What is Uniform convergence: Definition and 162 Discussions
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions
(
f
n
)
{\displaystyle (f_{n})}
converges uniformly to a limiting function
f
{\displaystyle f}
on a set
E
{\displaystyle E}
if, given any arbitrarily small positive number
ϵ
{\displaystyle \epsilon }
, a number
N
{\displaystyle N}
can be found such that each of the functions
f
N
,
f
N
+
1
,
f
N
+
2
,
…
{\displaystyle f_{N},f_{N+1},f_{N+2},\ldots }
differ from
f
{\displaystyle f}
by no more than
ϵ
{\displaystyle \epsilon }
at every point
x
{\displaystyle x}
in
E
{\displaystyle E}
. Described in an informal way, if
f
n
{\displaystyle f_{n}}
converges to
f
{\displaystyle f}
uniformly, then the rate at which
f
n
(
x
)
{\displaystyle f_{n}(x)}
approaches
f
(
x
)
{\displaystyle f(x)}
is "uniform" throughout its domain in the following sense: in order to guarantee that
f
n
(
x
)
{\displaystyle f_{n}(x)}
falls within a certain distance
ϵ
{\displaystyle \epsilon }
of
f
(
x
)
{\displaystyle f(x)}
, we do not need to know the value of
x
∈
E
{\displaystyle x\in E}
in question — there can be found a single value of
N
=
N
(
ϵ
)
{\displaystyle N=N(\epsilon )}
independent of
x
{\displaystyle x}
, such that choosing
n
≥
N
{\displaystyle n\geq N}
will ensure that
f
n
(
x
)
{\displaystyle f_{n}(x)}
is within
ϵ
{\displaystyle \epsilon }
of
f
(
x
)
{\displaystyle f(x)}
for all
x
∈
E
{\displaystyle x\in E}
. In contrast, pointwise convergence of
f
n
{\displaystyle f_{n}}
to
f
{\displaystyle f}
merely guarantees that for any
x
∈
E
{\displaystyle x\in E}
given in advance, we can find
N
=
N
(
ϵ
,
x
)
{\displaystyle N=N(\epsilon ,x)}
(
N
{\displaystyle N}
can depend on the value of
x
{\displaystyle x}
) so that, for that particular
x
{\displaystyle x}
,
f
n
(
x
)
{\displaystyle f_{n}(x)}
falls within
ϵ
{\displaystyle \epsilon }
of
f
(
x
)
{\displaystyle f(x)}
whenever
n
≥
N
{\displaystyle n\geq N}
.
The difference between uniform convergence and pointwise convergence was not fully appreciated early in the history of calculus, leading to instances of faulty reasoning. The concept, which was first formalized by Karl Weierstrass, is important because several properties of the functions
f
n
{\displaystyle f_{n}}
, such as continuity, Riemann integrability, and, with additional hypotheses, differentiability, are transferred to the limit
f
{\displaystyle f}
if the convergence is uniform, but not necessarily if the convergence is not uniform.
The below proposition is from David C. Ullrich's "Complex Made Simple" (pages 264-265)
Proposition 14.5. Suppose ##D## is a bounded simply connected open set in the plane, and let ##\phi: D \rightarrow \mathbb{D}## be a conformal equivalence.
(i) If ##\zeta## is a simple boundary point of...
Here is the solution I came up with
Consider the sequence of functions ##\{f_n\}=\left \{ \frac{x}{n(1+nx^2)}\right \}## defined on ##\mathbb{R}##.
By differentiating ##f_n(x)## and equating to zero we find critical points at ##x=\pm \frac{1}{\sqrt{n}}##.
By checking the second derivative we...
I have previously shown that the function series is differentiable at ##x\neq 0##. The series converges uniformly (thus pointwise) on ##\mathbb R## and the term wise differentiated series is uniformly convergent on any interval ##d\leq |x|##, where ##d>0##. Moreover, the terms are continuously...
Hey! 😊
Let $0<\alpha \in \mathbb{R}$ and $(f_n)_n$ be a sequence of functions defined on $[0, +\infty)$ by: \begin{equation*}f_n(x)=n^{\alpha}xe^{-nx}\end{equation*} - Show that $(f_n)$ converges pointwise on $[0,+\infty)$.
For an integer $m>a$ we have that \begin{equation*}0 \leq...
I'm not too sure how to use the hint here. What I had so far was this: an odd extension of ##f## implies ##f = \sum_{k=1}^\infty b_k \sin(k x)##. Notice for ##m>n## $$ \left|\sum_{k=1}^m b_k\sin(k x) - \sum_{k=1}^n b_k\sin(k x)\right| = \left| \sum_{k=n+1}^m b_k\sin(k x)\right| \leq...
Homework Statement
This is a translation so sorry in advance if there are funky words in here[/B]
f: ℝ→ℝ a function 2 time differentiable on ℝ. The second derivative f'' is bounded on ℝ.
Show that the sequence on functions $$ n[f(x + 1/n) - f(x)] $$ converges uniformly on f'(x) on ℝ...
Homework Statement
Let ##X \subset \mathbb{C}##, and let ##f_n : X \rightarrow \mathbb{C}## be a sequence of functions. Show if ##f_n## is uniformly Cauchy, then ##f_n## converges uniformly to some ##f: X \rightarrow \mathbb{C}##.
Homework Equations
Uniform convergence: for all ##\varepsilon >...
Homework Statement
The series is uniformly convergent on what interval?
Homework EquationsThe Attempt at a Solution
[/B]
Using the quotient test (or radio test), ##|\frac{a_{n+1}}{a_{n}}| \rightarrow |x^2*\sin(\frac{\pi \cdot x}{2})|, n \rightarrow \infty##.
However from here I'm stuck...
Hello everyone!
I'm a student of electrical engineering, preparing for the theoretical exam in math which will cover stuff like differential geometry, multiple integrals, vector analysis, complex analysis and so on. So the other day I was browsing through the required knowledge sheet our...
Hi Physics Forums,
I have a problem that I am unable to resolve.
The sequence ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}## of positive integer powers of ##\mathrm{sinc}(x)## converges pointwise to the indicator function ##\mathbf{1}_{\{0\}}(x)##. This is trivial to prove, but I am struggling to...
Let me give some context.
Let X be a compact metric space and ##C(X)## be the set of all continuous functions ##X \to \mathbb{R}##, equipped with the uniform norm, i.e. the norm defined by ##\Vert f \Vert = \sup_{x \in X} |f(x)|##
Note that this is well defined by compactness. Then, for a...
Hello! Can someone explain to me in an intuitive way the difference between pointwise and uniform convergence of a series of complex functions ##f_n(z)##? Form what I understand, the difference is that when choosing an "N" such that for all ##n \ge N## something is less than ##\epsilon##, in the...
Hi.
I have this power serie (2^n/n)*z^n that runs from n=1 to infinity, and I have to show whether it's uniform konvergence on [-1/3, 1/3] or not.
I hope someone can help me with this.
Hey! :o
I want to check the pointwise and uniform convergence for the following sequences or series of functions:
$f_n:[0, \infty)\rightarrow \mathbb{R}, f_n(x)=xe^{-nx}$ for all $n\in \mathbb{N}$
$f_n:[0, \infty)\rightarrow \mathbb{R}, f_n(x)=nxe^{-nx}$ for all $n\in \mathbb{N}$...
Hi, I started to study the function of Weierstrass (https://en.wikipedia.org/wiki/Weierstrass_function)
And in one part says that the sum of continuous functions is a continuous function.
i understand this but the Limiting case is a different history depend of the convergence, so what i need...
Homework Statement
##I## a set of real number and ##f_k : I \rightarrow \mathbb{R}## a succession of real functions defined in ##I##.
We say that ##f_k## converges punctually in ##I## to the function ##f : I \rightarrow \mathbb{R}## if
$$\lim_{k \to \infty} f_k(x) = f(x), \hspace{1cm} \forall x...
Hi everybody! I'm preparing an exam of "Analysis II" (that's how the subject's called in German), and I have trouble understanding how to find the limit of a multivariable function, especially when it comes to proving the uniform convergence. Here is an example given in the script of my teacher...
Hello;
I'm struggling with pointwise and uniform convergence, I think that examples are going to help me understand
Homework Statement
Consider the Fourier sine series of each of the following functions. In this exercise de not compute the coefficients but use the general convergence theorems...
Homework Statement
Find an example of a sequence ##\{ f_n \}## in ##L^2(0,\infty)## such that ##f_n\to 0 ## uniformly but ##f_n \nrightarrow 0## in norm.
Homework Equations
As I understand it we have norm convergence if
##||f_n-f|| \to 0## as ##n\to \infty##
and uniform convergence if there...
Find the Range of Uniform convergence of $ \zeta\left(x\right) = \sum_{n=1}^{\infty}\frac{1}{{n}^{x}} $
Using the Weierstrass-M test, I get this converges for $ 1 \lt x \lt \infty $
But the book's answer is $ 1 \lt s \le x \lt \infty $? I have scoured the book but can't see why they say it...
I need to prove that the complex power series $\sum\limits_{n=0}^{\infty}a_nz^n$ converges uniformly on the compact disc $|z| \leq r|z_0|,$ assuming that the series converges for some $z_0 \neq 0.$
*I know that the series converges absolutely for every $z,$ such that $|z|<|z_0|.$ Since...
Homework Statement
I came across a problem where f: (-π/2, π/2)→ℝ where f(x) = \sum\limits_{n=1}^\infty\frac{(sin(x))^n}{\sqrt(n)}
The problem had three parts.
The first was to prove the series was convergent ∀ x ∈ (-π/2, π/2)
The second was to prove that the function f(x) was continuous...
So I'm reading "An Introduction to Wavelet Analysis" by David F. Walnut and it's saying that the following sequence
" (x^n)_{n\in \mathbb{N}} converges uniformly to zero on [-\alpha, \alpha] for all 0 < \alpha < 1 but does not converge uniformly to zero on (-1, 1) "
My problem is that isn't...
on page 4, example 9 in this link, http://www.personal.psu.edu/auw4/M401-notes1.pdf, they show a sequence of functions is not uniformly convergent. To show this, you need to show that for some epsilon, there is no 'universal' N.
But they didn't pick a particular value of $z$, they chose...
Homework Statement
Define
f_n : \mathbb{R} \rightarrow \mathbb{R} by
f_n(x) = \left( x^2 + \dfrac{1}{n} \right)^{\frac{1}{2}}
Show that f_n(x) \rightarrow |x| converges uniformly on compact subsets of \mathbb{R}
Show that the convergence is uniform in all of \mathbb{R}...
A uniformly convergent sequence of continuous functions converges to a continuous function.
I have no problem with the conventional proof. However, in Henle&Kleinberg's Infinitesimal Calculus, p. 123 (Dover edition), they give a nonstandard proof, and they use the hyperhyperreals to do it. I...
Homework Statement
Find the range of uniform convergence for the following series
η(x) = ∑(-1)n-1/nx
ζ(x) = ∑1/nx
with n ranging from n=1 to n=∞ for both
Homework Equations
To be honest I'm stumped with where to begin altogether. In my text, I'm given the criteria for uniform...
Hi,
I have to prove the following theorem:
Let $f_n:[0,1] \to \mathbb{R}, \forall n \geq 1$ and suppose that $\{f_n|n \in \mathbb{N}\}$ is equicontinuous. If $f_n \to f$ pointwise then $f_n \to f$ uniformly.
Before I start the proof I'll put the definitions here:
$f_n \to f$ pointwise if and...
I have a question where I am supposed to show that a series does not converge uniformly, I get the majority of the question, but one part in the solution I can't see the rationale or how they decided on the result:
It has to do with the partial sum:
SN= (1 - (-x2)N+1)/ (1+x2)
The...
Homework Statement
Is the sequence of function ##f_1, f_2,f_3,\ldots## on ##[0,1]## uniformly convergent if ##f_n(x) = \frac{x}{1+nx^2}##?
2. The attempt at a solution
I got the following but I think I did it wrong.
For ##f_n(x) = \frac{x}{1+nx^2}##, I got if ##f_n \to0## then we must...
Homework Statement
Show that the sequence of functions ##x,x^2, ... ## converges uniformly on ##[0,a]## for any ##a\in(0,1)##, but not on ##[0,1]##.2. The attempt at a solution
Is this correct? Should I add more detail? Thanks for your help!
Let ##\{f_n\} = \{x^n\}##, and suppose ##f^n \to...
Homework Statement
Let f_{n}(x)=\frac{x}{1+x^n} for x \in [0,∞) and n \in N. Find the pointwise limit f of this sequence on the given interval and show that (f_{n}) does not uniformly converge to f on the given interval.
Homework Equations
The Attempt at a Solution
I found that the pointwise...
Homework Statement
Let \left[a,b\right] be a closed bounded interval, f : [a,b] \rightarrow \textbf{R} be bounded, and let g : [a,b] \rightarrow \textbf{R} be continuous with g\left(a\right)=g\left(b\right)=0. Let f_{n} be a uniformly bounded sequence of functions on \left[a,b\right]. Prove...
Homework Statement
http://gyazo.com/55eaace8994d246974ef750ebeb36069
Homework Equations
Theorem III :
http://gyazo.com/af2dfeb33d3382430d39f275268c15b1
The Attempt at a Solution
At first this question had me jumping to a wrong conclusion.
Upon closer inspection I see the...
Homework Statement
Suppose that ##s_n(x)## converges uniformly to ##s(x)## on ##[b, ∞)##.
If ##lim_{x→∞} s_n(x) = a_n## for each n and ##lim_{n→∞} a_n = a## prove that :
##lim_{x→∞} s(x) = a##
Homework Equations
##\space ε/N##
The Attempt at a Solution
I see a quick way to do this one...
Homework Statement
I would like to use the Weierstrass M-test to show that this family of functions/kernels is uniformly convergent for a seminar I must give tomorrow.
H_{t} (x) = \sum ^{-\infty}_{\infty} e^{-4 \pi ^{2} n^{2} t} e^{2 \pi i n x} .
Homework Equations
The Attempt at a...
Consider the sequence $\{f_n\}$ of complex valued functions, where $f_n=tan(nz)$, $n=1,2,3\ldots$ and $z$ is in the upper half plane $Im(z)>0$. I want to show two facts about this sequence:
1) it's uniformly locally bounded: for every $z_0=x_0+iy_0$ in the upper half plane, ther exist...
Let X be a set, and let fn : X---> R be a sequence of functions. Let ρ be the uniform metric on the space RX. Show that the sequence (fn) converges uniformly to the function f:X--> R if and only if the sequence (fn) converges to f as elements of the metric space (RX, ρ). [Note: the ρ's should...
Homework Statement
For question 25.15 in this link:
http://people.ischool.berkeley.edu/~johnsonb/Welcome_files/104/104hw9sum06.pdf
I have some questions about pointwise convergence and uniform convergence...
Homework Equations
The Attempt at a Solution
Our textbook says...
f is a continuous function on [0,infinity) such that 0<=f(x)<=Cx^(-1-p) where C,p >0 f_k(x) = kf(kx)
I want to show that lim k->infinity ∫from 0 to 1 of f_k(x) dx exists
so my idea is if I have that f_k(x) converges to f(x)=0 uniformly which I was able to show and that f_k(x) are all...
I'm wondering about uniform convergence. We're looking at it in my complex analysis class. We are using uniform convergence of a series of functions, to say that we can interchange integration of the sum, that is: \int\sum b_{j}z^{j}dz=\sum\int b_{j}z^{j}dz=\int f(z)dz
On an intuitive level I...
Homework Statement
I need to show that f_{n}=sin(\frac{z}{n}) converges uniformly to 0.
Homework Equations
So I need to find K(\epsilon) such that \foralln \geq K
|sin(\frac{z}{n})|<\epsilon
I'm trying to prove this in an annulus: \alpha\leq |z| \leq\beta
The Attempt at a Solution
I'm having...
Given a power series \sum a_n x^n with radius of convergence R, it seems that the series converges uniformly on any compact set contained in the disc of radius R. This might be a silly question, but what's an example of a power series that doesn't actually also converge uniformly on the whole...
Homework Statement
show that the integral of the poisson kernel (1-r^2)/(1-2rcos(x)+r^2) converges to 0 uniformly in x as r tend to 1 from the left ,on any closed subinterval of [-pi,pi] obtained by deleting a middle open interval (-a,a)
Homework Equations
the integral of poisson...
In Spivak's Calculus, there is a theorem relating the derivative of the limit of the sequence {fn} with the limit of the sequence {fn'}.
What I don't like about the theorem is the huge amount of assumptions required:
" Suppose that {fn} is a sequence of functions which are differentiable on...
Homework Statement
f(x)= {1, ‐1/2<x≤1/2}
{0, ‐1<x≤ ‐1/2 or 1/2<x≤1}
State whether or not the function's Fourier sine and cosine series(for the corresponding half interval) converges uniformly on the entire real line ‐∞<x<∞
Homework Equations
The Attempt at a Solution...
Homework Statement
f_{n} is is a sequence of functions in R, x\in [0,1]
is f_{n} uniformly convergent?
f = nx/1+n^{2}x^{2}
Homework Equations
uniform convergence \Leftrightarrow
|f_{n}(x) - f(x)| < \epsilon \forall n>= n_{o} \inN
The Attempt at a Solution
lim f_{n} = lim...
Homework Statement
Let f,g be continuous on a closed bounded interval [a,b] with |g(x)| > 0 for all x in [a,b]. Suppose that f_n \to f and g_n \to g uniformly on [a,b]. Prove that \frac{1}{g_n} is defined for large n and \frac{f_n}{g_n} \to \frac{f}{g} uniformly on [a,b]. Show that this is...