0
votes
0answers
36 views

Limit of measurable function is measurable

This question has been asked already here but I didn't get a satisfactory solution and didn't want to bring up an old question. Here is the question : Let $\{f_n\}$ be a sequence of measurable ...
0
votes
0answers
31 views

Tensor Product of Hilbert Spaces: incomplete?

Let $\mathcal{H}$ be an infinite dimensional Hilbert space and $\mathcal{H}\otimes_0\mathcal{H}$ its algebraic twofold tensor product. Define a scalar product on it as ...
0
votes
4answers
43 views

What is $f$? Finding where a function converges pointwise?

I have a question. Let $f_n(x) = x^{4n} + \frac1{n^2}$. AS $n \to \infty$, $f_n$ converges pointwise to a function $f$ on $[0,1]$ What is $f$? Now if I am understanding correctly, couldn't ...
0
votes
3answers
40 views

Does $L^2$ strong convergence and bounded $L^\infty$ imply convergence in $L^\infty$ along subsequences?

Take a sequence of functions $f_n \in L^2([0,1])$ such that $f_n \rightarrow f$ in $L^2([0,1])$ (convergence in $L^2$ norm) and $|f_n|_{L^\infty},|f|_{L^\infty} <R$ (uniformly bounded in ...
1
vote
1answer
23 views

Convergence and uniform convergence of a sequence of functions

Suppose that ($f_n$;$n \geq 1$) is a sequence of functions defined on an interval [a,b].We will say $f_n$ tends to $f$ uniformly on $[a,b]$ as n tends to infinity if for every $\epsilon>0$ there ...
1
vote
1answer
20 views

Proving that a function is a contraction map

I have a function defined by: $F(X) = a \left( \frac{X-A}{\|X-A\|_2} - \frac{B-X}{\|B-X\|_2} \right) $ with $X,A,B \in \mathbb{R}^3 $ $a\in \mathbb{R}_+ $. Is this a contraction map? If yes I want to ...
1
vote
1answer
46 views

Do one-sided limits exist for this functi0n?

Given a function $f$ with $f(x) := \begin{cases} \\ 0, & \text{if }x\in\mathbb{R}\setminus\mathbb{Q}\\ 1, & \text{if }x\in\mathbb{Q}\\ \end{cases}$ , what is $\lim\limits_{x\to0^+}f(x)$ and ...
1
vote
3answers
51 views

Prove $\frac{x}{1+n^2x^2}$ is uniformly convergent

I need to prove that \begin{equation}f_n(x)=\frac{x}{1+n^2x^2}\end{equation} Converges uniformly to $0$. I've tried a solution: (Scratchwork) Want ...
3
votes
1answer
41 views

Where does $f_n(x)=\frac{x^n}{1+x^{2n}}$ converge uniformly?

I'm given the function \begin{equation} f_n(x)=\frac{x^n}{1+x^{2n}}, \end{equation} where I can assume $f_n:[0,\infty)\to \mathbb{R}$. I need to determine and show the sets over which the ...
2
votes
1answer
66 views

Pointwise convergence and uniform

There is a norm $\Vert .\Vert$ in the space $C([a,b],\mathbb{R})$ such that convergence pointwise implies convergence in norm $\Vert .\Vert$ ? I think not because if there would be the natural ...
0
votes
1answer
44 views

Convergence of series of continuous bounded functions

Let $\mathcal{C}$ be the set of continuous bounded mappings from $\mathbb{R}^n$ to $\mathbb{R}^m$. Let $F: \mathcal{C} \rightarrow \mathcal{C}$ be a continuous (with respect to the $\sup$ norm) ...
0
votes
2answers
23 views

Searching for a function

I'm searching for a funcion with this values: f(0)=0 f(1)=1 f(2)=3/2 f(3)=7/4 ... lim from x to infinity: f(x) = 2 I don't want the recursive way to define f.
1
vote
3answers
47 views

Need to know how: $L^{p}$ convergence for function.

I feel terrible at the moment as I have exhausted everything possible to understand this and I am still stuck. I have looked everywhere for some sort of resource to the concept at hand, and yet, they ...
2
votes
3answers
88 views

How to read $\bigcap\limits_{r=1}^{\infty}\bigcup\limits_{k=1}^{\infty}\bigcap\limits_{n,m\ge k}^{\infty}\{x:|f_n-f_m|<\frac{1}{r}\}$

Can somebody explain me step by step why does this $\bigcap\limits_{r=1}^{\infty}\bigcup\limits_{k=1}^{\infty}\bigcap\limits_{n,m\ge k}^{\infty}\{x:|f_n-f_m|<\frac{1}{r}\}$ represents the set of ...
0
votes
1answer
33 views

Absolute and conditional convergence of a series with $\sin(x)$

I have to explore absolute and conditional convergence of this function series I tried to find $a(n)$ and $a(n+1)$ terms of the series and then divide it and take a limit. But I've got nothing. ...
1
vote
2answers
35 views

Convergent integral of divergent function

On one of my calculus lectures i was told that exist convergent inmproper integrals (in infinity) of divergent function. I was searching for an example in the internet, but I didn't find any. Has any ...
0
votes
1answer
43 views

Convergence rate of exponential function

If I have two exponential function, say $f_1(t)=4e^{-3t}+6e^{-7t}$ and $f_2(t)=\frac{2e^{-3t}+5e^{-7t}}{e^{-3t}+9e^{-7t}} - 2$ who are all converge to $0$. Then, the convergence rate of $f_1(t)$ can ...
1
vote
6answers
86 views

Prove that the succession $x_{n}$ where $x_{1}=1$ and $x_{n+1} = \sqrt{3x_{n}}$ is convergent

Well, ive been having a weee bit of problem solving this homework, can anyone give me a hand? Prove that the sequence $x_{n}$ where $x_{1}=1$ and $x_{n+1} = \sqrt{3x_{n}}$ is convergent and calculate ...
0
votes
5answers
65 views

What function, f(x), converges to linear function when x -> inf as well as when x -> -inf

I know that $\sqrt{1+x^2}$ does, but it is not good enough, because I need it to converge to $a+b x$ when $x\to\infty$ and $c+d x$ when $x\to-\infty$, with $a,b,c,d\in\mathbb{R}$. How to find the ...
0
votes
0answers
18 views

When does the argument of a function converge given that the image converges?

Suppose $x_n \to x$ and $f(v_n)=x_n,f(v)=x$. Under what (minimal) conditions on $f$ does $v_n \to v$?
3
votes
1answer
156 views

Mathematical Analysis help

Let $f$ be differentiable on $(0,\infty)$ and suppose $\lim_{x\rightarrow\ \infty}(f(x)+f'(x))=L$. Then, $\lim_{x\rightarrow\ \infty}f(x)= L$ and $\lim_{x\rightarrow\ \infty}f'(x)= 0$ My approach: ...
0
votes
1answer
90 views

Does convergence in $C(\Bbb R)$ imply convergence in $\mathcal D'(\Bbb R)$

Sequence of continuous functions $(f_k)$ which converge to $f$ in $C(\Bbb R)$. Does it imply that $f_k$ will converge to $f$ in $\mathcal D'(\Bbb R)$ as well? Or we need something more to make this ...
1
vote
1answer
50 views

Approximating Lipschitz funtion by $C^1$ function.

Let $G:\mathbb{R}\to\mathbb{R}$ be a Lipschitz function and $L$ its Lipschitz constant. Suppose that $G(0)=0$. It is known that $G$ is almost everywhere differentiable and $G'\in L^\infty$ with ...
0
votes
2answers
80 views

Show uniform convergence of indefinite function series

How can i show uniform convergence on function series like this one: $\sum\limits_{k=1}^{\infty} (\sqrt{1-x^{n}}-1)$ ? I have a given interval of [0 / 0.5] I thought about using the Weierstrass ...
3
votes
5answers
241 views

Sequence of continuous functions which converges to a continuous limit [duplicate]

Any help with this: construct a sequence of continuous functions defined on $ [0,1] $ which converges pointwise but not uniformly to a continuous limit ? Thank you.
1
vote
1answer
92 views

Check convergence of $f_{n}(x)=x^{n}-x^{2n}=x^{n}(1-x^{n})$

Check convergence of $f_{n}(x)=x^{n}-x^{2n}$ where $x\in(0,1)$ Please verify my answer, I'm not sure I'm doing it correctly. Thanks in advance!
0
votes
1answer
194 views

Can a sequence of functions converge to different functions pointwise and on average?

Is it possible for a sequence of functions in $\mathcal{C}\left[0,1\right]$ to converge to one function pointwise (not necessarily to a continuous function) and to a different function in average ...
2
votes
0answers
146 views

Scaling function and dilation equation in MRA

In the book I am reading, the author explains how we can use the dilation equation to obtain the scaling function through the process of iteration. In particular, we use the dilation equation, which ...
1
vote
2answers
119 views

Does this series approach an actual value?

Does this series converge to an actual number for any value of $x$? $$\sum_ {k = 1}^{\infty} \frac{\ln(k)\sin( 2\pi kx) }{ k}. $$ I tried summing the series for $x=2/3$ on wolfram alpha, and it seems ...
2
votes
2answers
50 views

Study of a series of functions

I've to study this series: $$\sum_{n=1}^\infty e^{\sqrt n\,x}$$ My teacher wrote that with the asymptotic comparison with this series: $$\sum_{n=1}^\infty\frac{1}{n^2}$$ My series converges ...
2
votes
0answers
115 views

uniformly convergent

Define functions $f_n\colon [0,1]\to\Bbb R$ by $f_n(x)=n^px\exp(-n^qx)$ where $p$, $q>0$ and $f_n\to 0$ pointwise on $[0,1]$ as $n\to \infty$ and $\sup|f_n(x)|=(n^{p-q})/e$. Assume that ...
1
vote
6answers
107 views

limit of convergent series

What is the limit of $U_{n+1} = \dfrac{2U_n + 3}{U_n + 2}$ and $U_0 = 1$? I need the detail, and another way than using the solution of $f(x)=x$, as $f(x) = \frac{2x+3}{x+2}$ because I can't show ...
2
votes
3answers
688 views

Meaning of functions that vanish at a point

I realize this may be a very thick question, but I have been wondering for some time. Sometimes I am asked to prove or read proofs involving "functions that vanish at a point" or "every point" or ...
4
votes
1answer
98 views

Problem with uniform convergence

Let $f_n$ ($n=1,2,\dots$) be a sequence of functions $f_n\colon \mathbb R\to \mathbb R$ of class $C^1$ such that $f_n \rightrightarrows 0 $, $f_n' \rightrightarrows 0 $. Assume moreover that functions ...
0
votes
1answer
187 views

Convergence in measure and pointwise convergence in continuity points

Hi can you help me with the following: $\{f_n\}$ a sequence of increasing functions with $f_n\to f$ in measure on $[a,b]$. Show that $f_n(x)\to f(x)$ at every $x$ where $f(x)$ is continuous. ...
12
votes
2answers
178 views

About the solution of the infinite recurrence $f(x,f(x,f(x,f(x,f(…))))=a$

On internet I found some recreational problems as $$3=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$$ $$\frac{1}{2}=\frac{1}{x+\frac{1}{x+...}}$$ $$2=x^{x^{x^{...}}}$$ And the trick to solve them was just ...
1
vote
1answer
102 views

Convergent rate for a class of functions

I want to find a class of function $h(\tau)$ which makes the following limitation converges to zero. $\lim_{t\rightarrow \infty }\int_{0}^{t}e^{-M\left( t-\tau \right) }h\left( \tau ...
3
votes
1answer
308 views

Laurent series for an infinite sum of functions

I have an infinite sum of analytic functions that is guaranteed to converge for every $x$, except for $x=0$: \begin{equation} g(x) = \sum_{n=1}^\infty f_n (x) \end{equation} I want to expand the ...
3
votes
1answer
501 views

Nested radicals

Let $S$ be the set of functions $f:\mathbb{R}\to \mathbb{R}$ such that $\sqrt{f(1)+\sqrt{f(2)+\sqrt{f(3)+\dots}}}$ converges. A function $q(x)$ dominates $p(x)$ if there exist an m such that $q(x)\gt ...
1
vote
2answers
192 views

Limit of function of a set of intervals labeled i_n in [0,1]

Suppose we divide the the interval $[0,1]$ into $t$ equal intervals labeled $i_1$ upto $i_t$, then we make a function $f(t,x)$ that returns $1$ if $x$ is in $i_n$ and $n$ is odd, and $0$ if $n$ is ...