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Questions tagged [uniform-convergence]

For sequences of functions, uniform convergence is a mode of convergence stronger than pointwise convergence, preserving certain properties such as continuity. This tag should be used with the tag [convergence].

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Uniform Convergence of Matrix Series

I have a question about the uniform convergence of matrix series. In Peter D. Lax's Linear Algebra and Its Applications (Page 129), he mentions: $e^A = \sum_{k=0}^{\infty} \frac{A^k}{k!}, e_m(A) = \...
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Is this set dense in the space of continuously differentiable functions on $[0,1]$. [closed]

We consider the set $$\left\{u \in C^2([0,1]) : u''(0)=\alpha_0 u'(0)+\beta_0 u(0)+\gamma_1 u(1), \quad u''(1)=\alpha_1 u'(1)+\beta_1 u(1)+\gamma_0 u(0) \right\}$$ where $\alpha_i,\beta_i,\gamma_i$ ...
walid hidda's user avatar
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Uniform convergence and differentiability, question about the proof

I have a question to the following proof of the well-known theorem. Theorem. Let $\{f_n \}$ be a sequence of functions converging to $f$ pointwise on $[a,b]$. If each $f_n$ is differentiable with ${...
serpens's user avatar
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P(sup_t $\vert X^{t}_n - X^{t}\vert$) vs sup_t P($\vert X^{t}_n - X^{t}\vert$) in probability theory

I have a question regarding probability theory which has to do with the notion of uniform convergence in probability. One considers a sequence of random variables $X_n^{t}$ parametrized by a parameter ...
Riccardo's user avatar
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Rate of Uniform Convergence of Fourier Series to a Smooth Function?

I'm wondering if there are any known results on the rate of uniform convergence of a Fourier partial sum to a smooth function ?. More specifically, I am wondering ...
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On a particular uniform convergence

Let's consider a function $\phi(x)$ that we call "fairly good", meaning that $\phi(x)$ and all its derivatives of all orders are $O(|x|^N)$, for $|x|\to\infty$, with $N\in\mathbb{N}$ known. ...
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Given that $f(x)=\sum_{n=1}^{\infty} \frac{\sin(nx)}{n^2}$, express the integral $\int_0^1 f(x) dx$ as a series. [closed]

Following my last question (Study the uniform convergence of $f (x) = \sum_{n=1}^{\infty} \frac{\sin(nx)}{n^2}$ in $\mathbb{R}$.), the second part of the problem goes as it follwos down below. ...
Tiago Coelho's user avatar
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exchange order of limit of random variable

Suppose to have a sequence of discrete random variables $X_n(\lambda)$ depending on some parameter $\lambda \ \in [0,\infty]$ and $n \in \mathbb{N}$ and that the following limits hold almost surely: $...
Riccardo's user avatar
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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 ...
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Check uniform convergence for $F_n(x)=\sin(x)^n$ [duplicate]

So I have a small question, when I tried to prove the required; check point convergence and uniform convergence for $F_n(x)=\sin(x)^n$ for $A \in [0,\pi/2)$ I tried to apply the Supremum test but a ...
Bar's user avatar
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Investigate whether $f_n (x)=\dfrac{nx}{1+n^3x^3}$ on $[0,\infty),n\in \mathbb N$ converges uniformly or not.

Investigate whether $f_n (x)=\dfrac{nx}{1+n^3x^3}$ on $[0,\infty),n\in \mathbb N$ converges uniformly or not. Obviously, $f_n(x)$ pointwise converges to $0$. Since $f_n(0)=0$ when $x=0$ and $\...
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Does Riemann sum converge uniformly?

$f$ is a continuous function defined on [0,1]. $F_{n}$ is defined as $\sum_{i=1}^{n} \frac{x}{n} f(\frac{ix}{n})$. It's not too difficult to see that $F_{n}(x)$ converges to $F =\int_{0}^{x} f$, but ...
Sam's user avatar
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Prove carefully $C^1[0,1]$ is incomplete

This post shows how to prove $C^1 [0, 1]$ is incomplete in the uniform norm. But I want to get a deeper understanding, specifically how to come up with an example. Here's my understanding: I know $C^0[...
HIH's user avatar
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Proof of uniform convergence in $ [a,\infty)$ but not $[0, \infty)$

Let $f_n(x)=\frac{x}{(1+x)^n}$. I need to prove that $f(x)=\displaystyle \sum_{n=1}^{\infty}f_n(x)$ is uniformly convergent in $[a,\infty), \forall a>0$ but not uniformly convergent in $[0,\infty)$....
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If $f_n(x)\rightrightarrows f(x), ~(f_n(x)-f_n(x\frac{n}{n+1}))n\rightrightarrows g$ then $xf'(x)=g(x)$

I want to prove following statement: Let $f_n:[0,1]\to\mathbb{R}$ be sequence of continuous functions uniformly convergent to $f\left(x\right)$. If $\left(f_n\left(x\right)-f_n\left(x\frac{n}{n+1}\...
Jakub Pawlak's user avatar
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Proving a sequence of functions isn't uniformly convergent

Let $f_n(x) = x^n(1-x^n)$. I need to prove that the sequence is not uniformly convergent in $[0,1]$. I have already proven that there is a pointwise convergence to $f(x)=0$. However, according to my ...
talopl's user avatar
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Uniform convergence of a sequence of polynomials on the unit circle

Let $(P_n)_n$ be a sequence of polynomials in one variable with complex coefficients. Assume that $(|P_n|)_n$ converges uniformly to $1$ on the unit circle (concretely, the supremum of $| |P_n(z)|-1|$ ...
Arnaud PLESSIS's user avatar
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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 ...
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Pointwise convergence+uniformly continuous implies uniformly convergence [closed]

Consider $f:\mathbb{R}\times U\to\mathbb{R}^{n},(t,x)\mapsto f(t,x)$, where $U$ is an open set of $\mathbb{R}^{n}$. Suppose $\tau_n$ is a sequence in $\mathbb{R}$ with $\tau_n\to\tau$ as $n\to\infty$, ...
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4 votes
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Sufficient conditions for uniform convergence in probability

I have a sequence of continuous random variables $\{X_n\}$, with density $f_n(x \mid \theta)$ w.r.t. the Lebesgue measure. $X_n = o_{p_\theta}(1)$ for any $\theta \in \Theta$. For a fixed $\theta_0$ ...
statstats's user avatar
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1 answer
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Refute uniform convergence

Does the sequence $g_n(x)=\frac{-nx+x^3(n+1)}{-1+nx}$ converge uniformly on $R_{\geq 0}?$ The pointwise limit is given by $g(0)=0$ and $g(x)=x^2-1$ for $x>0$, so the limit is not continuous in $x=0$...
HelloEveryone's user avatar
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2 answers
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Uniform convergence of $f_n(x) = \frac{\ln(1+\frac{x}{n})}{x+1}$ [duplicate]

Basically i have a problem proving the sequence in the title 1. Uniformly converges on a closed interval $[0,a]$ where $a > 0$ and 2. Uniformly converges on $[0,\infty).$ So far i have found the ...
Timon Bubnič's user avatar
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Weierstrass' M-test and convergence of series

I came across with the following proof about the convergence of Dirichlet L-series but I have troubles understanding it: Let $\delta >0$ and $f:\mathbb{Z}\rightarrow \mathbb{C}$ satisfy $|f(n)|\leq ...
Ishigami's user avatar
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Prove function defined by series is continuous [duplicate]

I am working on this problem. I am studying for an exam. I find it hard to believe this has not been asked before, but I could not find it. Show that the function $f(x) = \sum_{n=1}^{\infty} \frac{1}{...
user123456's user avatar
2 votes
1 answer
35 views

Equicontinuity of a set

this is an excercise in an old test from my course of analysis. Let $\rho : \mathbb{R} \to [0,\infty]$ continuos such that $\int_{\mathbb{R}}\rho(t)dt=1$. Let $f_n$ secuence of real functions, bounded ...
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1 vote
1 answer
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a sequence of characteristic functions converge uniformly near t=0, then they are equicontinuous

I encountered a problem when reading A course in probability theory by Kailai Chung: If the sequence of ch.f.'s $\{f_n\}$ converges uniformly in a neighborhood of the origin, then $\{f_n\}$ is ...
Puppet's user avatar
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Two series converges pointwise to same limit, and one converges uniformly

We have $\sum{f_n}$ and $\sum{g_n}$. $f_n,g_n$:[0,1]->$\mathbb{R}$ and all f are non-negative and all g are continuous. Both series converges pointwise to the same limit. How to prove that if $\sum ...
AveriX's user avatar
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2 votes
1 answer
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Uniform convergence of $\sum_{n=1}^{\infty}x^n\sqrt{1-x}=\sum_{n=1}^{\infty} f_n(x)$ on $(0,1)$

I want to check whether this series of functions converges uniformly or not. $\textit{Attempt:}$ I have used Weierstrass M-Test to find the maximal value for $f_n(x)$ and I got $M_n=(\frac{2n}{2n+1})^...
Bowei Tang'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
2 votes
1 answer
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If $\sum_{n=1}^{\infty}g_n$ converges uniformly, then each $g_n$ is bounded by a real number $M_n$.

$\textbf{Question}:$If $\sum_{n=1}^{\infty}g_n(x)$ converges uniformly on a set $A\subset \mathbb{R}$, is it true that each $g_n(x)$ is bounded by a real number $M_n$? This question came into my mind ...
Bowei Tang's user avatar
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5 votes
1 answer
135 views

Convergence of Step Functions Generated by Uniformly Distributed Random Points

I have encountered a surprisingly complicated problem to solve and I'm looking for some help. It could be difficult because I don't have a background in probability and so don't know the appropriate ...
Jason Bramburger's user avatar
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0 answers
11 views

Uniform Convergence in Probability of Parameterized Functions

I want to show that $$\text{arg} \, \text{min}_x \; f_{\hat{\theta}_n}(x) \overset{P}{\rightarrow} \text{arg} \, \text{min}_x \; f_\theta(x),$$ where $f_{\hat{\theta}_n}$ is a function that is ...
foobar_98's user avatar
1 vote
1 answer
38 views

Does convergence in probability implies uniform convergencce in probability?

Let $(F_{n,X})_{n \ge 1}$ be a sequence of the empirical cumulative distribution function of distribution positive integer-valued distribution $F_{X}$, such that for all $k \ge 1$ we have $$ F_{n,X}(k)...
bnm's user avatar
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2 votes
1 answer
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Prove that $\{f_n\}$ converges to $f$ with respect to the uniform norm if and only if it converges uniformly to $f$.

Problem Let $f$ and $f_1,f_2,\dots$ be continuous functions on $[a,b]$. Prove that $\{f_n\}$ converges to $f$ with respect to the uniform norm if and only if it converges uniformly to $f$. Definition$...
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Uniform Convergence of a Sequence of Differentiable Functions

I'm currently studying real analysis and I've come across a problem that I'm having trouble with. The problem is as follows: Let $(\phi_n)$ be a sequence of differentiable functions on $[a,b]$ such ...
user avatar
0 votes
1 answer
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Uniform Convergence of a Sequence of Functions and Differentiability

I am currently studying the topic of uniform convergence and its implications on differentiability. I came across a problem that I have been trying to solve but I am stuck. The problem is as follows: ...
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2 votes
1 answer
42 views

Convergence and Differentiability of a Sequence of Functions

Question: I'm studying a sequence of functions $$f_n(x) = \sqrt{x^2 + \frac{1}{n}}$$ defined on the domain $[-1,1]$ and I'm trying to understand their behavior as $n$ approaches infinity. Context and ...
prob1 yuma's user avatar
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1 answer
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Does the sequence $(f_n)_{n=1}^{\infty}$ given by $f_n(x) = \frac{x^2}{x^2 + (1-nx)^2}$ converge uniformly to its pointwise limit?

Let $(f_n)_{n=1}^{\infty}$ be the sequence of functions $f_n : [0,1] \rightarrow \mathbb{R}$ given by $$f_n(x) = \frac{x^2}{x^2 + (1-nx)^2}, \text{ for all } x \in [0,1].$$ (i) Find the pointwise ...
Luke's user avatar
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1 vote
1 answer
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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
1 vote
2 answers
43 views

Understanding Uniform Convergence of a Sequence of Functions

I am currently self-studying and came across the following theorem: Suppose that a sequence of functions $\phi_n$ converges uniformly to $0$ on $[a,b]$. Now suppose we have a sequence of functions $...
prob1 yuma's user avatar
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0 answers
29 views

Prove a series is uniformly convergent, hence continuous. [duplicate]

Question: Show that the series $\sum_{n=1}^{\infty}(ne^{-nx})$ is continuous on $(0,\infty)$. What I have tried: Since $ne^{-nx}$ is continuous for all $x>0$, it suffices to prove the uniform ...
hulee ouo's user avatar
3 votes
0 answers
34 views

Convergence to a continuous distribution on $\mathbb{R}^2$ implies uniform convergence

Let $\mathcal{D}$ be a continuous distribution on $\mathbb{R}^2$, and let $\{\mathcal{D}_i\}_{i=1}^\infty$ be a sequence of distributions converging to $\mathcal{D}$ in the following sense: $\forall$ ...
N. S.'s user avatar
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1 vote
0 answers
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Does $f_n\sim g_n$ iff $\lim_{n\to\infty}f_n=\lim_{n\to\infty}g_n$ for uniformly convergent sequences of PDFs hold over these vector operations?

Let $S_{\mathbb{Z}}$ be the set of infinite sequences of functions $f_n: \mathbb{Z} \to (0,1)$ satisfying $$\sum_{k\in\mathbb{Z}} f(k) = 1,$$ such that any such sequence, $f_n$, converges uniformly to ...
Kirk Fox's user avatar
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1 vote
0 answers
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Approximation by smooth functions - do the derivatives converge locally uniformly?

Say I have a function $f:\mathbb{R} \to \mathbb{R}$ which is continously differentiable and has a bounded derivative. Then I know I can approximate $f$ with smooth functions $\phi_n$ by mollifications ...
Snildt's user avatar
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0 answers
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Converge uniformly on compact subset of $\mathbb{D}$

Let $\scr{F}$ be the family of functions that are holomorphic on $\mathbb{D}$ such that $f(\mathbb{D})\subset \mathbb{C}-(-\infty,0]$. If $f_n$ is a sequence in $\scr{F}$ such that $f_n(0)\to 0$ as $n\...
Remu X's user avatar
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0 answers
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Uniform convergence of sequence of continuous functions implies uniform continuity?

I recently encountered a question where I'm not sure if I answered it correctly, given that I didn't use one of the assumptions. So here's the question: Prove that if a sequence of continuous ...
Darrell Tan's user avatar
2 votes
1 answer
64 views

Uniform Convergence of integral with parameter

I need to prove that there is no uniform convergence of $I(\alpha) = \int_{0}^{1}{\sin(\frac{1}{x})\frac{1}{x^{\alpha}}dx}$ where $\alpha \in (0,2)$ what I've tried so far: Let's prove pointwise ...
Jane Doe's user avatar
  • 115
1 vote
2 answers
67 views

Function is uniformly convergent on interval?

My professor introduced the following definition: Let $f_n:(a,b) \rightarrow \mathbb{R}, n \in \mathbb{N}$ be a sequence of differentiable functions such that: $\sum_{n=1}^\infty f_n(c)$ converges ...
rudinable's user avatar
0 votes
1 answer
37 views

Monomials are NOT a (NOT necessarily Schauder) basis?

A basis for an infinite dimensional Banach space $X$ is a set of elements $\{e_k\}=B\subseteq X$, such that Each finite subset of $B$ is linearly independent. The closure of the span, i.e. $\{\lim_{N\...
Confuse-ray30's user avatar
2 votes
1 answer
33 views

Existence of uniformly convergent sequence of polynomials converging to an analytic function

Let $\Delta_1,\Delta_2,\dotsc,\Delta_n$ be mutually disjoint open discs in the complex plane. Question part 1: Does there exists a sequence of polynomials $(p_n)_{n\in \mathbb N}$ converging uniformly ...
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