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 for Functions $X \to [0,+\infty]$?
An exercise in Tao's Introduction to Measure Theory asks to show that if $(X,\mathcal{B}, \mu)$ is a measure space and $f_n : X \to [0,+\infty]$ is a sequence of unsigned $\mathcal{B}$-measurable ...
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Checking consistency of minimizer given uniform convergence of its objective function
Suppose the true parameter is
$$\theta_0 = argmin_{\theta \in \Theta} E[q(w, \theta)]$$
and $\theta_0$ is the unique minimizer, i.e.,
$$E[q(w, \theta_0)] < E[q(w, \theta)]$$
for all $\theta \in \...
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Uniform convergence integration [closed]
It is question in infinite series about uniform convergence,how do I prove it as when I tried it I am not able to proceed with a solution to it , please help
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operations with uniformly convergence sequence of functions
Let $K$ be a compact of $\mathbb C$ and $S$ be a subset of $K$. Suppose that one has $p\in\mathbb N$ sequences of functions $(f_{n,i})_n$ ($1 \le i\le p$) uniformly convergeing towards a function $f_i$...
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Alternate proof of Dini's theorem over a closed real interval
Let $f_n: [a, b] \to \Bbb R$ be continuous on $[a, b]$, and construct a monotone decreasing sequence of functions $(f_n)=(f_1, f_2, ...)$ converging pointwise to $f(x)=0$. Show that $f_n \to 0$ ...
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On properties of solution by Euler's method; uniform convergence, etc.
I'm reading about Euler's method to construct approximate solutions to ODEs in Ordinary Differential Equations by Andersson and Böiers. I have questions about properties of the approximate solution. I'...
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Test the sequence of functions $x_n={e^{-nt}}$ for convergence in $C[0,1]$,$L_{1}(0,1)$ and $L_{2}(0,1)$.
Test the sequence of functions $$x_n(t)={e^{-nt}}$$ for convergence in
$C[0,1]$,$L_{1}(0,1)$ and $L_{2}(0,1)$. In case of convergence, find
the limit function.
What I have done.
(1) In $C[0,1]$
For $...
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Uniform convergence of lateral derivatives
Suppose we have a sequence $f_n:[a,b)\to\mathbb R$ of right differentiable functions (i.e., the right derivative
$$
f_{n+}'(x)=\lim_{h\to0^+}\frac{f(x+h)-f(x)}{h}
$$
exists for all $x\in[a,b)$.) and ...
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Approximation of continuous function by uniform convergence of piecewise constant functions
I am reading Analysis 2 by Terence Tao, and I have a question regarding approximating continuous functions by piecewise constant functions.
Suppose that $f\in C(\mathbb{R})$. Can you find a sequence ...
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On uniform convergence of partial derivatives on a compact set
In Rudin's Functional Analysis, §1.46, he defines the space $C^\infty(\Omega)$, where $\Omega\subseteq \mathbb R^n$ is open. He defines a topology on this space, in part by introducing an increasing ...
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A doubt on uniform limit of a sequence of uniformly continuous functions
I was looking at this question is MSE.
Suppose $f_n:[0,1] \to \mathbb{R}$ is a sequence of uniformly continous functions and $f_n\to f$ uniformly.
These are the questions that I put myself.
What can ...
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If $f:[0,1]\to\mathbb R$ is continuous, then $f_n(x) = f(x^n)$ converges uniformly on $[0,a],$ $a < 1$ and $∫_0^1 f_n(x)\,dx \to f(0).$
Given f continues function in [0,1] we define a sequence of functions fn(x) = f(x^n) , given that for every number 0<a<1 the sequence of function is uniformly convergs in [0,a] to f(0) prove ...
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Monotone convergence theorem - what does non-decreasing exactly mean?
I understand that if a sequence is bounded by supremum and is strictly increasing it will converge. It is intuitive because the sequence is strictly increasing.
I do not really understand the weaker ...
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Weak convergence does not imply joint weak convergence?
Suppose that $X_n\Rightarrow X$ and $Y_n\Rightarrow Y$ as $n\to\infty$ where "$\Rightarrow$" means convergence in distribution. We know that it does NOT imply that $(X_n,Y_n)\Rightarrow (X,Y)...
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The limit of the sequence of functions $\sqrt{x^2+\frac{1}{n^2}}$
I would like general feedback on my solution to this exercise, as well as a hint for arriving at the solution more directly. I suspect that my approach is sub-optimal.
Exercise
Does the sequence of ...
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Showing measurable function is zero a.e. [duplicate]
I'm trying to solve the following old exam question from my analysis class:
Suppose $f$ is a compactly supported measurable function, such that for all integers $n\ge0$,
\begin{equation}
\int_{-\infty}...
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Uniform convergence of specific sequence on a compact subset of the complex plane.
I would like to show the uniform convergence of two sequences $(f_n)$ and $(g_n)$ to $f : z \mapsto \frac{2z}{z^2 - \pi^2}$ and to $g: z \mapsto z$ on any compact K of the complex plane where the ...
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Counterexample to Egorov for functions valued in non-separable metric space
A general form of Egorov (e.g. https://www.ime.usp.br/~glaucio/mat6704/textos/GMTLecureNotes.pdf) states that:
Egorov Theorem : Let $\mu$ be an outer measure on the set $X$ and $(Y,d)$ a separable ...
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Do subdifferentials of a uniformly converging sequence of convex functions converge pointwise?
Consider a sequence of continuous convex functions, $f_n,f:A \rightarrow \mathbb{R}$, where $A \subseteq \mathbb{R}^n$, compact. The sequence $f_n$ converges uniformly to $f$. Does $\nabla f_n$ ...
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(Path) connected components of zero-sets of limiting functions.
$\newcommand{\set}[1]{\{#1\}}$
$\newcommand{\R}{\mathbb R}$
$\newcommand{\bd}{\text{Bd}}$
Let $X$ be a compact metric space and $f_n:X\to \R$ be a continuous function on $X$, one for each $n$.
Assume $...
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Uniform convergence for a quotient of two uniformly convergent sequences
I’m wondering if uniform convergence can hold if we have a $0/0$ indeterminate form for certain values of $x$ for the limit function.
More specifically, let $f_n$ and $g_n$ be two sequences of ...
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On uniform convergence of a series of functions [duplicate]
I'm stuck in proving the uniform convergence of the following series
$$
\sum_{k=1}^{+\infty}\frac{2x}{x^2+n^2}
$$
Note that if we set $f_n(x))=\frac{2x}{x^2+n^2}$, then $f_n(x)$ has a maximum at the ...
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$f_n=\frac{1}{x^{\frac{1}{n}}}$ uniformly converges to 1 [closed]
This is the problem that I am trying to prove. Show that $f_n(x)=\frac{1}{x^{\frac{1}{n}}}$ where $x\in (0,\frac{1}{2})$ is uniformly convergent to $f(x)=1$. It will be great if someone can tell me ...
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Proof check - small lemma related to Weierstrass Approximation Theorem
Let $K \subseteq \mathbb R$ be a compact set such that $0 \in K$ and
let $f: K \to \mathbb R$ be a continuous function such that $f(0) =
0$. By Weierstrass Approximation Theorem there exists a ...
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Big Rudin exercise 1.10: Convergence of bounded function sequence
Exercise 1.10: Suppose $\mu(X)<\infty$, $\{f_n\}$ is a sequence of bounded complex measurable functions on $X$, and $f_n \rightarrow f$ uniformly on $X$. Prove that
$$ \lim_{n\rightarrow \infty} \...
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Uniformly convergence of a functions sequence and Riemann sum
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function.
Suppose $\langle h_n \space \space | \space \space n \in \mathbb{N} \rangle$ is a sequence of real functions given by, $\forall x ...
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How to show that $f(x)=x^n$ does not converge uniformly in domain $D=[0,1)$.
Suppose $f_n:\mathbb{R}\to\mathbb{R}$ is defined by $f_n(x)=x^n$ where $n\in\mathbb{N}$, and $D=[0,1)$. I wish to show that $f_n(x)=x^n$ does not converge to $f(x)=0$ uniformly as $n\to\infty$, i.e.
$...
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How to show that $f(x)=x^n$ converges pointwise in domain $D=[0,1)$.
Suppose $f_n:\mathbb{R}\to\mathbb{R}$ is defined by $f_n(x)=x^n$ where $n\in\mathbb{N}$, and $D=[0,1)$. I wish to show that $f(x)=x^n$ converges to $f_n(x)=0$ pointwise as $n\to\infty$, i.e.
$\forall\...
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Does $\sum^\infty_{k=0}\frac{x^k}{1+x^k}$ converge uniformly on $[-1/2,1/2]$?
I have searched Approach Zero for a similar question, but without success. I am trying to verify the uniform convergence of $$\sum^\infty_{k=0}\frac{x^k}{1+x^k},$$ on $-\frac12\leq x\leq \frac12$. I ...
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How does $f$ being uniformly continuous imply $f_\epsilon \rightarrow f$ uniformly?
Let $j(x)$ be any positive infinitely differentiable function with support in $(-1, 1)$ so that
$$\int_{-\infty}^\infty j(x) dx = 1.$$
Define
$$j_\epsilon(x) = \epsilon^{-1} j(x/\epsilon).$$ If $f \in ...
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A general question about a model's behaviour by uniformly selecting a finite number of inputs from an infinite input space.
I am conducting an assessment of a model to draw conclusions about its behaviour. I uniformly select 100 natural numbers as inputs. By feeding these 100 inputs to the model, I obtain 100 corresponding ...
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Sequences convergence uniform convergence and impact of boundedness of the differentrial [duplicate]
Let $f_{n}:\left[0,1\right]\to\mathbb{R}$ be a sequence of differentiable functions converging to $f:\left[0,1\right]\to\mathbb{R}$ pointwise. Assume that there exists a constant $M>0$ such that $|...
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Uniform convergence implies convergence in expectation, Lebesgue integral
Assume that the sequence of random variables $(X_n)_{n\in \mathbb{N}}$ defined on some probability space $(\Omega,\Sigma, \mathbb{P})$ converges uniformly to $X$, that is $$\sup_{\omega \in \Omega}|...
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Trouble proving $\lim_{n\to\infty} x^n=0$ when $|x|<1$.
Intuitively speaking this is very clear, however when doing a formal proof I get stuck; this is what I have so far.
Let $\epsilon>0$, now, if $x<\epsilon$ the result follows immediately since $x^...
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Prove that the Bernstein polynomials converge uniformly
The problem
Define the Bernstein polynomial of $f$ by
$$B_nf(x) = \sum_{i=0}^n\binom n i x^i(1-x)^{n-i} f(i/n)$$
Prove without the use of the Stone-Weierstrass theorem that $\limsup_{n\to\infty}|B_nf -...
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Partial sums of the Riemann zeta function and uniform convergence
For every $\delta\gt 0$, the partial sums of the Riemann zeta function $\sum_{n=1}^N n^{-s}$ converge uniformly on $S_{1+\delta}=\{s\in\mathbb{C}:\Re (s)\gt 1+\delta\}$. But the convergence is not ...
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Rudin $7.14$ linked to $7.9$ why does Rudin make boundedness compulsory in uniform convergence?
In Rudin $7.14$, Rudin says
Theorem $7.9$ can be rephrased as follow: a sequence $\{f_n\}$ converge to $f$ with respect to the metric $\mathscr C(X)$ if and only if $f_n \rightarrow f$ uniformly on $...
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Pointwise convergence of a function where convergence to multiple values occur at a single point
Consider,
$$
f_n(x)=\frac{1-nx^2}{(1+nx^2)^2}
$$
where, $$x \in \mathbb{R}, n \in \mathbb{N}$$
It is clear to me that not including $x = 0$, each function point converges to $0$ as $n \to \infty$.
The ...
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How can we define the supremum norm on the space of continuous functions if they're not necessarily bounded?
I frequently see the uniform norm described as a norm on the space of continuous functions $C(Y, X)$ where $Y$ is a topological space and $X$ is a metric space, especially in the context of the ...
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Determine whether the function $g(x,y) = \sum_{k=1}^\infty \frac{(x-2y)^k \sin(kx + y)}{\sqrt{k!} (1 + x^{2k}y^{4k})}$is continuous on $\mathbb{R}^2$
Determine whether the function $$g(x,y) = \sum_{k=1}^\infty \frac{(x-2y)^k \sin(kx + y)}{\sqrt{k!} (1 + x^{2k}y^{4k})}$$is continuous on $\mathbb{R}^2$.
My attempt: I tried to use Weierstrass M-test, ...
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On a proof of the non-differentiability of the Blancmange curve
In my nameless lecture notes, in the construction of a continuous, nowhere differentiable function (the Blancmange curve), we encounter the definition of the sequential derivative and other real-...
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A continuous, almost nowhere differentiable function
Consider the function $$f(x)=\sum _{k=0}^{\infty }\frac{\sin \left(k^2\pi x\right)}{k^2}.$$ It converges uniformly on $\mathbb{R}$ according to the Weierstrass M-test and is continuous on $\mathbb{R}$ ...
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Uniform convergence,continuous functions with a function inside an integral [duplicate]
Given a continuous function $f:\mathbb{R}\to\mathbb{R}$, define the sequence $f_{n}(x)=\frac{n}{2}\int_{x-\frac{1}{n}}^{x+\frac{1}{n}}f(t)dt$. Show that the sequence $\{f_{n}\}_{n=1}^{\infty}$ ...
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Regarding uniform convergence of a series
Reading a textbook on basic complex analysis I stumbled upon the following result:
The series $\sum_{n=1}^{\infty}\tfrac{z^n}{n}$ converges uniformly and absolutely in the closed disk $\overline{D(0,r)...
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What is the usual meaning of this notation $f_n\rightrightarrows f$?
Let $f_n$ be a sequence of functions and $f$ be a function, all having the same domain and their range shall belong to the same metric space. What is the usual meaning of this notation:
$f_n\...
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$f_n'$ converges uniformly but bounded $f_n$ don't.
Let $[a,b]\subset\mathbb R$ be some compact non-degenerate interval. We know that if a sequence of differentiable functions $f_n:[a,b]\to\mathbb R$ converges pointwise in at least one point $c\in[a,b]$...
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How to prove $\sum_{n=0}^{\infty}a_n \cos{(nx)}$ does not uniformly converge in $\left( 0,2\pi \right)$ with the conditions below?
Let $a_n$ be a sequence of real numbers such that:
$a_n \ge 0$
$\forall n\in \mathbb{N}: a_n \ge a_{n+1}$
$\lim_{n \to \infty} a_n =0$
$\sum_{n=0}^{\infty } a_n =\infty $
Consider the sum $\sum_{n=0}...
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Class of Lipschitz Functions on the unit d-dimensional ball
Let $\mathcal{F} = \{f:\mathcal{B}_d \to \mathbb{R}\;:\; \text{f is Lipschitz}\}$, where $\mathcal{B}_d = \{x \in \mathbb{R}^d\;:\: \|x\|_2 \leq 1\}$ is the unit ball in $d$ dimension. Is the class $\...
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Contrapositive of uniform limit theorem for series
I am looking for the contrapositive of the uniform limit theorem, especially in regards to series. For functions we have (and I am quoting from another post on this site)
If a sequence of continuous ...
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Finding out where a complex function is holomorphic and relating this to its expansion as a power series.
I am a student whose Complex Analysis' knowledge is gone for a long time. With this in mind, I start by apologizing if my question is too trivial.
Let us consider the complex function
$$ \frac{1}{1 + \...
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