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].

A sequence of functions $f_n : X \to \mathbb R$ is said to converge uniformly to a function $f : X \to \mathbb R$ if $$ \lim_{n\to \infty}\sup_{x\in X}|f_n(x)-f(x)| = 0.$$ Roughly speaking, this means not only that $f_n(x)$ converges to $f(x)$ for all $x \in X$, but also that the rate of convergence is uniform over the whole of $X$.

Uniformly convergent sequences are well-behaved in certain ways that pointwise convergent sequences are not. For example, if $X$ is a topological space (such as a subset of $\mathbb R$), and if the functions $\{ f_n \}$ are continuous, then their uniform limit $f$ is also continuous. Furthermore, if $X$ is a bounded closed interval in $\mathbb R$, and if the functions $\{ f_n \}$ are Riemann integrable, then their uniform limit $f$ is also Riemann integrable, and $\int_X f= \lim_{n \to \infty} \int_X f_n $. These statements do not hold under the weaker assumption of pointwise convergence.

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Pointwise vs. Uniform Convergence

This is a pretty basic question. I just don't understand the definition of uniform convergence. Here are my given definitions for pointwise and uniform convergence: Pointwise convergence: Let $X$ be a set, and let $F$ be the real or complex numbers.…
Jeff
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Uniform convergence of $\sum_{n=1}^{\infty} \frac{\sin(n x) \sin(n^2 x)}{n+x^2}$

I'm not sure wether or not the following sum uniformly converge on $\mathbb{R}$ : $$\sum_{n=1}^{\infty} \frac{\sin(n x) \sin(n^2 x)}{n+x^2}$$ Can someone help me with it? (I can't use Dirichlet' because of the areas where $x$ is close to $0$)
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What are ways to compute polynomials that converge from above and below to a continuous and bounded function on $[0,1]$?

Background We're given a coin that shows heads with an unknown probability, $\lambda$. The goal is to use that coin (and possibly also a fair coin) to build a "new" coin that shows heads with a probability that depends on $\lambda$, call it…
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Uniform continuity, uniform convergence, and translation

Let $f:\mathbb R \to \mathbb R$ be a continuous function. Define $f_n:\mathbb R \to \mathbb R$ by $$ f_n(x) := f(x+1/n). $$ Suppose that $(f_n)_{n=1}^\infty$ converges uniformly to $f$. Does it follow that $f$ is uniformly continuous? Note: the…
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When does pointwise convergence imply uniform convergence?

On an exam question (Question 21H), it is claimed that if $K$ is compact and $f_n : K \to \mathbb{R}$ are continuous functions increasing pointwise to a continuous function $f : K \to \mathbb{R}$, then $f_n$ converges to $f$ uniformly. I have tried…
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Uniform convergence of derivatives, Tao 14.2.7.

This is ex. 14.2.7. from Terence Tao's Analysis II book. Let $I:=[a,b]$ be an interval and $f_n:I \rightarrow \mathbb R$ differentiable functions with $f_n'$ converges uniform to a function $g:I \rightarrow \mathbb R$. Suppose $\exists x_0 \in I:…
user42761
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Omitting the hypotheses of finiteness of the measure in Egorov theorem

I want to prove that if I omit the fact that $\mu (X) < \infty$ in Egorov theorem and place instead that our functions $|f_n|
alice
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If $f(x+1/n)$ converges to a continuous function $f(x)$ uniformly on $\mathbb{R}$, is $f(x)$ necessarily uniformly continuous?

Suppose $f:\mathbb{R}\mapsto\mathbb{R}$ is a continuous function. Define $f_n(x)=f(x+1/n)$ for all $n\in\mathbb{N}^+$. If $f_n(x)\to f(x)$ uniformly on $\mathbb{R}$, can we conclude that $f$ is actually uniformly continuous? If not, can you give a…
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Pointwise but not Uniformly Convergent

The Question: Prove that the sequence of functions $f_n(x)=\frac{x^2+nx}{n}$ converges pointwise on $\mathbb{R}$, but does not converge uniformly on $\mathbb{R}$. My Work: Prove Pointwise: First, $\lim\limits_{n\to\infty}…
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Settle a classroom argument - do there exist any functions that satisfy this property involving Taylor polynomials?

I'm going to apologize in advance; I might at some points say Taylor series instead of Maclaurin series. OK, so backstory: My calculus class recently went over Taylor series and Taylor polynomials. It seemed basic enough. Using the ratio test we…
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Intuitive explanation of proof of Abel's limit theorem

Assume the series $$f(x)=\sum_{n=0}^{\infty}a_n x^n$$ converges for $-r
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Does absolute convergence of a sum imply uniform convergence?

Suppose I have a series $\sum_{n = 0}^{\infty} f_{n}(x)$ which converges absolutely to a function $f(x)$. Does the series converge uniformly to $f(x)$? I want to say this follows from Dini's Theorem, but I can't seem to see how.
Shayla
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Weak convergence in probability implies uniform convergence in distribution functions

Exercise 1: Let $\mu_n$, $\mu$ be probability measures on $\left(\mathbb{R}, \mathcal{B}\left(\mathbb{R}\right)\right)$ with distribution functions $F_n$, $F$. Show: If $\left(\mu_n\right)$ converges weakly to $\mu$ and $F$ is continuous, then…
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How to prove a sequence of a function converges uniformly?

For $n \in \mathbb{N}$, define the formula, $$f_n(x)= \frac{x}{2n^2x^2+8},\quad x \in [0,1].$$ Prove that the sequence $f_n$ converges uniformly on $[0,1]$, as $n \to \infty$. I know that the definition says $f_n$ converges uniformly to $f$ if given…
Jaime
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If $\int_0^1 e^{- \frac{nx}{1-x}} f(x) \; dx =0$ for $n \geq 0$, then $f=0$ on $[0,1]$

Suppose that $f$ is continuous on $[0,1]$. If $$ \int_0^1 e^{- \frac{nx}{1-x}} f(x) \; dx =0 $$ for $n \geq 0$, show that $f(x)=0$ on $[0,1]$. I am unsure what I should be hoping to do to show this. I know $e^{-nx/(1-x)}$ converges pointwise to 0 on…
Kyle L
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