Questions tagged [pointwise-convergence]

For questions about pointwise convergence, a common mode of convergence in which a sequence of functions converges to a particular function. This tag should be used with the tag [convergence].

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Pointwise limit of continuous functions, but not Riemann integrable.

I am trying to find a simple example of a function $f:[0,1]\rightarrow\mathbb{R}$ which is a pointwise limit of continuous functions, but is not Riemann integrable. I know the classical example where ...
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366 views

Weak topology = topology of pointwise convergence?

In a book, I found the following theorem (slightly simplified here): Consider $\mathbb{R}$ with the Borel $\sigma$-field. Let $B$ be the set of all measurable, bounded, real-valued functions on $\...
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2answers
22 views

Convergence of Sequence of Indicator Functions

With $I \subset \mathbb{R}$, let $\chi_{i}$ denote the indicator function of I. Define a sequence of functions $\{g_{n}\}_{n=1}^{\infty}$ on $[0,1]$ as: $g_{1}(x) = \chi_{[0,1]}(x),\quad g_{2}(x) = \...
3
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38 views

Finding the Pointwise Limit of a Function

If I have a sequence of functions $f_n[0,2] \rightarrow \mathbb{R}$ where $f_n(x) = \frac{x^n}{2^n+n}$. If I attempt to find the pointwise limit, I work out that by taking $x \in [0,2]$: We can ...
3
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0answers
100 views

Prove the compactness and second countability of a subset of $\Bbb{R}^{L^1(G)\times D}$

I'm trying to fulfill the details in a proof whose aim is to guarantee the existence of a compact space with certain properties, but in this case I had no success. Let $X$ be a metric space (actually,...
3
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0answers
65 views

Pointwise convergence vector valued continuous functions

Let $(X,\|\cdot\|)$ be a (infinite dimensional) Banach space and $f_{n}:[0,1]\longrightarrow (X,\|\cdot\|)$ continuous, for each integer $n\geq 1$, a sequence of mappings. There is some "pointwise ...
3
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105 views

Show convergence of the series of functions $\sum_{n=1}^\infty \frac{x}{n^\alpha (1+nx^2)}$

Show convergence of the series of functions $\sum_{n=1}^\infty \frac{x}{n^\alpha (1+nx^2)}$ on $\mathbb{R}$, where $\alpha > \frac{1}{2}$ My attempt: It suffices to show that for every fixed ...
3
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44 views

Where to start? Sequence of functions.

I've been playing around with composing translations with the modulus function. Let $f_0(x) = |x|$, and $f_{n+1}(x) = \left| f_n(x)-\frac{1}{n+1}\right|$, where $n$ is a positive integer. I would ...
3
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337 views

Sequence of continuous functions over a compact set that converges pointwise and monotonically also converges uniformly

I have the following problem: Let $\{f_n\}_{n = 1}^\infty$ be a sequence of functions such that, for all $n \in \mathbb{N}$, $f_n:[0,1] \to [0,1]$ is continuous, and for all $x\in [0,1]$, we have ...
2
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1answer
44 views

Question of pointwise convergence of a sequence of functions

I have the following task: Define $f_n:[-1,1]\to \Bbb R$ by $$f_n(x)=\begin{cases}1 , \text{ for $-1 \leq x \leq -1/n$} \\ -\sin(n\pi x/2) , \text{ for $-1/n \leq x \leq 1/n$}\\-1 , \text{ for $1/n\...
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43 views

Weak and Pointwise Convergence question

I found an interesting property in some lecture notes on weak convergence: lemma 3.2 (2) on page 10: https://www.uio.no/studier/emner/matnat/math/MAT4380/v06/Weakconvergence.pdf . The idea is as ...
2
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0answers
35 views

Prove $∪_{k=1}^N[x_k − δ, x_k + δ] ⊃ [0, 1]$

Suppose ${f_n}$ is a sequence of nonnegative continuous functions defined on $[0, 1]$ and suppose $lim_{n→∞} f_n(x) = 0$ pointwise on $[0, 1]$. Prove that, $∀\varepsilon> 0$, there exist $δ ...
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69 views

Proof for problem similar to Dini's theorem

I am having trouble coming up with a proof for this problem about a sequence of continuous functions that converges (pointwise) to 0 on [0,1]. https://i.stack.imgur.com/FE7Wr.png Proving the first ...
2
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0answers
26 views

How to describe the process of adding more and more random non-overlapping intervals into a given interval?

Suppose I divide the interval $[0, 1]$ on the real axis by $N$ smaller randomly placed but non-overlapping intervals $[X_{i-1,N},X_{i-1,N} + \Delta_N],i=1,\cdots,N$, such that $$ N\Delta_N + \sum_{i=1}...
2
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66 views

Given a $f \in L^1(\mathbb{R})$ is there a $g : \mathbb{R} \to \mathbb{R}$ such that $g(f) = 0$ with $g \neq 0$?

Given a $f \in L^1(\mathbb{R})$ is there a $g : \mathbb{R} \to \mathbb{R}$ such that $g(f) = 0$? I would like to solve the problem myself but don't know how to start. Point of this question; I have ...
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0answers
23 views

Does taking the limit in one argument preserve regularity in another argument?

Preliminaries Let $T>0$ be a positive constant, $d,n \in \mathbb{N}_{\geq 1}$ be natural numbers and $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space. For $m \in \mathbb{N}_{\geq 0}$, ...
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0answers
89 views

Almost everywhere pointwise convergence vs pointwise convergence

Let $X$ and $Y$ be compact metric spaces. Suppose that $\{f^n\}$ is a sequence of continuous maps $f^n:X\to Y$, and suppose that for every Borel probability measure $\mu$ on $(X,\mathcal B(X))$, there ...
2
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1answer
59 views

Doubts about moving limit under integral

We have to count following integral: $$ \lim_{n\to\infty}\int_A\frac{\ln(n+y^3)-\ln n}{\sin(x/n)}d\lambda_2 $$ where $A=\{(x,y)\in\mathbb{R}^2_+:1<xy<4,1<y/x<4\}$ Now in order to find ...
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62 views

Why does $h\circ f_{n}\to h\circ f$ when $h$ is continuous and $f_n\to f$ pointwise?

Let $(X,d_{X})$ be a metric space, and for every integer $n\geq 1$, let $f_{n}:X\to\textbf{R}$ be a real-valued function. Suppose that $f_{n}$ converges pointwise to another function $f:X\to\textbf{R}$...
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0answers
22 views

Almost sure convergence for GARCH(1,1)-process

I'm proving the conditions for strict stationarity of GARCH(1,1)-process: $$X_t=\sigma_t Z_t\qquad \sigma_t^2 = \alpha_0 + \alpha_1 X_{t-1}^2 + \beta_1\sigma_{t-1}^2.$$ We can rewrite the process to a ...
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0answers
18 views

Mathematical intuition why the iterative Bellman update converges to the optimal solution

I know that the mathematical justification for using the Bellman-equation iteratively to find the optimal policy in Reinforcement Learning is based on convergence results. I wonder however whether ...
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1answer
31 views

Beta distribution converges in distribution to a binomial

I have the following problem. Let $X_n \sim \operatorname{Beta}(1/n,1/n)$ be a sequence of beta distributions, and $X \sim \operatorname{Bin}(1,1/2)$ which is equivalent to $\operatorname{bern}(1/2)$....
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0answers
41 views

Pointwise convergence and $L^{p}$ bounded implies weak convergence in $L^{p'}$.

I would just like to know if my working here is sound. I want to show that given a sequence of function $(u_{n})\in L^{p}(\Omega)$ (with $|\Omega|<\infty$) such that $u_{n}(x)\to u(x)$ for every $...
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22 views

Convergence on locally compact groups with an additional condition

This question concerns locally compact groups equipped with Haar measure, $(G,\lambda)$. For a class of such groups, there exists an approximate identity $F_\nu$ such that the map $f\in L^1(G)\mapsto ...
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19 views

joint convergence in distribution by convergence of conditional expectations

Let $(X_n,Y_n)$ be a sequence of random vectors in $\mathbb R^k\times \mathbb R^l$ such that $X_n\to X$ in distribution. Denote the conditional distributions $Y_n$ given that $X_n=x$ by $P_{n,x}$. I ...
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40 views

Uniform/Pointwise Convergence

So, I have a somewhat arbitrary question about this topic but first I need to state the theorem I am familiar with and from which I will base my question. Theorem: Let $(f_n)_n:[a,b] \to \mathbb{R}$ ...
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0answers
27 views

Convergence in distribution of empirical CDF

Without using the fact that the empirical cdf $F_{n}(x)$ converges to $F(x)$ under higher modes of convergence, I want to show that the empirical cdf converges to F(x) in distribution. In other words, ...
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0answers
37 views

prove pointwise convergence and uniform convergence

I have been buggin on this question for a few hours now. Q. for n = 1, 2, 3, ... let $f_n : [0, +\infty) \rightarrow \mathbb R$ be defined by $f_n = \frac{xn + sin (\frac{x}{n^2})}{2(n + \sqrt x )}...
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1answer
60 views

Show that $f_n(x)=n^2x^n(1−x)$ converges pointwise to $0$ using epsilon argument from definition

Let $f_n:[0,1]\to\Bbb R$ be defined by $f_n(x)=n^2x^n(1−x)$. Show that $f_n(x)$ converges pointwise to $0$ using an epsilon argument from the definition of pointwise convergence. I've got the ...
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1answer
57 views

Almost sure convergence characterisations

According to Wikipedia https://en.wikipedia.org/wiki/Convergence_of_random_variables $X_n \to X$ almost surely if and only if $\forall\epsilon>0$ $$P(\liminf_{n\to\infty}\{{\lvert X_n-X\rvert\lt \...
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0answers
39 views

Analysis of convergence (pointwise, local uniform, uniform)

How can we check if some series for example $$ \sum_{n=1}^{\infty}\frac{nx^2}{n^3+x^3}\qquad x>0 $$ is convergent pointwise / locally uniformed / uniformed? What are the methods / tools to check ...
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0answers
27 views

Is the dominated convergence theorem applicable whenever “THIS” theoem is applicable?

THIS theorem: Let $I =[a,b]$ be a closed and bounded interval and $\forall n\in \mathbb{N}$, $f_n:I \to \mathbb{R}$ be Riemann integrable on $I$. If the sequence $(f_n)$ converges uniformly to a ...
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0answers
42 views

Convergence of $\sum_{n=1}^{\infty}\frac{x}{[(n-1)x+1](nx+1)}$

Consider, $$\sum_{n=1}^{\infty}\frac{x}{[(n-1)x+1](nx+1)}$$ defined on $[0,1]$ Does this series converge point-wise and uniformly to a function on $[0,1]$? How can I approach this problem? Since I ...
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1answer
40 views

How to negate this statement

Let $\{f_n:A\to\mathbb{R}\}_{n\ge 1}$ is a sequence of functions that converges pointwise to some function $f:A\to\mathbb{R}$ and let $A_n:=\{x\in A:|f_n(x)-f(x)|\ge\alpha\}$ for some fixed constant $\...
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1answer
55 views

Uniform Convergence of a subsequence over an arbitrary interval

I am getting ready for an entrance exam that is in August and I am trying to get a head start on the analysis section. I recently came across this problem and it is giving me some trouble. Consider ...
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1answer
108 views

Pointwise and uniform convergence of a piecewise sequence of functions on the closed, punctured disk, $\overline{D}\prime(0,1)$.

Consider the sequence of functions $$f_n(z) = \begin{cases} n, & \text{if $0<|z|\leq\frac{1}{n}$} \\ \frac{1}{z^4}, & \text{if $\frac{1}{n}<|z|\leq1$} \end{cases} $$ for $n\geq 1$, on ...
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0answers
79 views

Prove $f(x,y)$ defined on $\mathbb{R}^2$ is Lebesgue Measurable if $f$ is continuous in each variable separately.

This problem is from Ziemer's Modern Real Analysis and there is a suggestion on the page which tells us to approximate $f$ in the variable $x$ by piecewise-linear continuous functions $f_n$ such that $...
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1answer
133 views

Can we interchange integral and limit if the sequence of function is uniformly integrable?

Assumptions: 1) $h(x,\omega)$ is defined on $\forall\omega\in\Omega, \forall x\in R\setminus\{x_0\}$ where $\omega$ is a random variable. 2) Let $h(x_0,\omega)=\lim\limits_{x\rightarrow x_0}h(x,\...
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1answer
65 views

Misunderstanding about uniform and pointwise convergence

Before I continue asking my question, I have seen this exact same question posted numerous times, but I don't feel like I seen any answers that were satisfactory in answering my complaint. For some ...
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2answers
196 views

Show that partial sums of a function converge pointwise but not uniformly

Let $g : (−1, 1) → R$ be the function $g(x ) := \frac{x}{(1−x)}$. With the notation as in $(b)$ show that the partial sums $\Sigma _{n=1}^N f^n$ converges pointwise as $N → ∞$ to $g$, but does not ...
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0answers
29 views

pointwise converging sequence of continuous functions on a compact: bounding oscilation from below

Assume It is known know that oscilation $\omega(f_n, U) > \epsilon$ for continuous functions $f_n$ and some open set $U$ starting from some $N$. Also it is known that $f_n \to \varphi$ pointwise. ...
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0answers
26 views

can limit of a function be a measure?

What is the limit when $\epsilon \rightarrow 0^+$ of $$ x\mapsto \sin(x/\epsilon) $$ ? It doesn't seem to be a function. Is it a measure ?
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0answers
100 views

Pointwise convergence of densities convolved with Gaussians

I am trying to understand the proof of the entropic central limit theorem presented here, but I am stuck on the step where the convergence in distribution is upgraded to convergence of the probability ...
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0answers
334 views

Proving pointwise convergence of a function sequence by definition

I'm writing a school paper about convergence. I am a bit stuck on proving pointwise convergence by definition for a particular function sequence related to the delta function: Let $T_n(x)$ be a ...
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0answers
133 views

Equivalent definitions for pointwise and uniform convergence

So for a sequence of functions $\{ f_n\}_{n \in \mathbb{N}}$ then $f_n$ converges pointwise to some function $f$ if $$\forall x \in I, \forall \epsilon > 0, \exists N > 0 :n\geq N \implies |f_n(...
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0answers
53 views

Does the following functional series converge? If so, Does it converge normally? uniformly?

Please check my attempt and result in solving the following question: Let $f_n(x)= x^n(1-x)^2$, $x\in [0,1]$. Does $\sum f_n$ converge? If so, does it converge normally? uniformly? My attempt: ...
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2answers
386 views

Finding the pointwise limit and checking for uniform convergence

I am wondering if I'm thinking properly regarding this question: First, according to the pointwise limit, I did this informally: As $n$ becomes very large, the interval $\left[\frac {1}{n}, \frac {2}...
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1answer
337 views

Continuity and pointwise convergence doesn't imply contiunity

We have a function $f$ and a sequence of functions $f_n$, both on $[a,b] \to \mathbb{R}$. $f_n$ is continuous for each $n \in \mathbb{N}$, and $f_n$ converges pointwise to $f$. I am asked to give an ...
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1answer
83 views

If a series converges uniformly on $[a+\varepsilon,b) $ for all $\varepsilon>0$ and pointwise on $[a,b)$, is then the convergence uniform on $[a,b) $?

The question arises from the specific series $\sum_{k=0}^\infty \dfrac{k (ex)^k}{(k+1)^2} $, which converges pointwise on $[-1/e,1/e) $ and uniformly on $[-1/e,1/e-\varepsilon] $ for every $\...
0
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0answers
30 views

Application of Egoroff's thm on infinite measure

This is what I have solved. My professor actually said this question need a condition for which m(E) is finite. He is keep saying that m,k cannot be fixed after reordering. But I think that there is ...