# 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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### 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 ...
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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. ...
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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 ...
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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 ...
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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 $| |P_n(z)|-1|$ ...
1 vote
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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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### 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 $\theta_0$ ...
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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 $x=0$...
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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 ...
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### 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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### 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 ...
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• 1,513
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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 $\{g_{n}\}$ ...
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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 ...
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### 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 ...
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### 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 ...
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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 ...
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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 $n\... • 1,071 0 votes 0 answers 28 views ### Uniform convergence of sequence of continuous functions implies uniform continuity? I recently encountered a question where I'm not sure if I answered it correctly, given that I didn't use one of the assumptions. So here's the question: Prove that if a sequence of continuous ... • 117 2 votes 1 answer 64 views ### Uniform Convergence of integral with parameter I need to prove that there is no uniform convergence of$I(\alpha) = \int_{0}^{1}{\sin(\frac{1}{x})\frac{1}{x^{\alpha}}dx}$where$\alpha \in (0,2)$what I've tried so far: Let's prove pointwise ... • 115 1 vote 2 answers 67 views ### Function is uniformly convergent on interval? My professor introduced the following definition: Let$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)$converges ... 0 votes 1 answer 37 views ### Monomials are NOT a (NOT necessarily Schauder) basis? A basis for an infinite dimensional Banach space$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\...
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 ...