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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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Using the sequential definition of uniform continuity to show $\sin(x)$ is uniformly continuous on $\mathbb{R}$

I want to show $\sin(x)$ is uniformly continuous on $\mathbb{R}$. Let $\{a_{n}\}$ and $\{b_{n}\}$ be sequences such that $\lim_{n\to\infty}[b_{n} - a_{n}] = 0$. Then, we need to show $\lim_{n\to\infty}...
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1answer
41 views

Is it true that $f(x) = x^{2}$ is uniformly continuous on $\mathbb{N}$?

I think that the answer is yes. If $f(x) = x^{2}$ were uniformly continuous on $\mathbb{N}$, then for every pair of sequences $\{u_{n}\}$ and $\{v_{n}\}$ satisfying $$\lim_{n\to\infty} \left( u_{n} - ...
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1answer
25 views

Establish uniform convergence

Let $\{f_n\}$ be a decreasing sequence of continuous functions, which converges to a continuous function $f$ on a compact set $E$. Prove that $f_n \to f$ uniformly on $E$.
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Converges Uniformly. [duplicate]

We have to show that $f_n(x)=x^n$ converges uniformly on $(-1,1)$. So, i find pointwise then $f_n\to 0$ , so consider $|f_n(x)-f(x)|=|x^n|$. Is way to choose $N\le n$ such that $|x^n|<\epsilon,$ ...
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1answer
24 views

Uniformly approximating $f\in \mathcal{C}([1,\infty))$ with polynomials where $\underset{x\rightarrow +\infty}{\lim} f(x)=a$

Suppose $f\in \mathcal{C}([1,\infty))$ and $\underset{x\rightarrow +\infty}{\lim} f(x)=a$. Show that $f$ can be uniformly approximated on $[1,\infty)$ by functions of the form $g(x)=p(1/x)$, where $p$ ...
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11 views

Prove that the sequence of derivative functions converges uniformly on every interval [-M,M].

The sequence is: $f_n(x) = \dfrac{nx^2+1}{2n+x}$ with derivative $f_n'(x) = \dfrac{4n^2x+nx^2-1}{4n^2+4nx+x^2}$. We know that $f'(x) = x$. We are asked to show that the sequence of derivatives, $f_n'(...
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26 views

Convergence of sequence $f'_n(x)$

Consider the sequence of real-valued functions $\{f_n\}$ defined by $$f_n(x)=\frac{1}{1+nx^2}. $$ Assuming the fact that $\{f_n\}$ converges uniformly to a function find out all real numbers $x $ ...
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3answers
60 views

Is $\lim_{n\to\infty}\int_0^1f_n(x)=\int_0^1f(x)$ in this case?

Let $f_n(x)=\begin{cases}\frac{e^{x^2}}{x^2} &x\in\left[1/n,1\right]\\ 0 &x\in(-\infty,1/n)\cup(1,+\infty)\end{cases}.$ This converges to $f(x)=\begin{cases}\frac{e^{x^2}}{x^2} &x\in\...
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1answer
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if $f_n(z)\rightarrow f(z)$ uniformly, then $ \frac{f'_n(z)}{f_n(z)}\rightarrow\frac{f'(z)}{f(z)}$ uniformly?

I am reading "The Taylor Series: An Introduction to the Theory of Functions of a Complex Variable", Dienes P.N., Dover (1957). In the proof of a theorem (Hurwitz) on page 351 it says that, if $f_n(...
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2answers
20 views

Determine pointwise and uniform convergence of a given sequence of functions and calculate an integral

Let $f_n:[0,\infty)\to\mathbb{R},f_n(x)=\frac {ne^{-x}+xe^{-n}}{n+x},\space\space\space\forall n\in\mathbb{N}$. Study the convergence and calculate: $A_n=\int_0^1f_n(x)dx.$ My attempt: For the ...
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1answer
24 views

If $\sum g_n(x)$ converges uniformly and absolutely and $|f_n(x)|\leq |g_n(x)|$ show that $\sum f_n(x)$ converges uniformly and absolutely.

I do not know how to prove if the statement above is true. I know i can use the Cauchy criterion i.e. $|\sum_{n\rightarrow m}f_n(x)|\leq\sum_{n\rightarrow m}|f_n(x)|\leq \sum_{n\rightarrow m}|g_n(x)...
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2answers
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Question on uniform convergence of sum of continuous functions.

i) I have been stuck on this for quite some time. Can anyone explain how $h(x)$ converges uniformly(and absolutely) given the inequality. I don't think I can use Weierstrass M-Test. ii) Secondly, ...
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What is the rate of convergence of the argmins of a sequence of uniformly convergent strictly convex functions?

From Theorem 2.1 and 2.2 in Kanniappan (1983) "Uniform convergence of convex optimization problems", if a sequence of strictly convex functions $f_n$ uniformly converge to a strictly convex function $...
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0answers
23 views

Sequence of bounded sequences converge uniformly to bounded function

Let $(f_n)$ be a sequence of bounded functions on a set $S$, and suppose $f_n \to f$ uniformly on $S$. Prove $f$ is bounded on $S$. By uniform convergence $\forall \epsilon \ \exists N>0 : \forall ...
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1answer
18 views

Sequence $(f_n) \to f_n$ on $S \subseteq \mathbb{R}$ converges uniformly iff $\lim_{n \to \infty} \sup \{f(x)-f_n(x)| : x\in S\} = 0$

I am having trouble proving the converse. Namely, if $$\lim_{n \to \infty} \sup \{|f(x) - f_n(x)| : x \in S \subseteq \mathbb{R}\} = 0$$ then the sequence $(f_n)$ converges uniformly to $f$ Any ...
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2answers
30 views

Applying the uniform boundedness principle

I try to solve the following: Let $X,Y$ be Banach spaces and $B:X\times Y\to \mathbb{C}$ a linear map such that $$x_n\to 0\implies B(x_n,y)\to 0\ \forall y\in Y$$ $$y_n\to 0\implies B(x,y_n)\to 0\...
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1answer
35 views

Can I use the extreme value theorem to prove uniform convergence of a sequence of functions on a compact interval?

For example $f_n(x)=(1+x/n)^n$ converges pointwise on $\Bbb R$ to $f(x)=e^x$, but not uniformly because $f_n(n)\to+\infty$ and $f_n(-2n)$ has no limit. Is it logically sound to say that the ...
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How to test the uniform convergence of a function series, without Weierstrass M-test

A function series $\sum f_n(x)$ is pointwise convergent in $A_p$ if $\forall x\in A_p$, $\sum f_n(x)$ converges. It is totally convergent in $A_p$ if it passes the Weierstrass M-test. If a series ...
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1answer
22 views

Extending uniform convergence of analytic functions on larger domains

Let $f_k, f: ]-\infty , 1 [ \to \mathbb {R}$ be analytic functions. Suppose $f_k $ converges uniformly to $f $ on $]-\infty,0] $. Is it true that $f_k$ converges to $f$ on $]-\infty, \epsilon [$ for ...
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1answer
49 views

Let $f$ be Riemann Integrable on $ [0,2\pi]. $

Let $ g(t) = \int_{0}^{2\pi} f(x)\cdot \sin(tx) dx, $ where $ t \in \mathbb{R}. $ Show that $g$ is uniformly continuous on $ \mathbb{R} $ and that $ \lim_{n \to \infty} g(n) = 0. $ I don't even ...
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1answer
28 views

Uniform Convergent on $\mathbb{R}$

Suppose $f_n:\mathbb{R}\rightarrow \mathbb{R}$ is a sequence of continuous functions, such that the sequence $f_n$ converges to $f$ uniformly on $[-N,N]$ for every fixed positive integer $N$. Which of ...
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convex functions and weak convergence

Let$ f_n$ and $f$ be convex $2\pi-$periodic real functions such that : We have the uniforme convergence of $f_n$ to $f$ by Alexandrov's theorem $f_n$ and $f$ are twice differentiabl so we have ...
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1answer
12 views

Convergence of complex sequence checkup

I have to find the region of convergence and analyze the the pointwise and uniform convergence of the following sequence: $$\psi_n(z)=\frac{e^{-inz^2}}{(n+1)\sqrt{n}}$$ What I did: What I did first ...
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1answer
19 views

The series $\sum_{n=1}^\infty(z^{2^n}-z^{-2^{n}})^{-1}$ converges compactly in $\mathbb{C} \setminus(\{0\} \cup \mathbb{D} \})$

I need to show that the series $\sum_{n=1}^\infty(z^{2^n}-z^{-2^{n}})^{-1}$ converges uniformly on compact subsets of $\mathbb{C} \setminus(\{0\} \cup \mathbb{D} \}$, where $\mathbb{D} = \{ z \in \...
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0answers
23 views

Term by term differentiation of series of functions.

Show that the series for which the sum of first $n$ terms $f_n(x) = \frac{nx}{1+n^2x^2}$ cannot be differentiated term-by-term at $x = 0$. What happens at $x \neq 0$? As- Here we are given with the ...
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0answers
20 views

Uniform boundedness of expectation

Let $X_{ij}$ $i,j=1, \dots N$ be a sequence of iid rvs. bounded by a constant across $j \neq i$. Let us define $f_{in}=\frac{1}{N}\sum_{j=1, j\neq i}^N (X_{ij} - E(X_{ij}))$. Then, I'm trying to show ...
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1answer
33 views

Picard Iteration, existence of a solution to an IVP

This a follow-up question from my previous post: ODE analysis problem Although it may not be useful, I still place the link above. Questions: An iterated sequence is given: $x_0(t)=0$ $...
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1answer
38 views

Uniform convergence of $\sum_{k=2}^\infty \frac{\sin(kx)}{k \ln(k)}$?

I want to determine if following series is uniformly convergent and on what interval if it is: $$\sum_{k=2}^\infty \frac{\sin(kx)}{k \ln(k)}$$ I see that $ \frac{|\sin(kx)|}{k \ln(k)} \leq \frac{1}{...
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0answers
48 views

What notion of convergence guarantees interchange of limits and improper Riemann integrals?

If $f_n$ converges to $f$ pointwise, then the Riemann integral of $f_n$ over a bounded interval $[a,b]$ need not converge to the Riemann integral of $f$ over $[a,b]$. But if $f_n$ converges to $f$ ...
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1answer
26 views

Convergence in $C([0, 1])$

Let's consider the space of continuous functions on the interval $[0, 1]$. We'll denote it by $C([0, 1])$. Now we can define a sequence: $$f_n(x) = x(1-x^n).$$ It's easy to find it's pointwise limit: $...
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0answers
31 views

Extraneous Condition in the Hypothesis?

If $\{f_n\}$ is a sequence of continuously differentiable functions on $[a,b]$, $f_n \to f$ uniformly on $[a,b[$, and there is a function $g : [a,b] \to \Bbb{R}$ such that $f'_n \to g$ uniformly on $[...
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2answers
180 views

Proving the limit of a sequence of functions must be polynomial.

Let $n \geq 1$ and let $\left(f_n (z)\right)_n$ be a sequence of polynomials whose coefficients are in $\mathbb{C}$. Suppose that $(f_n (z))_n$ converges uniformly to a function $f(z)$ on $\mathbb{C}$....
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1answer
56 views

Pointwise and uniform convergence of a piecewise sequence of functions on the closed, punctured disk, $\overline{D}\prime(0,1)$.

Consider the sequence of functions $$f_n(z) = \begin{cases} n, & \text{if $0<|z|\leq\frac{1}{n}$} \\ \frac{1}{z^4}, & \text{if $\frac{1}{n}<|z|\leq1$} \end{cases} $$ for $n\geq 1$, on ...
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0answers
19 views

How does uniform continuity and continuity affect the convergence of a function

Say we are working in $L^{\infty}(\mathbb{R})$ In the below cases what can we say about the convergence? 1) $f_{n},f$ are continuous and $\lim_{n\to \infty}f_{n}=f$ 2) $f_{n},f$ are uniformly ...
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0answers
28 views

Uniform Limit of Integrable Functions

If $\{f_n\}$ is a sequence of bounded measure functions on $[a,b]$ and $f_n \to f$ uniformly, then $f$ is integrable. This is actually proved in the book I am working through, but I came up with a ...
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2answers
27 views

Does uniform convergence and pointwise continuity of a sequence of functions on $[0,1]$ imply continuity of the limit function?

If $(f_n)$ is a sequence of real functions on $[0, 1]$ converging uniformly to a function $f$ on $[0, 1]$, and if $f_n$ is continuous at $x_n ∈ [0, 1]$ for each $n$ with $x_n\stackrel{n\to\infty}\...
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1answer
42 views

Uniform convergence of sequence of function $f_n(x) = \frac{nx^4+1}{nx^4+2x+3}e^{-nx^2}$ on the interval $(1,+ \infty)$

I need to prove that the sequence of functions $f_n(x)$ is uniformly convergent to $f$ on the interval $(1,+ \infty)$. I've already shown that $f_n(x) = \frac{nx^4+1}{nx^4+2x+3}e^{-nx^2}$ is ...
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1answer
60 views

Is the series $\sum_{n=1}^{\infty} \frac{\sin nx}{n}$ is uniformly convergent on $[0,1]$? [duplicate]

Is the series $\sum_{n=1}^{\infty} \frac{\sin nx}{n}$ uniformly convergent on $[0,1]$? My work : since $|\sin nx|\le 1$ so $\sum_{n=1}^{\infty} \frac{\sin nx}{n} \le \sum_{n=1}^{\infty} \frac{1}{n}...
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1answer
44 views

Prove that $\sum_{n=1}^{\infty}\frac{x^2}{1+n^3x^4}$ converges uniformly for $x\in \mathbb{R}$.

Prove that $$\sum_{n=1}^{\infty}\frac{x^2}{1+n^3x^4}$$ converges uniformly for $x\in \mathbb{R}$. My try: for $x=0$, of course it converges. for $x\neq 0$ , then $\frac{x^2}{1+n^3x^4}\le \frac{1}{n^...
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1answer
23 views

Lebesgue convergence

Let $f_{n}=n^{-1/p} \chi[0,n]$ show that the sequence $(f_{n})$ converges uniformly to the $0$ function but that it does not converge in $L_{p}(\mathbb{R},B,\lambda)$ My attempts, I was trying to use ...
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0answers
48 views

Uniform convergence $\sum_{n=1}^\infty \frac{n^2-n^4}{n^5+n^3+1}\left(x^2\sin\left(\frac{\pi x}{2}\right)\right)^n$. Troubling finding the interval.

I have to find the interval of uniform convergence, for the series $$\sum_{n=1}^\infty \frac{n^2-n^4}{n^5+n^3+1}\left(x^2\sin\left(\frac{\pi x}{2}\right)\right)^n$$ Now my attempt has been to use ...
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0answers
15 views

the convergence of the first and second derivative of a sequance of bounded functions

Let $f_n$ be a sequence of continuos function defined as $f_n:\mathbb{S}^{n-1}\rightarrow \mathbb{R} $ such that $||f_n-f ||_{\infty} \rightarrow 0 $ as $n \rightarrow +\infty$ I want to prove ...
0
votes
1answer
20 views

Is convergence of elementary (simple) functions always uniform?

Say I have a bounded continuous function $g(t)$ on the interval $[a,b]$ Let $$g_{n}(t)=\sum_{j=1}^{n}g(t_{j})X_{[t_{j},t_{j+1}]}(t)$$ Where X is the characteristic function, and $t^{n}_{j}$ a ...
1
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2answers
99 views

For each $\epsilon$, for which $\delta$ does $d(x,y)<\delta\implies d(f(x),f(y))<\epsilon$ hold?

I have the definition : A function from a metric space to a metric space is uniformly continuous if for all $\epsilon>0$ there exists $\delta$ such that $d(x,y)<\delta\implies d(f(x),f(y))<\...
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1answer
18 views

outer measure of a sequence

I am reading measure in the analysis independently, and got stuck on a problem. Hope you guys will help me out. Let $f_n:[a,b]\to \Bbb R$ be a sequence of measurable functions such that $f_n\to f$. ...
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1answer
26 views

Convergence of the following sequence of functions.

For $n \ge 1$, let $$g_n(x) = \sin^2 \left (x + \frac 1 n \right ), x \in [0,\infty)$$ and $$f_n(x) = \int_{0}^{x} g_n (t)\ \mathrm {dt}.$$ Then $(1)$ $\{f_n \}$ converges pointwise to a ...
4
votes
1answer
108 views

Uniform convergence of $\sum\limits_{n=1}^∞n^{-x}(e^{\frac{x}{n^2}}-1)$

Pointwise and uniform convergence of the following series of functions: $$\sum_{n=1}^{\infty} n^{-x}\left(e^{\frac{x}{n^2}}-1\right).$$ Now, the series of function converges pointwise as $x \in (-1,...
0
votes
1answer
18 views

Continuity through the lens of the uniform norm

So I was refreshing my knowledge of uniform convergence in the space of $C(X)$, continuous functions on a metric space, and wanted to ask whether the following characterization of continuity at a ...
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3answers
49 views

Uniform convergence of $f(x) = \sum_{n=0}^\infty x^n(1-x)^n$ in $(0,1)$

How to show uniform convergence of $$f(x) = \sum_{n=0}^\infty x^n(1-x)^n$$ for $x$ in $(0,1)$ ? I know I have to find the convergence radius but I don't know how.
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1answer
31 views

Series of continuous positive functions: uniform convergence?

Consider a functional series $f(x) = \sum_{n=0}^\infty f_n(x)$. All functions $f_n$ and $f$ are $[a, b] \to \mathbb R^+$. In addition, all functions $f_n$ are continuous. Can it be assured (thanks to ...