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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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Limits of functions in $L^p$ spaces and Hölder inequality

I have a severe problem understanding $L^p$ spaces and everything related. For example, see my thoughts on the following exercise: Let $f_n \in L^1(0,1) \cap L^2(0,1)$ for $n = 1, 2, 3, \ldots$ and ...
arridadiyaat's user avatar
2 votes
0 answers
52 views

Study the series of functions $\sum_{n=1}^{\infty} \frac{\cos(nx)}{n^2 + n}$ regarding pointwise and uniform convergence. [closed]

I'm studying Real Analysis and I have stumbled upon the following question. Study the series of functions $$\sum_{n=1}^{\infty} \frac{\cos(nx)}{n^2 + n}$$ regarding pointwise and uniform convergence ...
justtryingtohelp's user avatar
3 votes
0 answers
43 views

Study the pointwise convergence of the series $\sum_{n=1}^{\infty} \frac{n}{1+(x-1)^2 n^3}, x \in \mathbb{R}$.

I was studying Real Analysis and I was doing the following exercise. Study the pointwise convergence of the series of functions $\sum_{n=1}^{\infty} \frac{n}{1 + (x-1)^2 n^3}, x \in \mathbb{R}.$ I'm ...
Tiago Coelho's user avatar
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If $f_n \rightarrow f$ almost everywhere and all $f_n$ are measurable, how can I redefine $f$ on a null set to make it measurable?

I am self-studying real analysis out of Folland, and I am confused at the beginning of his proof of the Dominated Convergence Theorem. It reads as follows, where all $f_n$ map from measure space $(X,M,...
Lightbulb's user avatar
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1 answer
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Pointwise convergence of $f_{n}(x) = \sin(\frac{x}{n})$

I'm trying to prove wether $f_{n}(x) = \sin(\frac{x}{n}), f_{n}:\mathbb{R}\to\mathbb{R}$ has pointwise convergence or not. My first idea is it is pointwise convergent to the function $f = 0$ because ...
Bidon's user avatar
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Convergence in comparison principles of parabolic pdes

I assume that the following equations all have sufficiently smooth strong solutions. I have demonstrated that the solution to the (imaginary-time) Schrödinger equation: \begin{equation*} \begin{...
J.J.Zou's user avatar
0 votes
2 answers
53 views

Why is this sequence of functions pointwise convergent to $0$

Consider the following sequence of functions $f_n(x):[0,1] \mapsto \mathbb{R}$ $$f_n(x)= \left\{\begin{array}{cc} n^2 x(1-n x), & x \in\left[0, \frac{1}{n}\right] \\ 0, & x \in\left(\frac{1}{n}...
sleeve chen's user avatar
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1 vote
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On Luzin's generalization of Egoroff's Theorem: the case of domain of infinite measure

According to https://en.wikipedia.org/wiki/Egorov%27s_theorem#CITEREFSaks1937 Luzin's generalization of Egoroff's Theorem reads as follows: If a measurable set A is the union of a sequence of ...
Andrija's user avatar
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1 answer
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Proving that convergence of norms and convergence a.e. implies strong convergence

I have in my notes the following theorem Theorem $(Y,\mathcal{F},\mu)$ $\sigma-$finite measure space, $p\geqslant 1$, $\{f_n\}\subset L^p(Y)$ sequence of functions, $f\in L^p(Y)$ such that $$\lim_{n\...
Mr. Feynman's user avatar
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1 answer
48 views

Alomost sure convergence

I was studying the various types of convergence and I met a question for almost sure convergence. Consider random varibales $X_n,X:\Omega\to\mathbb{R}$, $\Omega$ is a probability ($\mathbb{P}$) space. ...
R-CH2OH's user avatar
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1 answer
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$f_n \rightarrow f$ p.w. $\Rightarrow$ $X_n \xrightarrow{d} X$

Given $(X_n)$ a sequence of random variables with associated probability density functions a sequence of random variables with associated probability density functions $(f_n)$, which converge ...
clementine1001's user avatar
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Suppose that $\{f_{n}\}$ converges uniformly to $f$, and let $g_{n}(x)=f_{n}(x+1/n)$. Show that $\{g_{n}\}$ converges to $f$.

This problem is from my calculus textbook about uniform and pointwise convergence. Problem. Suppose that $\{f_{n}\}$ converges uniformly to $f$, and let $g_{n}(x)=f_{n}(x+1/n)$. Show that $\{g_{n}\}$ ...
legogubben's user avatar
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0 answers
42 views

Pointwise convergence of $\frac{x^n}{1+x^n}$ [duplicate]

It is a question of pointwise convergence and I want to know whether I did it in the correct way or not. $\lim\limits_{x \to \infty} \frac{x^n}{1+x^n}$ By dividing with $x^n$, i got $\lim\limits_{x \...
TinCan's user avatar
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5 votes
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Show $\lim\limits_{n \to \infty}\frac{f\left(nx\right)}{n^2}=0$ a.e. $x\in\mathbb{R}$ if $f \in L^1\left([0,T]\right)$ and periodic.

Show $\lim\limits_{n \to \infty}\frac{f\left(nx\right)}{n^2}=0$ a.e. $x\in\mathbb{R}$ if $f \in L^1\left([0,T]\right)$ and periodic in $\mathbb{R}$, where $T>0$ is the period. My idea is as follows:...
guoran guan's user avatar
2 votes
1 answer
62 views

Agreement of a pointwise, weak limit and a weak-$\ast$ limit of Bochner-integrable functions

Let $T$ be a positive, real number and $\Omega \subset \mathbb{R}^d$ a bounded, connected, open set. For a given $p \in [2,\infty)$, I have a sequence of functions $(f_n)_{n \in \mathbb{N}} \subset L^\...
squilliam's user avatar
1 vote
1 answer
61 views

For which $a$ does $f_n(x) = \frac{x(1-x^2)^n}{n^a}, n=1,2,3,...$ converge pointwise and uniformly on $[0,1]$?

For which $a$ does $f_n(x) = \frac{x(1-x^2)^n}{n^a}, n=1,2,3,...$ converge pointwise and uniformly on $[0,1]$? I start with $\lim_{{n \to \infty}}f_n(x) = \lim_{{n \to \infty}} \frac{x(1-x^2)^n}{n^a} =...
Karl Johan's user avatar
2 votes
1 answer
64 views

Is the set of functions with positive derivative a.e. sequentially compact over the topology of pointwise convergence?

Consider the following set $H = \{ f :[0,1]\to[0,1] | \frac{df}{dx} \geq 0 \text{ a.e.} \}$, I was wondering if this set could be a sequentially compact set over the pointwise convergence topology, ...
Igor Soares's user avatar
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0 answers
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Sequence of functions in $C^\infty ([0, 1])$ which satisfy $\int_0^1|f_i(t)|dt = 1$ and converge pointwise to 0

I think taking bump functions supported in neighborhoods of $1/n$ should work, but I wonder if there's a more explicit construction. Here's my answer so far: For every $n\in\mathbb{N}$ and real ...
danimalabares's user avatar
0 votes
0 answers
41 views

Improving proof: $f_n\longrightarrow f$ pointwise, $\{f_n\}$ uniformly Lipschitz, show $f_n\overset{}{\rightrightarrows} f$ not using Arzela-Ascoli

The problem is that, for $\{f_n\}$ real-valued compactly supported sequence of functions with $f_n\longrightarrow f$ pointwise, $\{f_n\}$ uniformly Lipschitz, say $|f_n(x)-f_n(y)|\leq M|x-y|$ for any $...
CCQ's user avatar
  • 127
0 votes
1 answer
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continuity for limit of pointwise convergent functions

We Know that "The uniform limit of Continuous Functions is continuous." and here is its proof: We want to show that $f$ is continuous at a point $x_0$, say. The condition for continuity says ...
A12345's user avatar
  • 159
2 votes
0 answers
24 views

Convergence a.e. plus convergence of $L^1$ norms implies convergence in $L^1$ [duplicate]

Let $(X, \mathcal{X}, \mu)$ be a measure space. Suppose $(f_n)_{n\in\mathbb{N}} \subset L^1(\mu)$ is such that: $f_n \to f \in L^1(\mu)$ a.e. $ \int_A | f_n | \ \mathrm{d} \mu \to \int_A |f| \ \...
Caio Lins's user avatar
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Uniform convergence impies pointwise convergence. Am I missing something?

I'm reading 'Calculus 3rd Ed.' by M. Spivak, specifically a Corollary in Chapter 24. In it, the following is stated: Let $\sum_{n=1}^\infty f_n$ converge uniformly to $f$ on $[a,b]$. (1) If each $f_n$...
A P's user avatar
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0 answers
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Show that the sequence $(f_n)_{n \in \mathbb{N}} \in (C^0([0,2]),||\cdot||_1)$, is a Cauchy-sequence, but does not converge

We consider the sequence $(f_n)_{n \in \mathbb{N}}$ defined by $f_n : [0, 2] \rightarrow \mathbb{R}$ as follows: $f_n(x) =\begin{cases} x^n, & \text{if } 0 \leq x \leq 1 \\ 1, & \...
j.primus's user avatar
3 votes
1 answer
31 views

limit of convergent series in point wise convergent series of continuous functions

Hi I was looking for $f_n : [0,1] \to \mathbb{R}$ such that every $f_n$ is continuous and $f_n$ converges pointwise to a function $f : [0,1] \to \mathbb{R}$ that is continuous as well. Then I wanted ...
Jip Helsen's user avatar
2 votes
1 answer
147 views

how do you compute the value of $\sum\limits_{n=1}^{\infty} \dfrac{(-1)^n}{4n-3}$

I know that the series $\sum\limits_{n=1}^{\infty} \dfrac{(-1)^n}{4n-3}$ is convergent by Leibniz's law. However, finding the exact sum of this series can be quite challenging. I try to evaluate out ...
ToThichToan's user avatar
0 votes
0 answers
27 views

Ensure convergence of a sequence of thresholds

Let $f$ be an increasing and continuous real function with exactly one root. Let $\tau=\min\left\{k \in \mathbb{Z}, f(k) \geq 0\right\}$ be the first integer such that f is nonnegative (i.e. the first ...
Skywear's user avatar
  • 192
1 vote
1 answer
89 views

Alternative proof for Dini's theorem. Is it correct?

Let $(X,d)$ be a compact metric space, $f_n : X \to \mathbb{R}$ a sequence of continuous functions, $f_{\infty} : X \to \mathbb{R}$ a continuous function. Now suppose that $f_n \to f$ pointwise, and ...
Rick Does Math's user avatar
1 vote
0 answers
22 views

Pointwise convergence of heat kernels on perturbed Riemannian manifolds

Thank you in advance for your comments! Let $(M,g)$ be a Riemannian manifold (in general non-compact, connected). Then the associated Dirichlet Laplacian $\Delta_g$ generates the heat semigroup $(e^{s\...
crimsonmist's user avatar
3 votes
1 answer
45 views

Uniform convergence of a series of functions depending upon a parameter

I'm unable to prove if the following series converge uniformly on $[0,+\infty)$ for the values of the parameter $\beta$ between $2$ and $3$. $$ \sum_{n\geq 1}\frac{n^\beta x}{x^4+n^4} $$ The maximum ...
Mathland's user avatar
  • 526
2 votes
1 answer
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What is the limit function of $f_n(x) := \begin{cases} (x-n+1)(n+1-x): n-1 < x < n+1\\ 0: x \leq n-1 \lor x \geq n+1 \end{cases}$?

I have a question about the convergence of $(f_n)_{n \in \mathbb{N}}$ with $$ f_n(x) := \begin{cases} (x-n+1)(n+1-x): n-1 < x < n+1\\ 0: x \leq n-1 \lor x \geq n+1 \end{cases}. $$ Does this ...
Felix Gervasi's user avatar
0 votes
0 answers
30 views

Rate of convergence of a threshold defined with sequences of functions

Let $f$ be a real decreasing function (resp. $g$ a real increasing function), and $(f_n)$ be a sequence of real decreasing functions (resp. $(g_n)$ be a sequence of real increasing functions) such ...
Skywear's user avatar
  • 192
0 votes
1 answer
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If $f_n\to f$ pointwise on $[0,1]$, then $\sup_{[0,1]}f_n\to\sup_{[0,1]}f$ pointwise?

Let $\{f_n(x)\}_{n=1}^\infty$ be a sequence of bounded functions that converge pointwise on $[0,1]$ to some bounded function $f(x)$. I am trying to determine whether it is true that $\sup_{x\in[0,1]}...
Ray Siplao's user avatar
0 votes
1 answer
43 views

Exchange minimum and limit for a point-wise convergent function

I have a conjecture like this $$\lim_{n\rightarrow\infty}\min_{m\in\{1,2,3\}}a_n(m)=\min_{m\in\{1,2,3\}}a(m)$$ where $a(m)=\lim_{n\rightarrow\infty}a_n(m)$ for any $m\in\{1,2,3\}$ and $n$ is interger ...
jerry's user avatar
  • 135
1 vote
0 answers
62 views

Pointwise limit of functions on $[0,1]$

i was thinking on the problem bellow but I couldn't fully solve the problem, here is the statement: Let $f:[0,1] \rightarrow [0,1]$ be a continuous function such that $\forall x \in [0,1]$ there ...
Amir Mg's user avatar
  • 124
0 votes
1 answer
60 views

Convergence in $L^p_{loc}$ implies convergence of a subsequence in $L^\infty$

Let $\Omega \subset \mathbb{R}^n$ be bounded or unbounded. Suppose we have a sequence $\{f_n\} \in L^p_{loc}(\Omega)$ such that $f_n \rightarrow f$ in $L^p_{loc}(\Omega)$ for $f \in L^p_{loc}(\Omega)$....
CBBAM's user avatar
  • 6,275
0 votes
1 answer
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Product topology and intersection of basis elements

I was wondering if you could help me settle a discussion with a co-author. Neither of us is a hard-core topologist but our current paper forced us into this, so we figured this forum was a good place ...
Paul_S's user avatar
  • 13
1 vote
0 answers
62 views

Sufficient condition for continuity of conditional expectation

Suppose we have a conditional expectation, that is, an integral of this form $\int g(x,y)f(y|x)dy$, where $g(x,y)$ and $f(y|x)$ are continuos functions in their arguments, and $f(y|x)$ is a ...
Ldt's user avatar
  • 45
0 votes
0 answers
95 views

Variants of Dirichlet function in pointwise continuity

Suppose $f$ is a function from a complete metric space $X$ to a metric space $Y$, and suppose $Y$ has points $y_{0}$, $y_{1}$ such that the subsets $f^{-1}(y_{0})$ and $f^{-1}(y_{1})$ are both dense ...
hmeng's user avatar
  • 335
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0 answers
60 views

Where does the series $\sum_{n = 1}^{+\infty} \frac{(n!)^2}{(2n)!}z^n$ converge?

$z \in \mathbb C$ here. It's a power series with coefficints $c_n := \frac{(n!)^2}{(2n)!}$, hence part of the work is the computation of the convergence radius. Here it is: $$ \frac{c_{n+1}}{c_n} = \...
user665110's user avatar
1 vote
2 answers
62 views

If a sequence of quadratic forms converges pointwise, then does the associated sequence of matrices also converge?

Consider the sequence of functions $f_k : \mathbb N \times \mathbb R^n \to [0,\infty)$ defined as $f_k(x) = x^TP_kx$, where $P_k : \mathbb N \to \mathbb S_+^n$ is a sequence of symmetric and positive ...
mhdadk's user avatar
  • 1,468
0 votes
1 answer
64 views

pointwise convergence definition

I need some help to clarify with the idea of pointwise convergence. The condition for pointwise convergence is usually given that if the limit of $f_n$ exists for each $x$ $\in$ $A$. In Stephen Abbott'...
tt99999's user avatar
  • 11
0 votes
1 answer
43 views

Uniform convergence of $g_n(x) := \frac{(x+1)^2 e^{-x} \sin x}{n^2x^2+n\sqrt n x +1}$ over $[0, +\infty)$

This is part of a larger exercise: does $$g_n : [0, +\infty) \to \mathbb R ,\ g_n(x) := \frac{(x+1)^2 e^{-x} \sin x}{n^2x^2+n\sqrt n x +1}$$ converge uniformly for $x\ge0$? The sequence of functions ...
user665110's user avatar
-1 votes
1 answer
61 views

Prove that the sequence of functions $x^nLogx$ is uniformly convergent on [0,1] [closed]

Here is an image of the question Consider the sequence of functions $(f_n)_n$ defined on [0,1] by: $f_n(x) = x^nLogx$ if $x≠ 0$ & $f_n(0)=0$ Prove that the sequence of functions $(f_n)_n$ ...
Nabil El Houssein's user avatar
0 votes
1 answer
44 views

Question on pointwise convergence and norms

We know that pointwise convergence is a necessary condition of uniform convergence. Is this also a necessary condition in general if we consider another norm different from $\Vert\cdot\Vert_{\infty}$? ...
Philipp's user avatar
  • 4,564
1 vote
0 answers
53 views

Convergence of a series of the product of two functions when one of them is shifted by one

Suppose $f(x)$ is a smooth function and $g_n(x)$ is a sequence of functions such that $$ \sum_{n=1}^\infty f^{(n)}(x) g_n(x) $$ converges for $x \in [a,b]$, where $f^{(n)}(x)$ denotes the $n$-th ...
Megatron's user avatar
  • 111
0 votes
1 answer
83 views

How to show that the series of functions $\sum_{k = 0}^\infty(-1)^k \frac{x^{2k + 1}}{2k + 1}$ is pointwise convergent to a function? [duplicate]

The problem I'm given is as follows: Consider the sequence of functions $(f_k)_k$, where $f_k(x) = (-1)^k \frac{x^{2k + 1}}{2k + 1}$ as well as the resulting series: $$A(x) = \sum_{k = 0}^\infty f_k (...
johnjorgenson's user avatar
0 votes
2 answers
93 views

Does fourier series for $f(x)=\begin{cases}-1\ \text{for}\ x \in (-\pi, 0) \\0 \ \text{for}\ x=0 \\ 1 \ \text{for} \ x\in (0,\pi)\end{cases}$ converge

Does the Fourier series for the $2\pi$ periodic function $$f(x) = \begin{cases} -1 \ \text{for} \ x \in (-\pi, 0) \\\\ 0 \ \text{for} \ x = 0 \\\\ 1 \ \text{for} \ x \in (0,\pi) \end{cases} $$ ...
Carl's user avatar
  • 539
1 vote
1 answer
46 views

Show the sequence of functions (which can graphically be described as shrinking width triangles of height 1) converges pointwise to the zero function.

I would like some help in checking the following epsilon-delta proof I have written on a case-by-case basis for the pointwise convergence of the following function. I am trying to prove that the ...
palt34's user avatar
  • 139
0 votes
1 answer
115 views

Whats the point of Pointwise Convergence

I just finished taking my intro Real Analysis class and our final unit was on sequences of functions. We talked about how a function can be pointwise convergent (ex: fn(x) = x^n on [0,1] converges PW ...
mpear617's user avatar
  • 431
0 votes
1 answer
33 views

Proving numerical method 1st-order convergence

Let $u(t)$ be the numerical solution of a differential equation $y'(t)=f(t,y)$ for $t \in (0, T]$ with initial condition $y_{0}$. Let $|w(t)|=|u(t)-y(t)|$ be the error function. I have proven that ...
Redsbefall's user avatar
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