Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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].

0
votes
0answers
23 views

Preserving Bijectivity under pointwise limit

The following question I have tried to solve and have also tried to find reference for. A reference would be appreciated. Suppose that $f_i, g_i :\mathbb{R^n} \to \mathbb{R^n}$ are continuous ...
0
votes
1answer
12 views

Convergence of complex sequence checkup

I have to find the region of convergence and analyze the the pointwise and uniform convergence of the following sequence: $$\psi_n(z)=\frac{e^{-inz^2}}{(n+1)\sqrt{n}}$$ What I did: What I did first ...
0
votes
1answer
25 views

$n\chi_{[0, 1/n]}$ converges pointwise

I am confused why the function $ f_n = n\chi_{[0, 1/n]}$ converges pointwise. As I remember from my first analysis course, when we prove the pointwise convergence, we should fix $x$ first. Then, for ...
0
votes
1answer
19 views

a proof that the pointswise limits of lower semicontinuous (lsc) functions is lsc

I have a question regarding a proof regarding lower semicontinuous functions (the proof of the claim below). Definition: We call a function $f\colon \mathbb{R}^n\to\mathbb{R}\cup \{\infty\}$ lower ...
1
vote
1answer
56 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 ...
0
votes
1answer
26 views

Convergence of the following sequence of functions.

For $n \ge 1$, let $$g_n(x) = \sin^2 \left (x + \frac 1 n \right ), x \in [0,\infty)$$ and $$f_n(x) = \int_{0}^{x} g_n (t)\ \mathrm {dt}.$$ Then $(1)$ $\{f_n \}$ converges pointwise to a ...
4
votes
1answer
108 views

Uniform convergence of $\sum\limits_{n=1}^∞n^{-x}(e^{\frac{x}{n^2}}-1)$

Pointwise and uniform convergence of the following series of functions: $$\sum_{n=1}^{\infty} n^{-x}\left(e^{\frac{x}{n^2}}-1\right).$$ Now, the series of function converges pointwise as $x \in (-1,...
0
votes
2answers
24 views

Strongly convergent subsequence $+$ point-wise convergence $\Rightarrow$ strong convergence?

Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ has a strongly convergent subsequence, say $f_{n_k}$. Also, assume $f_n\to ...
1
vote
0answers
44 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 $...
1
vote
1answer
30 views

If $\int f_nd\mu \to \int f d \mu <\infty$ then for all measurable $E$, $\int _E f_nd\mu \to \int _E f d\mu$

Let $f_n:X\to [0,\infty]$ be measurable functions such that $f_n\to f$ pointwise. Then prove that if $\int f_nd\mu \to \int f d \mu <\infty$ then for all measurable sets $E$ we have $$\int _E f_nd\...
0
votes
1answer
38 views

$\mu(\bigcup_{n=N}^\infty\omega:|g_n(\omega)-g(\omega)|>\varepsilon)<\varepsilon$ then $g_n\to g$ pointwise a.e.

Let $(\Omega, \mathcal{A}, \mu)$ be a measure space and $g_n : \Omega \to \bar{\mathbb{R}}, n \in \mathbb{N}$ and $g : \Omega \to \bar{\mathbb{R}}$ be measurable functions. Prove that if $$\forall \...
3
votes
1answer
69 views

Pointwise limit of continuous functions is continuous on a dense set

I'm stuck in understanding the proof of the following theorem given during a course: Let $X$ be a Baire space, and $(Y,d)$ a metric space. Let $f_n:X\to Y$ be a sequence of continuous function, ...
-1
votes
2answers
39 views

If $f_n(s) \rightarrow f(s)$ for all s. Is it correct to say that $\lim_{n\to\infty}(\min f_n)=\min f$?

If $f_n(s) \rightarrow f(s)$ for all $s$, is it correct to say that $\lim_{n\to\infty}(\min f_n)=\min f$? Are minimums of $f_n$ converging to minimum of $f$?
1
vote
1answer
23 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,\...
1
vote
1answer
61 views

Why $L^2$ convergence, with Riemann Integral, does not imply pointwise convergence at any point?

Let $L^2[0,1]$ be the space of continuous square integrable functions, where we use the Riemann Integral, no Lebesgue allowed. Let $(f_n)_n$ be a sequence of continuous functions on $[0,1]$ and $f$ a ...
0
votes
0answers
29 views

Showing $\sqrt[4]{n}(Y_n/\sqrt{n} - 1) \xrightarrow{D} N(0,1)$ using characteristic functions, for $Y_n \sim Bin(n, 1/\sqrt n)$

Let $Y_n \sim Bin(n, 1/\sqrt n)$. Show that \begin{equation} \sqrt[4]{n}(Y_n/\sqrt{n} - 1) \xrightarrow{D} N(0,1) \end{equation} using that \begin{equation} X_n \xrightarrow{D} X \quad \text{ if ...
1
vote
1answer
22 views

Pointwise convergence and convergence of integral

Let $\Omega \subset \mathbb R^n, n \in \mathbb N,$ be a bounded domain. Suppose $f_n \in L^1(\Omega)$ converge pointwise to a function $f: \Omega \to \mathbb R$ and $(\int_\Omega f_n)_{n \in \mathbb N}...
2
votes
1answer
90 views

What exactly is the contradiction in proving that $h_n(x)$ does not converge uniformly on any bounded interval?

I am currently going through this pdf https://www.csie.ntu.edu.tw/~b89089/book/Apostol/ch9.pdf and in Exercise 9.2b, page 3, we have the following question. Prove that $h_n(x)$ does not converges ...
0
votes
0answers
18 views

Converge in norm and converge point wise are not equivalent.

I have known that for sequences of functions in $C[0,1]$, converge point wise is not equivalent with converge in norm for that two special norms: Integral and Suprem. However, when it comes to any ...
2
votes
1answer
54 views

Does $\sum^{\infty}_{n=1}xe^{-nx}$ converge uniformly on $[0,\infty)$?

I was trying to solve this problem with a friend but we got stuck. TRIAL We claimed that the series does not converge uniformly which implies that it is not uniformly Cauchy, i.e., $\exists\,\...
0
votes
1answer
20 views

Interval of convergence, pointwise and absolute

Give the series $$\sum_{n=0}^{\infty} \dfrac{(x + 10)^n}{3^n (n+1)},$$ find the intervals which result in point-wise and absolute convergence. Applying the root test we have, $$L(x) = \lim\limits_{n ...
0
votes
0answers
57 views

Convergence of $\sum_{n=1}\frac{1}{1-x^n} $

Consider $$ f(x)=\sum_{n=1}^\infty\frac{1}{1-x^n} $$ I am trying to find all $x\in\mathbb{R}$ at which the above series converges, absolutely converges, and all intervals of $\mathbb{R}$ on which the ...
0
votes
1answer
26 views

Difference between convergent in Measure and convergent a.e.

I’m thinking about two problems: In $L^2([0,1],dx);$ If $f_n\rightarrow f$ In $L^2$, then $f_n\rightarrow f$ In measure. If $f_n\rightarrow f$ In $L^2$, then $f_n\rightarrow f$ Almost everywhere. ...
7
votes
3answers
64 views

Pointwise convergence of sequence $(f_n)_n$ of functions to $f$ and changing limits

My analysis notes contains the following question: if $(f_n)_n$ is a sequence of functions of $A \subset \mathbb{R} \to \mathbb{R}$ and $a \in \mathbb{R} \cup \{-\infty, +\infty\}$ an accumulation ...
2
votes
1answer
37 views

Why do we need to use dominated convergence theorem?

I was thinking of this problem: If $f_n\rightarrow f$ pointwise a.e. and $\lvert f_n\lvert \leq g$ for some $g\in L^p$, then prove that $f_n \rightarrow f$ in $L^p$. To use the dominated convergence ...
0
votes
1answer
39 views

Convergence of $h_{n}(x) = x^{ 1 +\frac{1}{2n-1} }$ defined on [-1,1]

1. $$ h_{n}(x) = x^{ 1 + \frac{1}{2n-1} } = x^{ \frac{2n-1+1}{2n-1} } = x^{\frac{2n}{2n-1}} = (x^2)^{\frac{n}{2n-1}} = (x^2 )^{ \frac{1}{2-\frac{1}{n}} } $$ then $$ \lim_{n\rightarrow \infty} h_{n}(...
0
votes
1answer
23 views

Convergence in measure of a bounded sequence in $L^{2}[0, 1]$ implies weak convergence

Suppose a sequence $\{f_{n} \}$ of functions in $L^{2}[0, 1]$ converges in measure to $f$, and furthermore, assume there exists constant $K$ such that $||f_{n}|| \leq K$ for all $n$. Show that $\{f_{n}...
6
votes
2answers
74 views

Does an integral inequality imply a pointwise inequality?

$\newcommand{\R}{\mathbb{R}}$ Let $(f_n)_n \subset L^1(\R^N)$. Suppose that for any nonnegative function $\phi \in C_c^{\infty}(\R^N)$, we have: $$ 0 \leq \liminf_{n \rightarrow \infty} \int f_n \, \...
0
votes
0answers
42 views

Does weak convergence imply existence of an a.e. convergent subsequence? [duplicate]

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\Dpr}{\mathcal{D}^{1,\vec{p}}(\R^N)}$ I'm reading this article by El Hamidi and Rakotoson. On page 745 (page 5 of the PDF), they construct a bounded sequence ...
0
votes
2answers
64 views

Pointwise convergence to $\exp(-x^2)$ implies uniform convergence in set of functions of compact support [closed]

Let $C_c (\mathbb{R}) = \{f : \mathbb{R} \to \mathbb{R} \mid f$ is continuous and there exists a compact set $K$ such that $f = 0$ on $K^c\}$. Let $g(x) = \exp(-x^2)$. Is the following statement ...
2
votes
1answer
68 views

Prove that $\lim \limits_{n\to \infty}\int_{n}^{\infty} f_{n}(x)dx\neq \int_{n}^{\infty} f(x)dx $

Let $\{f_n\}$ be a sequence defined by $$f_n(x)=\begin{cases}1, & \text{if}\;x\geq n,\\0 & \text{if}\;x< n,\end{cases}$$ Prove: $\{f_n\}$ is monotone decreasing and ...
3
votes
4answers
507 views

Does pointwise convergence to a continuous function on a closed interval imply uniform convergence? [duplicate]

Let's suppose that $(f_n)_{n}$ is a sequence of continuous functions that converges pointwise to a continuous function $f(x)$ on a closed interval $[a, b]$. Is then the convergence uniform, too? If ...
2
votes
2answers
39 views

Why $\frac{\varepsilon}{x^2+\varepsilon^2}$ converges in the sense of distributions to a constant times the Dirac delta

The integral of $f_\varepsilon(x)=\frac\varepsilon{x^2+\varepsilon^2}$ is the tan inverse, which is well behaved anywhere on $\mathbb{R}$, and so $f_\varepsilon$ is in $L^1_\text{loc}(\mathbb{R})$. ...
2
votes
2answers
50 views

Any Sequence of a Measurable Sets Has a Convergent Subsequence?

For each $n\geq 1$, let $A_n$ be a measurable (Borel) set in the unit interval $I=[0, 1]$. Does there necessarily exist a subsequence $(A_{n_k})_{k=1}^\infty$ of $(A_n)_{n=1}^\infty$ such that the ...
1
vote
1answer
41 views

Uniform convergence for series of functions

When the sum of functions $$\sum_{n=0}^{+\infty} \frac{2^{nx}-1}{4^{nx}+1}$$ convergens pointwisely and uniformly? I found that this sum converges iff $x\in [0,+\infty)=I$ (otherwise the general term ...
0
votes
2answers
40 views

Proof that a sequence converges pointwisely, but not uniformly

I have the following task in my homework: We consider a sequence of functions $(f_n)_{n\in\mathbb{N}}$ given by $f_n: \mathbb{R} \to \mathbb{R}$ for all $n\in\mathbb{N}$. Prove that $f_n(x) = 1 - χ_{...
1
vote
1answer
48 views

Determining convergence of series of exponential functions

The functions i'm trying to determine the convergence properties of are a) $f_n(x)=xe^{-nx}$ on $[0,\infty)$ and b) $f_n(x)=nxe^{-nx}$ on $[0,\infty)$ for (a), $f_n(x)=x/e^{nx}$, it looks to me ...
0
votes
2answers
33 views

$f_n$ converges pointwise to $g$

For $n \in N$, define $f_n(x) = e^{-3nx} - \frac {e^x}{n}$. The sequence $(f_n)_{n=1}^{\infty}$ converges to a function $g$ pointwise on $[0,1].$ Find $g(x)$ for all $x \in [0,1].$ When $x = 0$, $...
0
votes
1answer
28 views

$\frac{x^n}{x^n+1}\to f$ not uniformely, but $\lim_{n\to\infty}\int_0^2\frac{x^n}{x^n+1}dx=\int_0^2f(x)dx$

Prove $f_n:\mathbb{R}\to\mathbb{R}$ defined by $f_n(x)=\frac{x^n}{x^n+1}$ converges pointwise to $f(x)=\frac{1}{2}1_{\{1\}}(x)+1_{(1,2]}(x)$. Prove that $\lim_{n\to\infty}\int_0^2\frac{x^n}{x^n+1}dx=\...
1
vote
1answer
24 views

Convergence of integral doesn't imply the uniform convergence.

Let $\{f_n\}_{n=1}^{\infty}$ and $f$ be integrable functions on $[0, 1]$ such that $$\lim_{n \to \infty}\int_0^1|f_n(x) - f(x)| dx = 0$$ Is there exists an example which shows that following ...
0
votes
0answers
14 views

Pointwise Convergence of a Piecewise Function with seemingly odd domain

I have the following function sequence: $$ f_n(x) = \begin{cases} -1 & -1 \leq x \leq -\frac{1}{n} \\ nx & -\frac{1}{n} < x < \frac{1}{n} \\ 1 & \frac{1}{n} \leq x \leq 1 \end{...
1
vote
3answers
36 views

Pointwise and Uniform Convergence on a specific intervall

I have the sequence of functions $$ f_n(x) = \frac{x}{x^2+ \frac{1}{n}} \quad x \in [0, \infty)$$ and I need to show that the sequence converges pointwise as well as uniformly, however only on the ...
0
votes
1answer
27 views

Pointwise and uniform converges of a series

For $x \in \mathbb{R}$ observe the series: $$\sum_{n=1}^{\infty} \frac{1}{n^2} cos\Big( \frac{x^3}{n} \Bigr)$$ Show that the series converges pointwise for $ -\infty \lt x \lt \infty $ and that ...
0
votes
1answer
21 views

Limit of the series of functions $f_n$

$$f_n(x)= \begin{cases} 1&\text{if }\, x\geq 1/n\\ n|x|&\text{if }\, x< 1/n. \end{cases}$$ We want to find the pointwise limit of this function. I think the answer should be $f(x)=1\...
3
votes
1answer
58 views

$f \in L^1(\mathbb{R})$ and $\lim_{|x| \to +\infty} f(x) = 0$ implies $\lim_{|x| \to +\infty} \int_{\mathbb{R}} f(x-y) d\mu(y) = 0$

Let $f \in L^1(\mathbb{R})$ be a Lebesgue integrable function on $\mathbb{R}$. Assume that $$ \lim_{|x| \to +\infty} f(x) = 0. $$ Let $\mu$ be a non-singular (i.e., if $\mu = \mu_a + \mu_s$, where $\...
2
votes
1answer
58 views

How to show pointwise and uniform convergence of specific series

I am to show that the following series converges pointwise, then uniformly. I am aware that uniform convergence implies pointwise convergence, but I have to show the pointwise convergence first and ...
1
vote
1answer
74 views

Convergence of $\sum_{n=1}^{\infty} \frac{1}{n} \sin\bigl( \frac{x^2}{n} \bigr)$.

I have to show that the series $\sum_{n=1}^{\infty} \frac{1}{n} \sin\bigl( \frac{x^2}{n} \bigr)$ is pointwise convergent for $-\infty<x<\infty$, and that it is uniformly convergent on any ...
1
vote
1answer
23 views

Is $\sum_{j=0}^\infty \frac{x}{(1+x^2)^j}$ pointwise convergent or uniform convergent?

My initial thoughts for this question was to find the Chebyshev norm of $f_j(x) = \frac{x}{(1+x^2)^j}$. Doing this I get $\vert\vert f_j \vert\vert _\infty = 1$. Does this mean that the series is ...
2
votes
2answers
29 views

Given f such that $ f(0)=0, \lim_{x\to \infty} f(x) = 1$, is $f_n(x)=f(x+e^n)$ uniformly convergent?

Let $f:\Re \rightarrow \Re$ be a bounded function with the properties $ f(0)=0, \lim_{x\to \infty} f(x) = 1$ is $f_n(x)=f(x+e^n)$ uniformly convergent? Not really sure how to go about applying the ...
0
votes
1answer
33 views

Determine whether the functional sequence is uniformly convergent.

The functional sequence is given by; $$f_n(x)=ne^{-(x-n)^2}$$ So far all I have got is $$f(x)= \lim_{n\to \infty} ne^{-(x-n)^2} = 0 $$ i.e. it is pointwise convergent to zero. Question: Am I correct ...