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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Convergence of a mollifier function.
Let $\gamma : [0,1] \rightarrow \mathbb{R}^{2}$ continuous function such as $\gamma(0)=\gamma(1)$.
Let $\rho_{n}$ unit mollifier. Let, for $n\geq 1$,
$A_{n}:=\gamma \star \rho_{n}([0,1])$ and $A=\...
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Limit of functions with modulus of continuity has modulus of continuity itself
If we have a sequence in $\mathbb{R}$ such that $f_n:\mathbb{R}\to\mathbb{R}$ and every $f_n$ has some finite modulus of continuity $\omega_{f_n}$ and $f_n \to f$ for some $f$, then $f$ should also ...
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Monotone approximation of integrable functions
It is well known that every $f \in L^1(0,1)$ (Lebesgue integrable functions on $(0,1)$) can be approximated by continuous (or even smooth) functions, in the sense that there exists a sequence of ...
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Does not exist any $x \in [0, 1]$ for which the sequence $f_{n}(x)$ converges?
Does there exist a sequence of continuous functions $f_{n}: [0,1] \rightarrow [0,\infty)$ such that $$\lim_{n \rightarrow \infty }\int_{0}^{1}f_{n}(x)dx = 0$$
but there does not exist any $x \in [0, ...
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Pointwise limit and limsup of sets
I don't think this problem should be too difficult, but I'm not sure on some details. Give a sequence of functions $f_{n} : \mathbb{R} \rightarrow \mathbb{R}$ converging for any $x \in \mathbb{R}$ to $...
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Prove that an announcing sequence converges point wise to a stopping time
In the proof of Theorem 1 in this post of George Lowther's Blog (proof of implication $3\Rightarrow 4$) we have a sequence of stopping times $\{T_k\}$ and a stopping time $T$ and we know that $T_k\leq ...
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Find conditions for function f for which the limit will be zero
I have limit of this expression:
$$ \lim_{t\to 0^{+}} \frac{1}{e^{t}\sqrt{\pi}} \int_{-\infty}^{\infty } (f(x-t+\sqrt{4t}s) -f(x)) \, e^{-s^2} ds \, , $$ where $ \textit{f}:\mathbb{R} \rightarrow \...
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Convergence sequence of functions
Consider the sequence of functions $f_n(x) = 3x^n + 2x$. For what values of $x$ there exists a
pointwise limit of this sequence as $n → ∞$? For what values of $a ∈ \Bbb R≥0$ this sequence converges ...
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Proof that this sequence of functions converges pointwise.
Consider the sequence of functions $f_{1}, f_{2}, \ldots$ on $[0,1]$ defined by $f_{k}(x) = x^{k}$ for all $x \in [0,1]$ and $k \in \mathbb{N}$. Show that this sequence converges pointwise to the ...
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Proving Pointwise Convergence of $f_n(x) = \frac{x}{1 + x^n}$
I am currently trying to show that the sequence of functions defined by $f_n(x) = \frac{x}{1 + x^n}$ converges pointwise on $U = [0, \infty)$. I have found the limits for the three specific cases and ...
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Pointwise convergence of Fourier transform in L2
I am wondering the following problem: given $f_n\in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$ and $f\in L^2(R)$, with $f_n\to f$ in $L^2$, how does $\hat{f_n}$ converge to $\hat{f}$? The $L^2$ convergence ...
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Show pointwise and uniform convergence on an example
I wrote down my proof, however it may not be formal. Can you please suggest improvements?
Let ${g_n}(x) = \begin{cases}
0, & \text{if x in [-n,n]} \\
f(x), & \text{if otherwise}
\end{cases}$
...
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Derivative of a pointwise limit of a sequence of functions
It is easy to construct a sequence of differentiable functions $f_n(x)$ converging pointwise to a function that is not differentiable (a simple example is $\tan^{-1}(nx)$). And it is a theorem that if ...
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$\sigma$-linearity of the Radon-Nikodym operator.
Let $(X,\mathcal X,\nu)$ be a probability space, let $\{ \mu_n \}_{n\in\mathbb N}$ be a set of finite positive measures on $(X,\mathcal X)$ that are all absolutely continuous with respect to $\nu$. I ...
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Showing that the limits are not interchangeable in $\lim\limits_{n \to \infty} \int_{0}^{1} f_{n}(x)$
The problem : Let $f_{n}:[0,1] \to \mathbb{R}$ be defined for $n \ge 1$ by
$$
f_{n}(x):=
\begin{cases}
2n^{2} x & \text{for} & 0 \le x \le \frac{1}{2n},\\
-2n^{2} (x - \frac{1}{n}) & \...
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Show the existence and Calcualte the $\lim_{n \rightarrow \infty} \int_{(0,\infty)} (\frac{|\sin x|}{x^2}\frac{x^{1/n}}{1+x^{1/n}})dλ(x)$
so I have to determine whether the limit for $n \rightarrow \infty$ exists and, if so, calculate it.
The integral is:
$L_n=\int_{(0,\infty)}(\frac{|\sin x|}{x^2}\frac{x^{1/n}}{1+x^{1/n}})dλ(x)$
So I ...
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Show the existence and compute $\lim_{n \rightarrow \infty} \int_{(0,\infty)}(\frac{ne^x+1}{ne^{2x}+4x^2})dλ(x)$
I have to determine whether the limit for $n \rightarrow \infty$ exists and, if so, I have to compute it
The Integral is:
$$K_n= \int_{(0,\infty)}\left(\frac{ne^x+1}{ne^{2x}+4x^2}\right)dλ(x)$$
I want ...
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Show the existence and calculate the $\lim_{n \rightarrow \infty}\int_{[0,n]}(\frac{1}{n}(1+\frac{x}{n})e^{\frac{-x}{n}})dλ(x)$
I wanted to know if I am right with this problem.
I have to determine whether the limit for $n \rightarrow \infty$ exists and, if so, I have to calculate it.
The Integral is:
$J_n=\int_{[0,n]}(\frac{1}...
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Pointwise convergence with convergence in measure to some function
I have the following question about measure theory:
Let $(X,\mathcal A,\mu)$ be a measure space, and let $A$ be a measurable set. We say that $(f_n(x))_n$ convergences in measure $\mu$ to the ...
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Show that if $X_n$ converges to $X$, also $|X_n|$ converges to $|X|$ almost surely. [closed]
Show that if $X_n$ converges to $X$, also $|X_n|$ converges to $|X|$ almost surely.
I know the fact that this will surely hold in the case of the convergence in probability, but I don't know how to ...
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Subsequence in convergence of integrals
In an article I'm currently reading, a reasoning is used that I don't understand.
We have an integral of a function over a domain with both depending on the same $\epsilon>0$. They show that $$\...
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How can I prove that this subset $C$ is closed in $C[0,1]$?
Let $C=\{f\in C[0,1]: f(0)=f(1)\}$. I need to show that $C$ is closed in $C[0,1]$ with respect to $||f||_{\infty}=\max(f)$.
I know that I can define $\phi(f)=f(0)-f(1)$ and then argue that $\phi$ is ...
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Uniqueness of the limit and finiteness of the measure space
We know that the convergence almost surely/everywhere implies the convergence in probability/measure.
But is this also true when the measure space is not sigma finite since this can lead to different ...
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Why doesn't any pointwise convergent imply uniform?
Roughly speaking, the difference between the pointwise convergence and the uniform convergence is the N's dependence on x. So for pointwise convergence in domain $E$:
$$ \exists N \text{ s.t. } |f_n(x)...
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Uniqueness of the limit in measure on a NON-sigma-finite space
Limits in measure on a non-$\sigma$-finite measure space $(X, \mathcal{A} , \mu)$ need not
be unique.
May someone explain me why, better with an example?
Is it possible that, in the case of non $\...
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Convergence of arguments in optimization
This question is motivated by non-parametric maximum likelihood estimation is statistics but I guess it applies more generally to any optimization problem.
Let $\{x_1,x_2,\dots,x_n\}$ be a data sample ...
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Fourier series pointwise convergence
As far as I know, there are theorems like Jordan-Dini’s that ensure pointwise convergence of the Fourier Series of a periodic function to the function itself given that one-sided limits and ...
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Uniform integrability and $L^1$ convergence of $(1/X_n)$
Let $X_n \to c > 0$ almost surely, where $c$ is a constant and $X_n > 0$ for all $n$. Also, let $(X_n)$ be uniformly integrable, so in particular $X_n \to c$ in $L^1$.
Question:
Do we have ...
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If $(f_n)$ converges pointwise to $f$, then $f$ is uniformly continuous
I am preparing for my Analysis $2$ final and am looking over some problems on past homeworks and exams that I could not solve. This is one of them:
Prove that if $(f_n)_{n\in\mathbb{N}}$ is ...
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intuition for $\lim_{n \to \infty} f_n'(x) \ne f'(x)$ (Tom Apostol's exercise $11.7.18$)
Let $f_n(x) = \frac{\sin nx}{n}$ and $f(x) = \lim_{n \to \infty} f_n(x)$. Therefore, for all fixed $x$, $f(x) = 0$
The derivatives are then (for all fixed $x$): $f_n'(x) = \cos nx$ and $f'(x) = 0$.
I ...
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If $f_n \to f$ and for every $n > 0$ $f_n$ is uniformly continuous then f is uniformly continuous. Where did my proof fail?
Let $[a,b] \subseteq \Bbb R$, let $f_n:[a,b] \to \Bbb R \space\space$ such that $\forall n>0 \space\space f_n$ is uniformly continuous on $[a,b]$. Let $f:[a,b] \to \Bbb R$, Suppose that $f_n \to f \...
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Finding the interval of convergence of the infinite series $\sum_{n=1}^{\infty}\frac{2^n + x^n}{1+3^n x^n}$
Exercise problem 3.2.1(c) in Problems in Mathematical Analysis, by Kaczor and Nowak asks to find where the following infinite series converges pointwise:
$$\sum_{n=1}^{\infty} \frac{2^n + x^n}{1+3^n x^...
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Interval of convergence of the infinite series $\sum_{n=1}^{\infty} \frac{x^n}{1+x^n}$
I am self-learning Real Analysis and solving some exercise problems from Problems in Mathematical Analysis, by Kaczor and Nowak. I would like someone to verify if the arguments below are technically ...
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How do we show that $\sin{x}$ is in fact equal to the infinite series $\sum\limits_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}$?
Taylor's Theorem tell us that (given some assumptions) if we write a function $f(x)$ as the sum of a Taylor polynomial and a remainder term, the remainder term has a specific form
$$R_{n,a}(x)=\frac{f^...
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$\Gamma$-convergence vs. uniform convergence: conditions for convergence of minimizers of functionals
In variational problems, sometimes we want to show that minimizers of a sequence of functionals $F_n$ converge to the minimizers of some $F$. To do this, we usually show 1) $\Gamma$-convergence of $...
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Let $f_n\to f$ pointwise for measurable functions $f_n$. Is $f$ measurable?
The following is a standard result of real-valued measurable functions:
Theorem: let $\{f_n\}$ be a sequence of measurable functions $(X,\Sigma_X)\to(\mathbb{R},\mathcal{B}_{\mathbb{R}})$, where $\...
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Spivak, Ch. 24, Problem 1a: Does $\sqrt[n]{x}$ converge pointwise to $f(x)=1, x\in (0,1]$, $f(0)=0$ on $(0,1]$ or on $[0,1]$?
I have a question about pointwise convergence of a sequence of functions.
The question is very specific and simple, and is at the last section below.
Here is my understanding of the concept (based on ...
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Is converge pointwise with this property implies uniform convergence? [duplicate]
Let $(f_n)_{n\in\mathbb N}\in (\mathbb C^{[a,b]})^{\mathbb N}$ which validates the property :
$\forall \varepsilon >0, \exists\delta>0, \forall x,y\in[a,b], |x-y|\le\delta \implies \forall n\in \...
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Interval of pointwise convergence of a function series
$$ \sum _{n=0}^{\infty }\:\left(\frac{x}{3}\right)^n\sin\left(\frac{\pi \:x}{6}\right) $$
The problem is (fill blanks):
The interval of pointwise convergence of the series above is the union of the ...
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Uniform convergence of sequence of differentiable functions
Show that the sequence of differentiable functions $x^n/n$ in $[0,1]$ converges uniformly to a differentiable function $f$ in $[0,1]$. Also, show that the sequence $f'_n$ converges to a function $h$ ...
- 1,143
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Point wise and uniform convergence of sequences of functions [closed]
Analyze the uniform convergence of the following sequences of functions:
$x+(1/n)$. What can we conclude about $(x+(1/n))^2$?
$1/(1+x)^n$ in $[0,1]$. Also, study the point wise convergence of this ...
- 1,143
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pointwise convergence of $n\sin(n^2x)$
Given the sequence of functions $$f_n : [0,1] \rightarrow \mathbb{R},f_n(x) =\left\{\begin{array}{ll} n \sin(n^2x), & 0 ≤ x ≤ \frac{\pi}{n^2},\\ 0, & \text{else.} \end{array}\right. $$
...
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If $\varphi_n$ converges pointwise to $f$, then does $\varphi'_n=\max_{1\leq j \leq n}\varphi_j$ converge pointwise to $f$?
Let $f:X\to [0,\infty]$ be a function, and let $\{\varphi_n\}_{n=1}^\infty$ converge pointwise to $f$, i.e., $\forall x\in X \ ; \ \displaystyle\lim_{n\to \infty}\varphi_n(x)=f(x)$.
If we define a ...
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Show the convergence of $X_k$ and $Y_k$
Consider two sequences $(X_0, X_1, \dots)$ and $(Y_0, Y_1, \dots)$, where $X_k\in \mathbb{R}^n$ and $Y_k \in\mathbb{R}^n$. Suppose two continuous nonlinear functions $f$ and $g$, so that the ...
- 159
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26
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Pointwise limit and norm limit in a sequence space
I have a situation where $(E,\|\cdot\|)$ is a complete normed sequence space. A sequence $a_n$ is cauchy in it and its pointwise limit is zero vector. I have a doubt whether $\|a_n\|$ will also ...
- 342
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3
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Visualizing where uniform convergence fails but pointwise convergence holds
I am well acquainted with the concepts of pointwise and uniform convergence, namely on the former one fixes a point $x$ and then investigates if a given sequence of functions converges. In the latter, ...
- 2,681
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119
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Pointwise convergence in a Sturm-Liouvlle problem.
Let's consider a Sturm-Louiville problem in $[0,1]$. For me it's clear that a Fourier series of eigenfunctions converges uniformely for any continously differentiable function $F$ with $F(0)=F(1)=0$ ...
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How to find the pointwise limit of $(f_n)$ if it is defined in terms of an arbitrary positive sequence $(a_n)$?
I'm trying Exercise 8.1.2(a) from Cesar O. Aguilar's Introduction to Real Analysis. It says:
Let $(a_n)$ be a sequence of positive numbers and define $f_n:[0,1]\to\mathbb{R}$ as
$$f_n(x)=\left\{\...
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How to prove the space ${g(x)=\int_{-c}^c F(w)e^{iwt}dw: F(w)\in L^2[-c,c]}$ with the norm $l_\infty$ is a Banach space.
I want to prove the norm $\|g\| = max_{x\in R }|g(x)|$ is complete in the space ${g(x)=\int_{-c}^c F(w)e^{iwt}dw: F(w)\in L^2[-c,c]}$. But I do not know how to prove that the Cauchy sequence $g_n$ ...
- 41
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When pointwise convergence implies local uniform convergence [duplicate]
Let $f_{n}:\mathbb{R } \rightarrow \mathbb{R }$ a sequence of continous functions and $f:\mathbb{R } \rightarrow \mathbb{R }$ a continous function. If $f_{n}$ convergens pointwise to $f$, than the ...
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