For questions about uniform convergence of a sequence or a series of functions on a set. Also used with the tag [convergence].

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0answers
14 views

Show uniform convergence of a sequence $f_n$ on $[a,b]$ if $f_n$ converges uniformly on $(a,b)$ and $f_n(a),f_n(b)$ converge.

Apparently, I need to show that if $\{f_n\}$ converges uniformly on $(a, b)$ and $\{f_n(a)\}$ and $\{f_n(b)\}$ converge, then $\{f_n\}$ converges uniformly on $[a, b]$. However, I don't even see how ...
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0answers
32 views

family of functions/sequences taken over reals instead of naturals

How does convergence for a sequence and a family of functions change when considering $n$ taken from $\mathbb{R}$ instead of the $\mathbb{N}$? for example, consider mollifiers which are defined as ...
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2answers
43 views

Show that $\{f_n\}_{n=1}^\infty$ is a decreasing sequence but the convergence is not uniform on $[0,1]$

Let $$ f_n(x) := \begin{cases} 1 &\text{for $x$ in } \left(0, \frac{1}{n}\right)\\ 0 &\text{$x$ elsewhere in } [0,1] \end{cases}. $$ Show that $\{f_n\}_{n=1}^\infty$ is a decreasing sequence ...
2
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0answers
30 views

When to Interchange Limit & Integral

I really got stuck in the proof of Cauchy's Integral Formula for higher derivatives: Under what Conditions over a function $f$, we can infer that : $\displaystyle\lim_{h\rightarrow ...
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1answer
24 views

Pointwise limit,$f$, of the sequence is not bounded

Question: Let $f_n(x) := \frac{nx}{1+nx^2}$ for $x \in A := [0, \infty)$. Show that each $f_n $is bounded on $A$, but the point-wise limit of $f$ of the sequence is not bounded on $A$. Does $(f_n)$ ...
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1answer
23 views

Uniform-norm of $f_n(x)=ne^{-nx^2}$

Consider a sequence of functions, $(f_n(x))=ne^{-nx^2}$. I think the uniform norm is $ne^{-n}$, but according to my solution, it is $n$. Why is this the case? Don't we just take out the $x$ for the ...
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1answer
15 views

Uniform convergence of functions and Hausdorff convergence of their graphs

Consider a sequence of continuous functions $f_n:[a,b] \to \mathbb{R}$. If their graphs $G_n$ converge to the graph $G$ of a continuous function $f$ (in the Hausdorff metric $d_H$), prove that $f_n$ ...
5
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1answer
49 views

Which properties do still hold for the limit of a sequence of functions?

I think it would be very useful to have a list of properties that are preserved for the limit of a sequence of functions, and was wondering if you could help me making the list more complete. Let ...
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1answer
13 views

Uniform Convergence of Series Help

Suupose the sequence $(b_k) , k\geq 0$ satisties $\sum k|b_k| < \infty$, then show that $\sum_{k=0}^\infty b_kx^k$ converges uniformly to a function $g$ on $|x| \leq 1$ and that $g'(x) = ...
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1answer
30 views

Mollification of a function continuous on $\mathbb{R}\backslash\{0\}$, need uniform convergence

We know that if $f:\mathbb{R} \to \mathbb{R}$ is continuous, then its mollification $f_\epsilon$ converges uniformly to $f$ on compact subsets of $\mathbb{R}$ as $\epsilon \to 0$. My question is, ...
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0answers
39 views

Bound on $f_n'$ implies uniform convergence of $f_n$?

Let $f_n$ be a sequence of functions that converge pointwise to a function $f$. Suppose I know that $|f_n'(x)| \leq C(x)$ where the constant doesn't depend on $n$. How do I conclude that $f_n \to f$ ...
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2answers
116 views

Real Analysis - Uniform Convergence Problem

So I screwed up a problem on my exam. I know that now. But pure mathematics is as difficult and terrifying as it is rewarding for me, and I can't let this go! If someone could tell me if the following ...
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0answers
22 views

An example of a sequence of continuous real functions pointwise convergent, but nowhere locally uniformly convergent? [duplicate]

I've been trying to come up with an example of a sequence of continuous real function which would converge pointwise everywhere, but nowhere converge locally uniformly, but I can't really think of ...
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1answer
33 views

functions of two variables with one variable defined on a compact set uniformly converge to zero

Let $f$ be a holomorphic function on $[0,1]\times \mathbb{R}$. If for each $x\in [0,1]$ fixed, $\lim_{y\to\infty}f(x,y)=0$, prove that $f$ is bounded. My idea: I do not know how to prove and I also ...
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3answers
69 views

Question on Uniform Conergence

I need to show that $\sum_{k=1}^\infty$$(\frac {x}{2})^k$ does not converge uniformly on (-2, 2) I know I have to show that $\sup_{x\in X}\lvert f_n(x)-f(x)\rvert\nrightarrow0 $ as ...
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0answers
17 views

Sufficient conditions for Uniform Law of Large Numbers

I would need a Uniform Law of Large numbers for $f_T(\theta)$ over $\Theta$ when $f$ is the indicator function and, thus, not continuous over $\Theta$. Do you know about any sufficient conditions?
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1answer
35 views

Uniform convergence and differentiation proof of remark 7.17 in Rudin's mathematical analysis

Rudin page 152 Theorem 7.17: Suppose $\{f_n\}$ a sequence of functions, differentiable on $[a,b]$ and such that $\{f_n(x_0)\}$ converges for some point x0 on [a,b]. If {f′n} converges uniformly on ...
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0answers
11 views

Uniform law of large numbers for discontinuous functions?

Do you know about any Uniform Law of Large numbers (see http://en.wikipedia.org/wiki/Law_of_large_numbers#Uniform_law_of_large_numbers) that work when f is the indicator function (and thus not ...
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0answers
32 views

proving uniform convergence for a uniformly continuous function

I've been stuck on this question (2a) for about a day now! Question and havn't really gotten anywhere, I think that the series is that integral of a step function that approximates f ie: $\Gamma_n ...
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0answers
16 views

Implication of uniform stochastic boundedness?

Let $\theta \in \Theta \subseteq \mathbb{R}^d$ be a parameter vector. Let $Q: \Theta \rightarrow \mathbb{R}$ be a function mapping from the parameter space to the real numbers. Let $Z_T$ be a a ...
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1answer
42 views

Understanding pointwise convergence vs. uniform convergence example

I'm trying to understand the difference between pointwise convergence and uniform convergence. I read this post and the last answer on it is the following: $f_n\to f$ pointwise on $(a,b)$ if for ...
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1answer
41 views

Bounded continuous functions from a topological space to $\mathbb R$ is complete

Let $X$ be a topological space and let $BC( X \to \mathbb R)$ be the space of bounded continuous functions from $X$ to $ \mathbb R$ equipped with the supnorm $||*||_{\infty}$. How to prove $BC( X \to ...
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1answer
19 views

In Egorov's theorem, remove the condition $\mu(E) < \infty$ and let the sequence be convergent in measure. The conclusion holds for subsequence

Let $(X,\mathscr{F},\mu)$ be a measure space, $E \in \mathscr{F}$, $\{f_n\}$ is a sequence of measurable functions on $E$, and the sequence converges to function $f$ in measure. Show that $\exists ...
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0answers
21 views

If periodic function has a discontinuity at $x_0$ its Fourier series cannot converge uniformly on any interval containing $x_0$, why?

I'm reading about Fourier series and in one point my book states the following: Suppose $f$ is a periodic function. If $f$ has a discontinuity at $x_0$, the Fourier series of $f$ cannot ...
0
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2answers
55 views

Prove uniform convergence using the M-test

Problem: Prove that the following series converges uniformly on the given interval using Weierstrass M-test. $$\sum\frac{nx}{1+n^5x^2} $$ on $|x| < +\infty$. My attempt has been to use $M_{n} = ...
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1answer
21 views

If a sequence of continuous functions converges pointwise to a continuous function on $ [a,b] $, it converges uniformly

If a sequence of continuous functions converges pointwise to a continuous function on $ [a,b] $, it converges uniformly. Looking at other theorems on the relationship between continuity and uniform ...
0
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1answer
34 views

uniform convergence sequence

The sequence of function $\{f_n\}$ defined on $\mathbb{R}$, every function is decreasing function (if $x \geq y$ then $f_1(x)\geq f_1(y)$, $f_2(x)\geq f_2(y)$,.......) and sequences of function is ...
0
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1answer
27 views

Complex Weierstrass M-test question.

Use the Weierstrass M-test to show $\forall\epsilon>0,\sum_{n=1}^{\infty} a_nn^{-z}$ converges uniformly if $Re(z)>=1+\epsilon$, where $a_{n}$ is bounded. This is what I've done: ...
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1answer
25 views

Does the following limit hold uniformly for all $f$ in a function space $X$?

Suppose I have the following for all $\epsilon > 0$ $$F(f,h) \leq C_0\epsilon + C_1(\epsilon)\sqrt{h}\quad\text{for all $f \in X$}$$ where $F:X\times \mathbb{R} \to \mathbb{R}$ and $X$ is some ...
0
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1answer
33 views

If $f_n \to f$ uniformly a.e. and $u_n \to u$ a.e., does $f_n(u_n) \to f(u)$ a.e.?

Here $f_n$ and $f$ are real-valued functions of 1 variable. Suppose that $f_n \to f$ uniformly on $\mathbb{R}\backslash \{0\}$. Let $u_n \to u$ pointwise a.e. on a manifold $X$. Does it follow that ...
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0answers
22 views

Uniform Convergence of Series Implies Uniform Convergence of the Related Sequence

Question: Prove that if the series $\sum_{i=1}^\infty u_n(x)$ converges uniformly on an interval, then $\lim_{n \to \infty} u_n = 0$ uniformly on that interval. What I have so far: We are given: ...
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1answer
47 views

modern analysis: uniform convergence of sequences

Consider $f_n(x) = \dfrac{\sin nx}{\sqrt n}$. Show that this converges uniformly but that the sequence $\{f'_n\}$ is unbounded.
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2answers
98 views

analysis: sequence of functions, uniform convergence

Give examples of each of the following: a.) A sequence of integrable functions (of R), {fn}, which converge pointwise to an integrable function, f but the limit of the integral of fn is not equal to ...
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3answers
64 views

modern analysis: limits, integrals, uniform

Suppose $\{f_n\} \to f$ uniformly on $[a, b]$ and both $f$ and the $f_n$ are integrable. Prove that $\lim_{n\to \infty}\int_{a}^bf_n(x)dx = \int_{a}^bf(x)dx$
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1answer
54 views

If $f_n \to f$ pointwise a.e. and $g_n \to g$ uniformly then $g(f_n) \to g(f)$ pointwise a.e?

Let $X$ be some space (eg. a manifold). Suppose that $f_n:X \to \mathbb{R}$ converges pointwise a.e. to $f:X \to \mathbb{R}$. Let $g_n:\mathbb{R} \to \mathbb{R}$ be continuous with $g_n \to g$ ...
1
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1answer
52 views

Showing the power series of $\cos(x)$ converges uniformly to $\cos(x)$ on every bounded interval

Show that the power series of $\cos(x)$ converges uniformly to $\cos(x)$ on every bounded interval. My attempt: The power series for $\cos(x)$ is $$\sum_{n=0}^{\infty} ...
1
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1answer
45 views

sequence uniformly convergent on the boundary of a bounded set in $\mathbb{C}$

Let $(f_n)$ be a sequence of functions which are analytic on a bounded region $A\subset\mathbb{C}$ and continuous on the closure $Cl(A)$. Suppose that the sequence is uniformly convergent on the ...
1
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1answer
48 views

Showing if $f_n \to f$ uniformly and each $f_n$ has at most $10$ discontinuities, then so does $f$

Suppose that $f_n:[a,b] \to \Bbb R$ and $f_n$ uniformly converges to $f$ as $n$ goes to infinity. How to prove that if each $f_n$ has at most ten discontinuities (the discontinuities for each $f_n$ ...
2
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1answer
33 views

Power series centered at $x =0$

I have this question in my advanced calculus textbook. Give an example of a power series centered at $x = 0$ that converges on $[-2,2]$ but not absolutely on the entire interval $[-2,2]$, and ...
0
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1answer
27 views

Uniform convergence to the derivative

Suppose $\phi \in C^{\infty}(\mathbb{R})$. Then is it true that $$ \frac{\phi(x+h)-\phi(x)}{h} \to \phi'(x) \quad (h \to 0) $$ uniformly on $\mathbb{R}$? It seems to me like this is trivially the case ...
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1answer
53 views

Question on uniform convergence of a subsequence

Any help would be great. Let $F,G$ : $\mathbb{R}\to (0,\infty ) $ be continuous. Let $f_n: \mathbb{R} \to \mathbb{R} $ be everywhere differentiable and satisfy $ |f_n(x)| \leq F(x) $ and $ ...
0
votes
1answer
23 views

A stronger concept than total boundness

A space, every proper principal filter of which is refined by a Cauchy filter, is called totally bounded. Is there a term (and theory) about a stronger concept: a space every proper filter of which ...
2
votes
2answers
85 views

Prove that $f_n$ converges uniformly on $[a,b]$

Let $f_n$ be a sequence of functions defined on $[a,b]$. Suppose that for every $c \in [a,b]$, there exist an interval around $c$ in which $f_n$ converges uniformly. Prove that $f_n$ converges ...
0
votes
0answers
19 views

Rate of convergence almost surely

I would like to see the rate (ordre) of convergene almost surely of a scheme they said from that $ \forall \beta \in ]0,1/2[,\qquad > Dt^{\beta}.\sup_{t\in[0,T]}|\widehat{X_{t}} -X_{t}| ...
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1answer
17 views

Different uniform spaces having the same set of Cauchy filters

I want to understand how Cauchy space is different than uniform space. For this I need an example: An example of two different uniform spaces having for both of them the same set of Cauchy filters?
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1answer
39 views

Compact convergence of series

I am trying to show that $$f(z) = \sum_{k=1}^{\infty} \frac{(-1)^{k}}{z+k}$$ converges compactly over $\mathbb{C}$ and starting to think that this statement is false after several attempts. If I ...
0
votes
1answer
29 views

Uniform convergence and Integrals

For $x \in \mathbb{R}$ and $n \in \mathbb{N}$, let $f_n(x) = \frac{x}{1+nx^2}$ and $f(x)=0$. (i) Prove $f_n \to f$ uniformly. (ii) Calculate $\displaystyle{\lim_{n\to\infty}\int_0^1} f_n(x)dx$ in ...
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1answer
31 views

Show that the upper envelope of diff EQ solutions is a solution

Assume $f(x,y)$ is continuous in the rectangle $$R:=\{(x,y)\in \mathbb{R}^2: |x-x_0|\leq a,\,|y-y_0|\leq b\},$$ and let $S$ be the set of functions $y=y(x)$ which are continuous, satisfy ...
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0answers
37 views

Normal convergence versus uniform convergence

I am wondering about the nature of uniform and normal convergence. I know that uniform convergence is a weaker condition than normal convergence and that normal convergence even implies uniform ...
0
votes
1answer
47 views

Does $f_n$ converge to $f$ uniformly?

Consider $f_n: [0,1]\longrightarrow\mathbb{R}$ is given by $$f_n(x) = \begin{cases} \sqrt{n}, & \quad 0<x<\frac{1}{n} \\ 1, & \quad \text{otherwise.} \end{cases}$$ 1.) ...