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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67 views

Does there exist the following theorem for complex case?

‎Consider the following theorem (see theorem 7.17 of [W. Rudin. Priniciples of Mathematical Analysis, Mcgraw-Hilly, 1976.])‎ Theorem. Suppose ‎$‎\{f_n\}‎$ ‎is ‎sequence ‎of ‎functions, ‎differentiable ...
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11 views

uniform convergence of cesaro sums of fourier series

Can someone help me with this exercise? Let f be a 1-periodic and continuous function in $\mathbb{R}$ .Show that $\sigma_{n}(f)(x)\xrightarrow{n\to \infty} f(x)$ uniformly in the interval [0,1) where ...
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Study point wise and uniform convergence. [closed]

Let $(f_n)$ be a sequence of function defined on $\mathbb{R}^+$ by $f_0(x)=x$ and $f_{n+1}(x)=\frac{x}{2+f_n(x)}$ for all $n\in \mathbb{N}$ Study point wise and uniform convergent of $(f_n)$ on $\...
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38 views

Proving that a sequence of hokomorphic functions does not converge uniformly on the unit disk

I am uncertain whether my argument to disprove the uniform convergence of a sequence of holomorphic functions are correct. Show that the sequence of holomorphic functions $f_n$ given by $f_n(z)=z^{2n}...
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26 views

How can I prove the stability of this system? [closed]

Consider the following system: $\dot{x}_1=x_2$ $\dot{x}_2=-x_2-sgn(x_1)$ where sgn(.) is the sign function. In simulations, both $x$ and its derivative converge to zero but I can not prove it ...
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28 views

Generalization of term by term differentiation theorem for function series

From mono-dimensional theory of function series, we know that the following term by term differentiation theorem holds: Theorem 1. Suppose that $u_k : [a, b]\to \mathbb{R}$, for each $k = 1, 2,\dots$, ...
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31 views

Show that $g_n$ converges to $g$ uniformly.

Problem Let $f:\Bbb{R}\times[0,1]\rightarrow\Bbb{R}$ be a continuous function and $\{x_n\}$ a sequence of reals converging to $x$. Define $g_n(y)=f(x_n,y),\hspace{0.5cm}0\le y\le1$ $g(y)=f(x,y),\...
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Why $\sum_{n=1}^{\infty}\frac{x^{\alpha}}{1+n^2 x^2}$ doesn't converge uniformly on $[0, \infty)$ for $\alpha > 2$?

I'm trying to understand why $\sum_{n=1}^\infty\frac{x^\alpha}{1+n^2 x^2}$ doesn't converge uniformly on $[0, \infty)$ for $\alpha > 2$. My book says that $\frac{x^\alpha}{1+n^2 x^2}$ is monotonic ...
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Uniform convergence of series of function defined by recurrence formula

I have the exercise. Let $ \{f_n\}_{n=1}^{\infty} $ be a series of functions on $I = [0,1]$ defined with $f_1(x) =x, f_{n+1}=\sqrt{f_{n}(x)+2}$ (1) Show $ \{f_n\}_{n=1}^{\infty} $ is bounded for each ...
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Prove the uniform convergence of ${\frac{1}{(1+ {\frac{y^2}{n}})^n}}$

How can I prove the uniform convergence for these two tasks: $${\frac{1}{(1+ {\frac{y^2}{n}})^n}} ⇉ e^{-y^2} $$ I understand that the function on the right is the limit function, because: $$\lim _{{...
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A sequence of functions $f_n : [0, 1] → R$ which converges uniformly to a discontinuous function $f(x)$.

Give an example or argue that such a request is impossible. I argued that such a request is impossible because by theorem of the continuity of the uniform limit, if $f_n$ converges uniformly then ...
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1answer
24 views

Convergence. Cauchy and uniform

I know that if function is uniformly convergent ($ |f_n(x)-f(x)|<\epsilon. \forall n > N(\epsilon)$), it is Cauchy convergent ($ |f_n(x)-f_m(x)|<\epsilon. \forall n,m > N(\epsilon)$) So my ...
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Uniform limit of a parameterized series

I'm dealing with the parameterized series defined by : $f(x,n)=\underset{i\geq 0}{\sum }\frac{\left( i+n\right) !q_{i}x^{i}}{i!n!}$ where $q_{i}>0$ is such that $\underset{i\rightarrow \infty }{\...
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34 views

Check the pointwise convergence and uniform convergence of a sequence of functions.

For each $n\in \mathbb N$, let $f_n(x)=\begin{array}{cc} \Bigg\{ & \begin{array}{cc} nx^2 & 0\leq x\leq \frac{1}{n} \\ x & \frac{1}{n}<x \leq 1 \end{array} \end{...
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About uniform convergence.

Let $f$ and $f_1, f_2, \dots, f_n, \dots$ be continuous functions on $[a, b]$. Suppose that $\lim_{n \to \infty} f_n(x) = f(x)$ for any $x \in [a, b]$. I think the following proposition is true: If $\...
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I need to determine the domain of convergence and whether the series converge uniformly.

Here is the series: $\Sigma^{\infty}_{n=1}\frac{x^{2n}}{1+x^{4n}} $ Here is what I did: Given that $1+x^{4n} > x^{4n}$ we use the series $\sum^{\infty}_{n=1} \frac{x^{2n}}{x^{4n}} = \sum^{\infty}_{...
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How do I prove that a sequence of functions is convergent?

For $ n \in \mathbb N$ $f_n (x) = \frac{n^2 }{(x^2 +n^2)}$ Let $x \in \mathbb R $. Prove that $f_n (x) \rightarrow 1$ as $n \rightarrow \infty$ My attempt: Let $\epsilon>0 $, then by definition of ...
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Uniform Convergence of the Complex Exponential Function

Here is an excerpt from a textbook I am reading: The prime example of a power series is the complex exponential function, which is defined for $z \in \Bbb{C}$ by $e^z = \sum_{n=0}^\infty \frac{z^n}{n!...
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Has “uniform non-convergence” ever been formulated or played an important role in any mathematical lines of inquiry?

Below I will: first define my notations for the general setup for notions such as "uniform convergence" (Sec. 1); for the sake of reference, recall notions of convergence and uniform ...
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33 views

Show that $f(s)=\sum \frac{1}{n^s}$ is continuous for $Re(s)>1$

Show that $f(s)=\sum \frac{1}{n^s}$ is continuous for $Re(s)>1$. In my attempt i try to use Weierstrass Test, $\frac{1}{n^s}=\frac{1}{n^{Re(s)+iIm(s)}}=\frac{1}{e^{\log n^{\Re(s)+iIm(s)}}}=\frac{1}{...
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54 views

Let $f_n(x):\mathbb R \to \mathbb R$ be defined by $f_n(x)=\frac{x}{1+nx^2}$. Which of the following statements are correct?

For $n$ a positive integer, let $f_n(x):\mathbb R \to \mathbb R$ be defined by $f_n(x)=\frac{x}{1+nx^2}$. Which of the following statements are correct? (a) The sequence $\{f_n(x)\}$ of functions ...
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Convergence in measure of $\cos(\frac{x}{n})$ for $x\in[0,\pi]$.

$X=[0,\pi]$, $\mu$ is the Lebesgue measure, $f_{n}(x)=\cos\frac{x}{n}$, $g(x)=1$. I want to see if this sequence of functions converges in measure. I will use an auxiliary set, namely $$ A(\varepsilon,...
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Converges uniformly on an arbitrary closed disc implies on every compact subsets

Suppose that we have a given sequence of functions $(f_n)_{n\geq 0}$. The goal is to show that it converges uniformly on every compact subsets of $\mathbb{C}$. Let $R>0$ be arbitrary, and define, ...
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1answer
37 views

Show that $f(x):=\sum\limits_{n=0}^{\infty}\frac{1}{n}h(2^{n}x),$ where $h$ is a piecewise function, converges uniformly on $[0,1]$

For $x\in\mathbb{R}$, consider a piecewise function defined by $$h(x):=\left\{ \begin{array}{ll} x,\ \ \ 0\leq x\leq 1\\ 2-x,\ \ 1\leq x\leq 2\\ 0,\ \ \text{otherwise}. \end{array} \right.$$ Now, ...
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Show $ f_n\xrightarrow{L^1}f\iff f_n\to f \text{ in measure} \iff f_n\to f \text{ almost uniformly} \iff f_n\to f \text{ a.e.}$

Consider $(f_n)_n$ an increasing sequence in $\mathcal{L}^1$ and $f\in \mathcal{L}^1$. Show that $$ f_n\xrightarrow{L^1}f\iff f_n\to f \text{ in measure} \iff f_n\to f \text{ almost uniformly} \iff ...
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37 views

Show that $\sum_{n=1}^{\infty}\frac{x^{2}}{x^{2}+n^{2}}$ does not converge uniformly on $(-\infty,\infty)$.

I am trying to prove that this infinite series $\sum_{n=1}^{\infty}\frac{x^{2}}{x^{2}+n^{2}}$ does not converge uniformly on $(-\infty,\infty)$. I can definitely show that this series converges ...
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53 views

Why Folland's Advanced Calculus is so strict about uniform convergence?

Folland's Advanced Calculus uses uniform convergence to justify the interchange of limits (i.e. to change order of integration and summation). But actually uniform convergence is far powerful than ...
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29 views

Show that the sum function $f(x) = \sum_{n=1}^\infty \frac{1}{ \sqrt{n} } (exp(-x^2/n)-1)$ is continous

Consider for $x \in \mathbb{R}$ the sum function defined as $$ f(x) = \sum_{n=1}^\infty \frac{1}{ \sqrt{n} } (exp(-x^2/n)-1) $$ I have shown that the series converges point wise by using that $$ |exp(-...
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Is the dominated convergence theorem applicable in this case?

I have the following integral: $$J(x_1,x_2) := \frac{1}{4} x_1^2\ x_2^2 \int_{-\infty}^{\infty} d\tau_3 \int_{-\infty}^{\tau_3} d\tau_4 \int_{-\infty}^{\tau_4} d\tau_5 \int_{-\infty}^{\tau_5} d\tau_6\ ...
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Application of Egoroff's thm on infinite measure

This is what I have solved. My professor actually said this question need a condition for which m(E) is finite. He is keep saying that m,k cannot be fixed after reordering. But I think that there is ...
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Uniform Convergence and Limit Inside Integral When Index Is Not a Natural Number

Suppose we have the quantity $$\lim_{\varepsilon \to 0} \int_{0}^{2 \pi} f\left(\varepsilon e^{ri\theta}\right)d\theta,$$ where $\varepsilon > 0, r>0$ and we want to know if we can move the ...
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79 views

Uniformly convergent Fourier serie

In my book it says that if F (of causeF'=f) is continuous and piecewise differentiable and if F'=f for all non-break-points then is the Fourier serie for F uniformly convergent. But do F have to be ...
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Question about proof of interchanging integral and derivative using uniform convergence.

Theorem: Assume that $\varphi(x,t):[a,b]\times[c,d]\to \Bbb R$ $\varphi(x,t)\in R[a,b]$ ( $\varphi(x,t)$ is Riemann integrable on $[a,b]$ )$\ \ \forall t\in [c,d]$. i.e. $$\int_a^b\varphi(x,t)dx$$ ...
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41 views

Fourier series and determining convergence

Let $g\in PC_{2\pi}$ given by $g(x)=e^{-x}$ for $x\in(-\pi,\pi)$ and $g(-\pi)=g(\pi)=1/2(e^\pi+e^{-\pi})$. How can I find the Fourier series for $g$ and determine if the series convergence pointwise ...
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Dini's theorem failing at a single point

Suppose we have a sequence of continuous functions $f_k: [0, 1] \to [0, 1]$ such that $f_{k + 1}(\beta) \leq f_k(\beta)$ for all $\beta$ (i.e monotone decreasing), and a continuous function $f: [0, 1] ...
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Convergence of hankel transform for polynomials

The hankel transform is related to fourier transforms that have some kind of spherical symmetry. The simplest is related to the 2D radially symmetric fourier transform. This transform $F(k)$ of $f(r)$ ...
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35 views

Uniform convergence with sine functions

Let $f_n = \sin(\frac{3n}{4n+5}x) $ as $f_n:\mathbb{C} \rightarrow \mathbb{C} $. I try to determine if $f_n$ is uniformly convergent on the open intervals $I_1=(0; 1)$ and $I_2(1; \infty)$. Let $\...
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56 views

Stone Weierstrass Theorem for non compact sets

In Rudin (Principles of Mathematical Analysis page 162), the Stone-Weierstrass theorem is proven assuming we have an algebra $\mathscr{A}$ of continuous functions on a compact set $K$ which vanishes ...
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1answer
32 views

Continuity of series sumfunction

Question regarding the series $$\sum_{n=1}^{\infty} e^{-n^2x}=\sum_{n=1}^{\infty} f_n(x)$$. I wished to show that the sumfunction $f(x)=\sum_{n=1}^{\infty} f_n(x)$ is continuous on $]0,\infty[$ (it ...
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1answer
25 views

Series with limit

The problem is the following. Let $(a_{n})_{n\geq0}$ such that $\lim_{n\to\infty}a_{n}=\alpha\in\mathbb{C}$ and let $(b_{n})_{n\geq0}$ a succession of positive real numbers. We know that the series $$...
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1answer
57 views

Weierstrass' M-test in reverse

Weierstrass' M-test says that the series of functions on some set $X$: $$\sum_{n=1}^\infty f_n(x)$$ if $\forall n \in \mathbb{N}, \exists M_n$, \forall x\in X where $M_n \geq |f_n(x)|$, so the ...
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1answer
44 views

Uniform convergence of analytic function on a closed and bounded interval

Let $f:[0, a] \to \mathbb{R}$ be an analytic function. That would mean (by my book definition) that for every $c \in [0, a]$ there exist a neighborhood $U_c \subset [0,a]$ such that $$f(x) = \sum_{n=0}...
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2answers
76 views

Uniform convergence of integral with parameter $\int_0^\infty x^a \cdot e^{-x^{a}}d x$ [closed]

Does the integral $$\int_0^\infty x^a \cdot e^{-x^{a}}d x$$ with parameter $a\in (0,\infty)$ converge uniformly? It is clear how to solve this problem if $ 0 < a < \infty$ But what about my ...
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26 views

Let $f_n:[1,2]\to[0,1]$ be given by $f_n(x)=(2-x)^n$ for all non-negative integers $n$. Let $f(x)=\lim_{n\to \infty}f_n(x)$ for $1\le x\le 2$.

Let $f_n:[1,2]\to[0,1]$ be given by $f_n(x)=(2-x)^n$ for all non-negative integers $n$. Let $f(x)=\lim_{n\to \infty}f_n(x)$ for $1\le x\le 2$. Then which of the following is true? (a) $f$ is ...
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1answer
22 views

Is the sequence $f_n(x)=\frac{x^{2n}}{1+x^{2n}}$ uniformly convergent?

Given $R\in\mathbb{R}$, $R>1$ investigate the sequence of functions $(f_n)_{n\in\mathbb{N}}$, given by $$f_n(x)=\frac{x^{2n}}{1+x^{2n}}, \qquad x \in\ [R,\infty)$$ with regard to uniform ...
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27 views

convergence of a succession of functions.

Can you help me with this problem? Let $f: \mathbb{R} \to \mathbb{R}$ a continuous function, and for all $n \in \mathbb{N}$ define $f_{n} : \mathbb{R} \to \mathbb{R}$ as $f_{n}(x)=f(\frac{x}{n})$, $...
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33 views

checking uniform convergence of series $\sum_{n=1}^\infty x^n$

I have a doubt in a question in which I need to check the uniform convergence of the series given by: $$\sum_{n=1}^\infty x^n$$ on (-$1,1$) Now if the series is uniformly convergent,then its ...
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1answer
27 views

Necessary and sufficient condition for $f_n$($x$) = $b_{n}x$+$c_{n}x^2$ to uniformly converge to zero

Have been trying some questions on uniform and point-wise convergence of sequence of functions. Got stuck in this. I have to prove the following:- Let ($b$$_n$) and ($c$$_n$) be sequences of real ...
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2answers
39 views

Uniform convergence of a sequence of functions which is integral of another sequence

I was going through some questions on pointwise and uniform convergence. Got stuck in one of those which says: Let $g_n(x) = \sin^2(x+\frac{1}{n})$ be defined on $[0,\infty).$ and $f_n(x) = \int_0^...
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1answer
88 views

Uniform convergence of $\sum _{n=1}^{\infty} \frac{\left(-1\right)^{n-1}}{n}x^n $

I treid to show that $$\sum _{n=1}^{\infty} \frac{\left(-1\right)^{n-1}}{n}x^n = \log(1+x)$$ for $\mid x\mid <1$ by showing that $$\sum _{n=1}^{\infty} (-1)^{n-1} x^{n-1} = \frac {1}{x+1}$$ ...

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