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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].

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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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Convergence of a countable sum of characteristic functions of half-open intervals with coefficients

Let $\mu$ be the Lebesgue measure on $\mathbb{R}$ and denote by $\chi_A$ the characteristic function of $A\subseteq\mathbb{R}$. Assume that $a_n\in\mathbb{R}$ and $I_n\subseteq\mathbb{R}$ is a bounded ...
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Yosida Transform's pointwise convergence.

Let $f = \Gamma-\lim_{j\to\infty} f_{j}(x)$, where $f_{j}:\mathbb{R}\rightarrow [0, +\infty] $ are convex and lower semicontinuous functions such that $\sup_{j} f_{j}(0)< \infty$. I have to prove ...
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1answer
39 views

Does $f_n\to0$ pointwise + $f_n$ integrable + $f_n$ uniformly bounded imply $\int f_n\to 0$?

If $\{f_n:[0,1]\to\mathbb{R}\}_{n\ge 1}$ is a sequence of Riemann-integrable functions that converge pointwise to the zero function and $\{f_n\}_{n\ge 1}$ is uniformly bounded by $M$ can we prove that ...
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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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1answer
22 views

Uniform convergence- Point-wise convergence. Doubt regarding the difference

I know the definition of "pointwise convergence" and "uniform convergence", nevertheless I have some difficulties understanding the difference between those two concepts. My book defines Uniform ...
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2answers
27 views

Uniform or pointwise convergence of a sequence of functions

My problem is that I find it kind of hard to contrast between uniform and pointwise convergence. For example with this proof I'm not quite sure whether I have proven uniform or poitwise convergence: $...
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18 views

Uniform boundedness of pointwise convergent functions

Suppose $(f_{n})_{n\geq 1}$ is a sequence of bounded functions on an interval $(a,b)$ (may be of infinite length) converging pointwise to a function $f$. Can one conclude that the sequence of $L^{\...
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1answer
39 views

Prove that $(f_n)$, $f_n =x^n$, $x \in (0,1)$ is not uniformly convergent on $(0,1)$

Question 1. Prove that $f_n:(0,1) \to \mathbb{R}$ is not uniformly convergent on $(0,1)$, where $f_n = x^n , n\in \mathbb{N}$ . Proof: We need to show that, $\forall \ k \in \mathbb{N} $, $\exists \...
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Show uniform convergence and pointwise convergence for $\sum_{n=1}^ \infty \frac{z^ {n-1}}{(1-z^n)(1-z^ {n+1})}$

Consider the series: $$\sum_{n=1}^ \infty \frac{z^ {n-1}}{(1-z^n)(1-z^ {n+1})}$$ show this converges to: (a) $\frac{1}{(1-z)^2}$ for $|z|<1$ (b) $\frac{1}{z(1-z)^2}$ for $|z|>1$ ...
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1answer
25 views

Pointwise convergence of $h_{n}(x)$ on [0,$\infty$)

I know that it converges pointwise to $1$ if $x>0$ and to $0$ if $x=0$ using limits . But I am struggling to show this formally. Any help would be greatly appreciated . Thanks
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Pointwise convergence of $x^n$ in $[0,1]$

I am reading a book where they state that $f_n(x) = x^n$ converges pointwise to $f(x) = \chi_{\{1\}}(x)$ with respect to the norm $\sup_{x\in[0,1]}|f_n(x)-f(x)|$ and two pages later they state that it ...
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19 views

Heuristic argument for pointwise convergence imply uniform convergence

I didn't want put the statement precisely in the title of the topic to not have a long title, but the items below describe precisely when pointwise convergence imply uniform convergence: Every ...
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1answer
34 views

Let $g_n : [0,\frac{ 1} {2} ] → \mathbb R$ by $g_1 = g$ and $g_{n+1}(t) = \int_0^t g_n(s) ds,$ for all $n ≥ 1.$ Show that $\lim_{n→∞} n!g_n(t) = 0,$

Let $g : [0,\frac{ 1} {2} ] → \mathbb R$ be a continuous function. Define $g_n : [0,\frac{ 1} {2} ] → \mathbb R$ by $g_1 = g$ and $g_{n+1}(t) = \int_0^t g_n(s) ds,$ for all $n ≥ 1.$ Show that $\...
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1answer
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A precise definition for a sequence of functions that “Converge Pointwise, but not Uniformly”

I am trying to lay down a precise deifinition of "Pointwise Convergent but NOT Uniformly Convergent" sequence of functions $(f_n)$ with domain $D \subset \mathbb{R}$ and codomain $\mathbb{R}$. We ...
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1answer
21 views

Pointwise convergence implies convergence in the norm

Let $A$ and $B$ be normed spaces with norms $||\cdot||_A$ and $||\cdot||_B$ respectively, and let $\mathcal L(A;B)$ be the normed space of linear transformations from $A$ to $B$, with the norm $$||T||...
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38 views

Prove $φ_n =2^{-1+2n}(-1+2^n)\to f$ pointwise

(Particular case of the general theorem and proof) Let $f:X\to[0,\infty]$ be measurable function defined on the interval $(k2^{-n},(k+1)2^{-n}]$ and $φ_n:X\to [0,\infty)$ monotone increasing ...
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1answer
27 views

Rigorous construction of the pointwise limit of a sequence of random variables

Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space and let $$X_1,X_2,X_3,... \: \Omega \rightarrow \mathbb{R} $$ be a sequence of random variables. Moreover, let there be an event $A \...
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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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1answer
26 views

is $f_n=x^{-n} $convergent point wise? uniformly?

I believe on [1,2] $$f_n=x^{-n} $$ is point wise convergent because in n tends to infinity, there can be a function like f=0, such that $$|f_n-f|$$ tends to zero. hence this function is point wise ...
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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 ...
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1answer
27 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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2answers
39 views

Uniform convergence of the following series:

Does the following series converge uniformly ?$$\sum_{n=0}^{\infty} \frac{-1^n} {x+n} \ for \ (x\in R^+)$$ My thought: for pointwise convergence: $$ \lim_{n \rightarrow \infty} \frac{-1^n} {x+n} =0 $$...
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1answer
24 views

Showing that the pointwise limit of continuous functions equals its supremum somewhere on compact domain

I'd appreciate hints on proving the following theorem: If $f(x) = \lim_{n\to\infty} f_n(x)$ for each $x \in [0,1]$ and $M = \sup_{x\in[0,1]} f(x)$, then there is $t \in [0,1]$ such that $f(t) = M$. ...
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1answer
34 views

What is the closure of the set of continuous bounded real valued functions on $\mathbb{R}^d$ under pointwise convergence?

What is the closure of the set of continuous bounded real-valued functions on $\mathbb{R}^d$ under pointwise convergence? How might one go about finding this closure? Also, are there common names for ...
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45 views

Fourier partial sums of Sawtooth wave are not equal its convolution with the Dirichlet kernel!

Let $f$ be the $2\pi$-periodic function relating \begin{equation} f(x) = \frac{\pi-x}{2} \end{equation} on $(0, 2\pi)$. The coefficients of its Fourier series are easily calculated [see (*), ...
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Proof Verification for Uniform Convergence on Sequence of Functions

just looking for a verification on a proof. Thanks in Advance Let $f_n$ be a sequence of functions such that $f_n=\frac{x^{2n}}{1+x^{2n}}$ defined on $[-2,2]$. Prove or Disprove Uniform Convergence ...
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3answers
76 views

Pointwise and uniform convergence of $\sum\limits_{n=1}^{+\infty}(-1)^n\frac{x}{x+e^{-nx}}$ $\quad$ $x\in R$ [closed]

How can I prove pointwise and uniform convergence of $\sum\limits_{n=1}^{+\infty}(-1)^n\frac{x}{x+e^{-nx}}$ $\quad$ for $x\in \mathbb{R}$? uniform convergence: the pointwise convergence is on $E=(-\...
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1answer
39 views

A difficulty in understanding an example of a remark on pointwise convergence of Fourier series.

The theorem and the remark and an example on a remark are given below: But I do not understand the example $f(x) = x$, specifically I do not understand: 1- what do the author mean by "$f(x) = x$ as ...
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31 views

Proof of Taylor series convergence

These problems deal with the Taylor series $f$ at the point $a$ which is $\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$ and let $A\subset \Bbb{R}$ be an interval, $a\in A$ and $f : A \rightarrow \...
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1answer
45 views

Prove uniform convergence of $\sum\limits_{n=1}^\infty \frac{-1}{n(nx+1)^2}$ on $[a,\infty)$

I need to prove that the limit function of a function series is differentiable on $[a,\infty)$, where $a>0$. I wanted to use the theorem that the function series has a derivative if the function is ...
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54 views

Does $L_1$ convergence of continuous functions imply pointwise convergence?

Suppose that $(f_n)$ is a sequence in $C[0,1]$ which is convergent with respect to the $L_1$ norm. Then is $(f_n(x))$ necessarily convergent for all $x\in[0,1]$? I'm pretty sure the answer is no, ...
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48 views

Pointwise convergence of $\sum_{n=0}^\infty \frac{1}{2^n\sqrt{1+nx}}$

Given this series : $$ \sum_{n=0}^\infty \frac{1}{2^n*\sqrt{1+nx}} $$ I have to prove for which $x \geq 0$ the series converges pointwise. if $x=0$ the series is : $$\sum_{n=0}^\infty \Big(\frac{1}{...
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1answer
54 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
13 views

With what extra condition interchange limit and integration allowed?

$f_n$ is sequence of continuous function which converges pointwise to continuous function. with what condition we can interchange limit and integration? I know that if sequence converges uniformly ...
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1answer
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Proving a sequence of functions converges, is differentiable, etc.

I have this question asking to prove that a sequence of functions is differentiable, pointwise and uniformly convergent, and something regarding the equality of limits of the function's derivative. ...
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Pointwise and uniform convergence $\sum_{n=1}^{\infty}\frac{(n+1)^n-n^n}{n!}x^{n^n}$

I am in deadlock studyibg the pointwise and uniform convergence of the following series: $$\sum_{n=1}^{\infty}\frac{(n+1)^n-n^n}{n!}x^{n^n}$$ Maybe should I handle it as a power series? But how? Any ...
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Pointwise limit function of a piecewise function

Find the pointwise limit function of: $$f_n(x)=\begin{cases} 0 & |x|> 1/n \\ nx+1 & x \in [-1/n, 0) \\ 1-nx & x \in [0, 1/n] \end{cases} $$ I think that in ...
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1answer
37 views

Bounded sequence of functions implies convergent subsequence

Here you can see my attempt at the proof. I am sure I did something wrong because my prof asked me to show it for rationals and I "somehow" showed it for all reals. I would appreciate it if someone ...
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4answers
351 views

What is the main difference between pointwise and uniform convergence as defined here?

I have a little confusion here. I have seen the following several times and seem to be a bit confused as to differentiating them. Let $E$ be a non-empty subset of $\Bbb{R}$. A sequence of functions $\...
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1answer
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Suppose that $f_n\to f$ and $g_n\to g$, as $n\to \infty,$ uniformly. Then, $f_n g_n\to fg,$ as $n\to \infty,$ pointwise on $E$.

Can you, please, check if the following proof is correct? Thanks for your time and effort. Suppose that $f_n\to f$ and $g_n\to g$, as $n\to \infty,$ uniformly on $E\subseteq \Bbb{R}.$ Then, $f_n ...
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In functional analysis, is there a commonly accepted short-hand notation for specific types of convergence?

In math literature on functional analysis I found various short-hand notations for specific types of convergence, e.g. a single right arrow for pointwise convergence $$f_n(x) \underset{n \to \infty}{\...
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1answer
30 views

Manipulations with convergence a.e.

Let functions $f_n$ be measurable, $n \in N$, $f_n\rightarrow f$ almost everywhere. Prove that $\operatorname{arctg}f_n \rightarrow \operatorname{arctg}f$ almost everywhere. Honestly speaking, we ...
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1answer
40 views

Upper semicontinuous function as a poinwise limit of continuous fuctions

The encyclopedia of mathematics claims, without proof, that an upper semicontinuous function on a completely regular topological space X is the pointwise limit of a decreasing sequence of continuous ...
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35 views

Question on Existence of Limit from Monotone Convergence Theorem for non-negative measurable functions

Note $E^{*}$ is the space of non-negative measurable functions. In our lectures, it is written: Let $(f_{n})_{n}\subseteq E^{*}$ and $0 \leq f_{n} \leq f_{n+1}$, $\forall n \in \mathbb N$. $\...
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1answer
38 views

Do the property of $f_n$ carry over to $f_n'$?

Suppose a sequence of differentiable functions $f _ { n } : \mathbb { R } \rightarrow [ 0,1 ]$ converges pointwise to the zero function. Does it follow that the derivatives $f _ { n } ^ { \prime }$ ...
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1answer
32 views

Proof that pointwise convergence can disrupt convergence

I'm trying to get a grasp on point-wise convergence and am hoping to prove something to give a concrete example of why it's weak. The lemma goes as follows . Suppose $f _ { n } : [ a , b ] \...
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1answer
359 views

Examples of some Pointwise Convergent Sequences of Functions

I have recently come across pointwise/uniformly convergent sequences of functions, and I am hoping if someone could give some examples of certain sequences of functions so that I could understand the ...
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2answers
103 views

Continuous function as pointwise limit but not as uniform limit of a sequence of continuous functions on $[0,1]$

I recentaly find an article where it is said that there is a sequence of continuous functions $\{f_n:[0,1]\rightarrow\Bbb R\}_{n\in \Bbb N}$ that converges pointqise almost everywhere to zero function ...
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2answers
53 views

Prove that the function $f_{n}(x) = \frac{1 - |x|^{n}}{1 + |x|^{n}}$ converges pointwise for $x\in \mathbb{R}$.

I want to show that the function $$f(x) = \frac{1 - |x|^{n}}{1 + |x|^{n}} $$ converges pointwise for all $x\in \mathbb{R}$. Furthermore, there are some intervals $(a, b)$ on which the function ...