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].
782
questions
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Convergence of $f_n(x) = \frac{\sqrt x (1-x)}{1+nx}$ and $g_n(x) = \sqrt n f_n(x)$
I have the following exercise:
Consider the following sequences of functions
$$f_n : [0,1] \to \mathbb R \,,\ f_n(x) := \frac{\sqrt x (1-x)}{1+nx}$$
$$g_n : [0,1] \to \mathbb R \,,\ g_n(x) := \sqrt n ...
- 540
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1
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Proving that convergence in RKHS implies pointwise convergence without using reproducing property
Let $(\mathcal{H}, \mathcal{K})$ be a reproducing kernel Hilbert space and denote $\mathcal{K}_x := \mathcal{K}(x, \cdot)$.
Is there a simple way to prove $f_n \to_\mathcal{H} f$ (shorthand for $\|f_n ...
- 21
-1
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Understanding the inner working of the strong LLN
Assume a probability space $(\Omega,F,P)$. Assume I have real-valued random variables $X_1,X_2,\ldots,X_n$ and $X$. We say that $X_n$ converges to $X$ pointwise if the
set $$S=\{\omega\in\Omega\vert \...
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75
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Proving that the infinite sum converges to $1$ as $k$ tends to infinity
While solving a PDE using the Eigenfunction expansion method, I approximated the following function with an infinite sum:
$$
f(x) = (1-x)e^{-kx}
= \sum_{n=0}^\infty A_n \cos(w_nx) \tag 1
$$
where $x ...
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Questions and observations regarding Putnam 2020 - A.6.
CONTEXT
My starting point is Question A.6 of Putnam 2020 competition, that goes like that
For a positive integer $N$, let $f_N$ be the function defined by
$$
f_N(x) = \sum_{n=0}^N \frac{N+1/2-n}{(N+1)...
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Does the sum of a sequence of functions $\{f_n(x)\}$ converge if the upper bound $h_n(x)$ with $f_n(x)\le h_n(x)$ converges pointwise to $0$?
I am dealing with a problem on the existence of the limit for a sequence of function $\{g_n(x),n\in\mathbb N\}$ with function $g_n(x):\mathbb R^n\to\mathbb R^m$, i.e. to judge the existence of limit $...
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3
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Confusion about "$:=$" notation in pointwise convergence definition
My instructor's lecture notes state:
Let $\{f_n:E\rightarrow\mathbb{R}\}_{n\in\mathbb{N}}$ be a sequence of functions defined on nonempty $E\subset\mathbb{R}$. We say that the sequence converges ...
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A non-pointwise convergent positive fucntion and its L1 limit.
I have a positive function $f(t,x,y)$ ($0\leq f(t,x,y)\leq 1$ for all $x,y\in\mathbb{R}$, $t\in[0,\infty)$) . Let us further assume that $\lim_{t\rightarrow\infty}f(t,x,y)$ does not exist. Now, let $\...
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Convergence and question on tutor's claim
In my analysis class we were given the following problem:
Let $(f_n)_{n\in \mathbb{N}}$ be a sequence of continuous functions in $[0,1]$ and assume they converge uniformly to some $f:[0,1] \to \mathbb{...
- 378
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2
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Uniform convergence of sequence of functions gn and fn
Given two functions
$$f_n (x)= x\left(1+\frac{1}{n} \right), x \in \mathbb{R}, n \in \mathbb{N}$$
And
$$g_n (x)= \begin{cases} \frac{1}{n} ,& x \space \text{is irrational} \space \text{or} \space ...
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I can't seem to find a formula for the partial sums of this infinite series.
I'm working on a problem set for my partial differential equations class. We are studying convergence of Fourier Series. One of the questions is asking me to compute whether
$$\sum_{n=0}^\infty (-1)^n ...
- 11
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70
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Series of functions $\log(1+x^{2n})$
Does the series of functions $\sum_{n=0}^{+\infty} \log(1+x^{2n})$ converges for $|x|<1$?
Obviously I've observed that for $x=0$ the series is convergent and for $x\ge1$ is divergent.
How should I ...
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"Measuring the Gap" in Fatou's Lemma
Tao's Introduction to Measure Theory asks me to prove the following result, named "Defect version of Fatou's lemma"
Let $(X,\mathcal{B},\mu)$ be a measure space, and let $f_1,f_2,\ldots : X ...
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2
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A weighted infinite sum of functions attains its maximum?
Let $f_n:[0,1]\to[0,1]$ be a sequence of continuous functions with $\sum_{n\ge 1}f_n(x)=1$ for all $x\in[0,1]$. Let $a_n$ be a sequence of numbers in $[0,1]$. Can we show that the function $$S:[0,1]\...
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2
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Setwise limit of measures is a measure, is that true for nets as well?
It is well known that the setwise limit of a sequence of measures is a measure.
Is the same true for nets? (Note that the proof given in the link above relies crucially on Radon-Nikodym, and so it ...
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Setwise convergence of measures, apparent paradox
I have come across an apparent contradiction while working with the setwise topology of probability measures. Can someone please point out the mistake to me?
Let $(X,\mathcal{A})$ be a standard Borel ...
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1
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Prove that the sequence of functions converges pointwise
Let
For $n\in \mathbb{N}$ and $k\in \{0, 1, 2, ..., 2^{n}-1 \}$ is defined
$$I_{k}^{n}=\left[\frac{k}{2^{n}}, \frac{k+1}{2^{n}}\right)$$
and $f_{n}:[0, 1) \rightarrow \mathbb{R}$ is defined by
$$f_{n}(...
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Sequence of bump functions with an additional integral condition on the gradient
My question is simple: Is there a sequence of functions $f_n\in C_c^\infty(\mathbb R^d)$ such that
$0\le f_n\le 1$ for all $n$
$f_n\to 1$ pointwise
$\int|\nabla f_n|^2\,dx\le 1/n$ for all $n$?
I ...
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Limit of continuous functions is Riemann integrable
Here is an analysis problem I'm stuck on:
Let $f\in C^0([0,1])$ with $f(0)=0$ and $f$ increasing and convex. Define:
$$ f_n(x) = n\big[f(x)-f(x-\tfrac{1}{n})\big] $$
Show:
$f(1-\tfrac{1}{n})\le\...
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75
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Weak convergence does not imply joint weak convergence?
Suppose that $X_n\Rightarrow X$ and $Y_n\Rightarrow Y$ as $n\to\infty$ where "$\Rightarrow$" means convergence in distribution. We know that it does NOT imply that $(X_n,Y_n)\Rightarrow (X,Y)...
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Constructing particular sequence of finitely supported measures converging to a given measure
Let $(\Omega, \mathcal{B}, \mu)$ be a measure space where $\Omega \subset \mathbb{R}^K$ is the unit sphere in the $K$-dimensional space, and $\mu$ is a finite, positive measure (not necessarily a ...
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Convergence in a complete measure implies a.e. convergence for a subsequence
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple functions from $X$ ...
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Let $\|f_n-f\|_{L^p_{\text{loc}}} \to 0$. There is a subsequence $(n_k)$ such that $f_{n_k} \xrightarrow{k \to \infty} f$ a.e.
Let $p \in [1, \infty)$ and $Y:= \mathbb R^d$. Let $L^p_{\text{loc}} (Y)$ be the space of measurable functions $f:Y \to \mathbb R$ such that
$$
\|f\|_{L^p_{\text{loc}}} := \sup_{y \in Y} \|1_{B(y, 1)} ...
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Do subdifferentials of a uniformly converging sequence of convex functions converge pointwise?
Consider a sequence of continuous convex functions, $f_n,f:A \rightarrow \mathbb{R}$, where $A \subseteq \mathbb{R}^n$, compact. The sequence $f_n$ converges uniformly to $f$. Does $\nabla f_n$ ...
- 1,461
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Weak convergence when different parts of the the support of the distribution "converge at different rates" (Corrected version)
This is a follow up of this question, which contained a mistake, as very helpfully pointed out by Snoop. Hence, the corrected version.
Let $F$ be the CDF of a distribution, supported on $[-1,1]$. $F$ ...
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1
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Weak convergence when different parts of the the support of the distribution "converge at different rates"
Let $F$ be the CDF of a distrubtion, supported on $[-1,1]$. It has a density $f$ (See footnote 1). Consider the sequence of finitely supported distributions defined as follows.
Take $t\in [-1,1]$, $n \...
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$f_n=\frac{1}{x^{\frac{1}{n}}}$ uniformly converges to 1 [closed]
This is the problem that I am trying to prove. Show that $f_n(x)=\frac{1}{x^{\frac{1}{n}}}$ where $x\in (0,\frac{1}{2})$ is uniformly convergent to $f(x)=1$. It will be great if someone can tell me ...
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Almost everywhere convergence in product measure and that in coordinate ones
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the ...
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1
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CDF of a convergent positive series
Let $Y_0, Y_1, \ldots$ be an i.i.d. random sequence such that
$$ \mathbb{P}(Y_k = 0) \;=\; 1 - \mathbb{P}(Y_k = 1) \;=\; p \qquad \text{for each $k\ge 0$}. $$
I am interested in the following random ...
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How to show that $f(x)=x^n$ does not converge uniformly in domain $D=[0,1)$.
Suppose $f_n:\mathbb{R}\to\mathbb{R}$ is defined by $f_n(x)=x^n$ where $n\in\mathbb{N}$, and $D=[0,1)$. I wish to show that $f_n(x)=x^n$ does not converge to $f(x)=0$ uniformly as $n\to\infty$, i.e.
$...
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2
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How to show that $f(x)=x^n$ converges pointwise in domain $D=[0,1)$.
Suppose $f_n:\mathbb{R}\to\mathbb{R}$ is defined by $f_n(x)=x^n$ where $n\in\mathbb{N}$, and $D=[0,1)$. I wish to show that $f(x)=x^n$ converges to $f_n(x)=0$ pointwise as $n\to\infty$, i.e.
$\forall\...
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Convergence of the perimeter of level sets
Suppose you have a sequence of $C^1$ functions $\{\phi_n\}_{n\in \mathbb{N}}$ defined on $\mathbb{R}^n$ that converges in $C^{1}_{\mathrm{loc}}$ to a function $\phi$, as $n \to +\infty$. By $C^1_{\...
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Definition of pointwise convergence for a difference scheme (numerical PDEs)
I am currently reading Numerical Partial Differential Equations: Finite Difference Methods by J.W. Thomas and I'm trying to get a better grasp of the following definition on page 42.
Definition 2.2.1....
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0
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Sequences convergence uniform convergence and impact of boundedness of the differentrial [duplicate]
Let $f_{n}:\left[0,1\right]\to\mathbb{R}$ be a sequence of differentiable functions converging to $f:\left[0,1\right]\to\mathbb{R}$ pointwise. Assume that there exists a constant $M>0$ such that $|...
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Pointwise convergence of a function where convergence to multiple values occur at a single point
Consider,
$$
f_n(x)=\frac{1-nx^2}{(1+nx^2)^2}
$$
where, $$x \in \mathbb{R}, n \in \mathbb{N}$$
It is clear to me that not including $x = 0$, each function point converges to $0$ as $n \to \infty$.
The ...
- 65
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1
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Monotone convergence theorem on a function of two variables (integrating over one variable)
Consider a measurable space $\big( E \times F, \mathcal{E} \otimes \mathcal{F} \big)$, an $(\mathcal{E} \otimes \mathcal{F})$-measurable and positive function $f$, and a measure $\nu$ on $\big( F, \...
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2
answers
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Uniform convergence of $\sin(nx) \cdot\frac{x}{n} + x$
I've shown that $f_n(x) = \sin(nx)\cdot \frac{x}{n} + x$ converges pointwise to $f(x) = x$. Now if I consider $|f_n(x) - f(x)| = |\frac{x}{n}\sin(nx)|$ it looks that it is dependent over $x$ so I ...
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15
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Applying convergence space to "convergence in measure" and processes
I read up on "Convergence space" on Wikipedia and An initiation into convergence theory - Szymon Dolecki. I think I get the general idea. However, I study within applied math, so I am trying ...
- 1,055
2
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1
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75
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Pointwise convergence of generalized inverse function [duplicate]
I am reading Resnick's Extreme Values, Regular Variation, and Point Processes. In chapter 0.2 he writes about the generalized inverse of a non-decrasing function F: $$F^{\leftarrow}(y):=\inf\{x:F(x)\...
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1
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19
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Bound for the difference between two functions
Consider $f_1, f_2, g_1, g_2$ four continuous functions defined on the real line.
I know that for every $x \in \mathbb{R}$
$$0 \leq |f_1(x)| \leq |g_1(x)|, \quad 0 \leq |f_2(x)| \leq |g_2(x)|$$
If I ...
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1
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35
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If a sequence of finite signed measures converges the sequence of their "maximizers" converges?
$F^1, F^2$ are the CDFs corresponding to two distributions corresponding to positive finite measures, supported in $[0,1]$, i.e. $F^i(x)$ is the measure of the set $[0,x]$ under the $i$-th measure. ...
- 1,461
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0
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45
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Weak Convergence implies Pointwise Convergence (on a Countable set)
Let $\mathcal{C} = \{x_1, x_2, \cdots\}$ be a countable set of $\mathbb{R}$. Let $\{\mathbb{P}_n\}$
and $\mathbb{P}$ be probability measures on $\mathcal{C}$. Prove that $\mathbb{P}_n \stackrel{w}{\...
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2
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68
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Probability convergence of a martingale defined as iid random variable product
Let $(\beta_n)_{n \geq 1}$ be positive independent and identically distributed random variables with $\mathbb{E}[\beta_1] = 1$ and $\mathbb{P}[\beta_1 < 1 ] > 0$.
Define the martingale $M_n = \...
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0
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90
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Fundamental theorem of $\Gamma$-convergence
In the paper "A handbook of Γ-convergence" I've read the following:
"This is the fundamental theorem of Γ-convergence, that is summarized by the implication
Γ-convergence + ...
1
vote
1
answer
32
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Pointwise and uniform convergence for particularly well-behaved functions
I'm asking myself...
if $f_n(x):\mathbb{R}\to\mathbb{R}$ are infinitely differentiable functions, and each $f_n$ is such that $f_n(x)\underbrace{=}_{|x|\to\infty}\mathcal{O}(|x|^{-N})$ for any $N\in\...
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0
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24
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prove there is no pointwise convergence of $\sin(nx)$ [duplicate]
I was reading a calculus book and there was this task:
prove that you cannot select a pointwise convergent subsequence from the sequence $$f_n(x) = \sin(nx)$$
consider that it's on $$[0, 1]$$
I've ...
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1
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116
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Initial Topology and Weak and Strong Operator Topology
Given a set $X$ and a family of topological spaces $(Y_i)_{i \in I}$ and functions $f_i : X \to Y_i$, we can define a topology on $X$ called the initial topology on $X$ with respect to the topological ...
- 7,770
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0
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30
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Convergence almost everywhere criterion
I was reading Loeve's Probability Theory and I saw the following criterion for determining convergence almost everywhere:
Convergence a.e. criterion. Let $X, X_n$ be finite measurable functions.
$ ...
- 11
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0
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75
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Summing Over Infinitely Many Frequencies
Given $a,b>0$, let $f_n:\mathbb{R}\to\mathbb{R}$ for $n>0$ be the random variable $t\mapsto(1/\sqrt{n})\sum_{k<n}\sin(t/(a+kb/n)+\phi_k)$ where $\phi$ is uniformly distributed in $[0,2\pi)^n$....
- 3,522
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0
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34
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Intuition in choosing finite set of points in proof that Pointwise convergence implies convergence in pointwise topology
I am trying to understand this answer. I am trying to understand this answer. In both directions of proof, I am confused why we take a finite subset of the indexing set. Someone seems to have had the ...
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