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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].

2
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1answer
85 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
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0answers
17 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
50 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
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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
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0answers
56 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
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1answer
24 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
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3answers
61 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
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1answer
35 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
37 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
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1answer
21 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
64 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
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0answers
37 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
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2answers
56 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
67 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
414 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
37 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
48 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
40 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
32 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
42 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
32 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
20 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
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0answers
13 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
35 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
25 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
56 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
53 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
71 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 ...
1
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1answer
30 views

There exists $\{f_n\}$ as above that converges to $0$ uniformly on $[0,1]$.

Q. Let $\{f_n\}$ be a sequence of functions defined on $[0,1]$. Suppose there exists a sequence of distinct numbers $x_n\in[0,1]$ such that $$f_n(x_n)=1.$$ Prove or disprove the following statements: ...
2
votes
2answers
34 views

Is the limit uniform on $R$? Is the limit uniform on $[0,1]$?

a)Find the pointwise limit $\frac{e^{x/n}}{n}$ for $x\in R.$ For given $x\in R$ and $\varepsilon>0,$ let $M \in N$ be such that $M=\frac{x}{\ln(\varepsilon)}$. I claim that the limit is $0$. ...
4
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2answers
53 views

Finding a pointwise convergent sequence that does not converge in the square mean

I am having a tough time finding a function sequence $(f_{n})_{n\in\mathbb N}$ of continuous functions of $[0,1] \to \mathbb K$, whereby $\mathbb K \in \{\mathbb R,\mathbb C\}$, such that $(f_{n})_{n\...
2
votes
2answers
39 views

Sequences pointwise convergent to 0 on dense sets

Let $X$ be a compact metric space and let $D\subset X$ be a dense set. Take a sequence of continuous functions $f_n\colon X\to \mathbb R$ such that $(f_n)$ is uniformly bounded and converges to 0 ...
2
votes
4answers
54 views

Difficulty with limit to show pointwise convergence (without L'hospital's rule)

I've come across a question which asks to show pointwise convergence on $[0,1]$ for the following sequence of functions defined for integers $n\geq0$ : $f_n(x)=n^2x^n(1-x)$. I know that the sequence ...
1
vote
2answers
31 views

Uniform convergence of piecewise sequence of functions

I stumbled upon this problem and I'm having hard time understanding the solution , mainly the part about the supremum of the function. Study the uniform convergence of the following sequence of ...
1
vote
1answer
22 views

Does simple convergence imply local uniform convergence

Consider $f_\nu:\Omega\subset\Bbb R^n\mapsto \Bbb R,\ \nu\in\Bbb N$ some functions Local uniform convergence is when $\forall x\in\Omega$ there exists an open neighborhood $U_x\ni x$ of $\Omega$ such ...
2
votes
0answers
62 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 ...
1
vote
1answer
54 views

Pointwise convergence of a sequence of piecewise functions {fn} [closed]

or any x∈[0,∞), there is always Nx∈ℕ s.t. Nx>x, thus we have fn(x)=e−x,∀n≥Nx, which means limn→∞fn(x)=e−x,∀x∈[0,∞) To show that the convergence is uniform, it suffices to show that limn→∞supx≥0|fn(x)...
0
votes
1answer
23 views

Pointwise convergence and continuity problem

I have the following function: $$f_{n}(x)=\frac{n}{1+nx}$$ and want to study its convergence on $[0,1]$. I let $x$ be fixed on $[0.1]$ and compute the limit of $f_{n}(x)$. I can rewrite it as $f_{n}(...
2
votes
2answers
46 views

Show pointwise convergence of this piecewise function.

My textbook is making an example of how uniform convergence of sequences of functions is important. As and important the author shows that, in general, pointwise convergence does not preserve some ...
0
votes
1answer
66 views

Finding an increasing sequence of step functions which converges pointwise everywhere to $χ_{[0,1]\cap\mathbb Q}(x)·x$

Find an increasing sequence of step functions which converges pointwise everywhere to the function$$ f(x)=\chi_{[0,1] \cap \Bbb Q} (x) · x. $$ I know how to approximate a non-negative measurable ...
0
votes
1answer
53 views

Pointwise convergence of an increasing sequence of step functions to a measurable function.

Give an example of a measurable non-negative function $f$ such that no incresing sequence of step functions which converges pointwise everywhere to $f$ exists. I have tried to the best of my ability ...
1
vote
0answers
31 views

Example of a convergent pointwise sequence of function and not uniformly convergent [duplicate]

We say that a sequence $f_n(x)$ is pointwise convergent to $f(x)$ on the set $E$ when for every $x\in E$ and $\epsilon >0$ there exists an integer $N$ such that the distance between $f_n(x)$ and $...
-1
votes
2answers
83 views

Why $\sum_{n=0}^\infty (-1)^nx^{2n}$ converge pointwise?

Consider the geometric series $\sum_{n=0}^\infty (-1)^nx^{2n}$. Does it converge pointwise in the interval $-1<x<1$? Solution. The series has ratio $-x^2,$ thus the $n^{th}$ partial sum is $...
0
votes
1answer
47 views

Pointwise convergence in limit of conditional probabilities implies almost sure convergence?

Consider two sequences of real valued random variables $\{X_n\}_{n\in \mathbb{N}}$ and $\{Y_n\}_{n\in \mathbb{N}}$, with, $\forall n \in \mathbb{N}$, $X_n$ and $Y_n$ defined on the probability space $(...
0
votes
0answers
26 views

Understanding Pointwise Convergence Implying Uniform Convergence

I'm having trouble with the solution to following problem: a) Suppose that $\{f_n\}$ is a sequence of continuous functions on [a,b] which approaches $0$ pointwise. Suppose moreover that we have $f_n(...