$ of strictly positive functions $f_k\equiv f_k(x,y)$ with $\|f_k(x,\cdot)\|_{L^1}=1$ for all $x\in\mathbb{R}$ and such that for each $...
- 51
2
votes
0
answers
1k
views
Is there an infinite product like this for $\cos x$?
There are infinite products of iterated square roots for $\log x$ and $\arccos x$ as functions of $x$. For example
$$\log x = \frac{x - 1}{\sqrt{x}\sqrt{\frac{1}{2} + \frac{1}{2}\left ( \frac{1 + x}{...
2
votes
0
answers
147
views
Interchanging limit and infinite product in Euler product for Dedekind function s=1
For an quartic (non-Galois) CM-field $K$ I have factors $v_p$ and for every prime $p$ found the following relation
$$v_p={\frac {\prod_{\mathfrak{p}|p;\mathfrak{p}\subset\mathcal0_{K}}(1-N_{{K/{\...
- 21
1
vote
0
answers
59
views
Asymptotic behavior of the Hermite functions
I would like to understand the asymptotic behavior of the Hermite function :
$$\psi_k(x) = \frac{1}{\sqrt{2^k k!}}H_k(x) e^{-\frac{x^2}{2}},$$
where $H_k(x)$ is the $k-$th Hermite polynomial. For ...
- 21
1
vote
0
answers
79
views
Asymptotics of ${2n \choose n+k} {2n \choose n}^{-1}$ when $k$ grows with $n$
The quotient $Q(n,k) := \frac{{2n \choose n+k}}{{2n \choose n}}$ clearly converges to one for $k \in \mathbb{N}$ fixed and $n \rightarrow \infty$. Simultaneously it converges to zero, if $k$ grows ...
- 1,295
1
vote
0
answers
96
views
Limit of alternating sum of factorial moments which diverge
Consider the non-negative, integer valued random variable $X$, and its $i^{\text{th}}$ factorial moment $E_{i}[X]$. Then we have that
$$
P(X=0) = \sum _{i=0}^{\infty} \frac{(-1)^i E_{r}[X]}{ i!}
$$
...
- 640
1
vote
0
answers
79
views
Convergence mode with inputs and functions varying in tandem
Given a sequence $(f_n)$ of functions between metric spaces, let's say that $f_n$ "converges flexibly" to $f$ if, whenever $x_n \to x$ is a convergent sequence of inputs, it follows that $...
- 143
1
vote
0
answers
78
views
A question about extension problem related to fractional laplacian
I am studying the article "An Extension Problem Related to the Fractional Laplacian" by L. Caffarelli and L. Silvestre. link. At page 2, for a function $f\colon\mathbb{R}^n\to\mathbb{R}$, we ...
- 339
1
vote
0
answers
120
views
Natural candidates for super-half-exponential which limit to half-exponential function from above
There are no closed form candidates for half-exponential functions "Closed-form" functions with half-exponential growth.
However super-half-exponentials (functions whose composition grows ...
- 1,826
1
vote
0
answers
59
views
Uniform lower bound on a sequence of functions on $[0,1]$
Consider the sequence of functions $\{F_n(\cdot)\}_{n \in \mathbb{N}}$ on $[0,1]$, where for each $n$, $F_n(\cdot)$ is defined as
\begin{equation}
F_n(x) = \sum_{i=2}^n \Big( x^{b_n} \frac{i}{(i+1)^{...
- 151
1
vote
0
answers
51
views
Mean value of a function with binomial coefficients as weights
Is the following true?
Let $a$ be a positive integer and let $t_n$ be a sequence of numbers. We define the binomial mean of $t_n$
$$
\beta_{t_n,a} = \frac{1}{2^n t_n}\sum_{r^a \le n} \binom{n}{r^a}...
- 5,470
1
vote
0
answers
51
views
Convergence acceleration of a series by using optimal parameters
One of the ways of accelerating the convergence of a series is by transforming into a faster series using optimal parameters. Examples of this approach can be found in this paper. I obtained a ...
- 5,470
1
vote
0
answers
314
views
Can Carlsons's iterative algorithm for $\arctan x$ be inverted to get one for $\tan x$?
In the article An algorithm for computing logarithms and arctangents, by B. C. Carlson, the following iterative algorithm for arctangents is given:
The algorithm uses that $2^n\tan(2^{-n}\arctan(x))=\...
1
vote
0
answers
71
views
Proving an Algorithm that generates minimal $\|x\|_0$ for the underdetermined system $Ax=b$
Let $A \in \mathbb {F}^{m \times n}$ with $m< n,$ $b \in \mathbb{F}^m$ and let $x$ be unknown in $\mathbb{F}^n.$ Assume $0<p<1.$ Then $$\arg \min\limits_{x: Ax=b} \|x\|_0 = \lim\limits_{p \to ...
- 259
1
vote
0
answers
150
views
Positivity of alternating series
Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then
$$
f(x)=\sum_{n\geq 0}a_n x^n
$$
converges absolutely for all $x$. Under ...
- 1,329
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 ...
- 51
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\...
- 791
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 ...
- 587
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
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_{...
- 2,306
0
votes
0
answers
67
views
LLN of random nearest neighbor function
There are two samples of iid random variates: $X=\{X_1,X_2,...,X_n\}$ and $Y=\{Y_1,Y_2,...,Y_n\}$. Further, $\forall i,j: X_i$ is independent of $Y_j$. The probability distributions $P,Q$ are unknown ...
- 89
0
votes
0
answers
168
views
Sequence of functions tending to zero in L^2
Let us consider a sequence of functions $f_n : (0,1)\times (0,1) \to \mathbb{R}$ in $L^2((0,1)\times (0,1))$ satisfying the following condition:
$$
\lim_{n \rightarrow \infty}\int_{1/j}^{1 - 1/j}\...
- 29
0
votes
0
answers
146
views
Does the following sequence $\{g_n\}$ converge?
Consider a function sequence $\{f_n(t)\}$ ($n\in\mathbb{N}^+$) defined on the interval $(\frac{1}{2},1)$, where
\begin{eqnarray}\label{eqn:constraint1}
f_n(t)=\frac{\exp\left(n\left(\log R(h_t) - th_t\...
- 550
0
votes
0
answers
91
views
Does $L^1$ convergence preserve the regularity of this sequence of functions?
Let $f_n$ be a sequence of $L^1(]0,1[)$ functions such that $f_n$ is non-decreasing, at least left-continuous, $f_n(0^+) <0$, $f_n(1^-) >0$, for all $n \in \mathbb N$. This sequence converges
$...
- 143
0
votes
0
answers
159
views
Limiting property of polylogarithm ratio
I try (without success) to figure out what could be the following limit if any...
For real $s$ strictly $> 1$ and $x \rightarrow +\infty$ (x real) the limit of the polylogarithm ratio $\frac{Li_{s-...
0
votes
0
answers
299
views
When convolution with exponential kernel is bounded
Let $g(t)=e^{-\omega t}$, $\omega>0$. What is, in terms of well-known function spaces, the space $X$, $L_{loc}^2(0,\infty)\subset X$, of all functions $f:\mathbb{R}^+\to \mathbb{R}^+$, satisfying
$...
- 395
-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:
\...
-1
votes
1
answer
236
views
Natural candidates for sub-half-exponential which limit to half-exponential function from below
There are no closed form candidates for half-exponential functions "Closed-form" functions with half-exponential growth.
However sub-half-exponentials (functions whose composition grows ...
- 1,826
-1
votes
1
answer
193
views
Limit of the convolution of derivative of Gaussian heat kernel
I'm looking for the following limit:
$$\lim_{\varepsilon\to 0^+}\int_{-\sqrt{\varepsilon}}^{\sqrt{\varepsilon}}\frac{1}{\sqrt{2\pi}\varepsilon^{3/2}}\left(-1+\frac{x^2}{\varepsilon}\right)e^{-\frac{x^...
-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>...
- 1