Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange
Join us in building a kind, collaborative learning community via our updated Code of Conduct.

Questions tagged [analysis]

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

0
votes
0answers
12 views

Calculating the influence of changing volumes and values on a single average

I'll try and explain this well, and hopefully someone can help. Looking for a mathematically sound methodology (obviously). I've got a sample population with a population size of $x$ and an average ...
0
votes
3answers
56 views

If $x'(t)$ is bounded for $x\geq0$ show that $\lim_{t\to\infty}{x(t)}=0$

Assume that $x(t)$ is nonnegative for $t\geq0$ and $\int_0^\infty{x(t)\;dt}<\infty$. If $x'(t)$ is bounded for $x\geq0$, then show that $\lim_{t\to\infty}{x(t)}=0$. I started with contradiction ...
1
vote
1answer
18 views

Uniformly convergence of a sequence defined in compact set.

Let $(f_{n})$ be a sequence of functions $[-1,1] \to \mathbb{R}$ defined by: $f_{0}(t)=0$ and $f_{n+1}(t)=f_{n}(t)+\frac{1}{2}(t^{2}-f_{n}^{2}(t))$. Show that $(f_{n})$ converges uniformly. ...
7
votes
3answers
101 views

For a continuous function $f$ satisfying $f(f(x))=x$ has exactly one fixed point

Let $f \colon [ 0, 1] \to [0, 1]$ be a continuous map such that $$ f\big( f(x) \big) = x \ \mbox{ for each } x \in [0, 1], $$ and $$ f(x) \neq x \ \mbox{ for at least one } x \in [0, 1], $$ ...
0
votes
1answer
21 views

Theorem stating the connection of limits of sequences and functions.

Following is the theorem I have been given for regarding the connection between limits of sequences and functions, in order to help decide whether a limit for a function does exist or not: Theorem: ...
0
votes
1answer
30 views

Prove that $\Vert f(x) -f(y)\Vert\geq (1-k)\Vert x-y\Vert,\;\text{and}\;\Vert f'(x)h\Vert\geq (1-k)\Vert h\Vert,\;\forall\,x,y,h\in\Bbb{R^n}$

Good day all! I'm preparing for a Graduate exam, so I need to solve this problem. Let $f:\Bbb{R}^n\to\Bbb{R}^n$ be a function of class $C^{1}$. We suppose that there exists $k\in ]0,1[$ such that $$\...
1
vote
2answers
44 views

Is $\dfrac{a^2+b^2}{6}\leq \dfrac{a^2+b^2}{3}+\dfrac{ab}{3}\leq \dfrac{a^2+b^2}{2}$ for any $a,b\in\mathbb R$?

Is $\dfrac{a^2+b^2}{6}\leq \dfrac{a^2+b^2}{3}+\dfrac{ab}{3}\leq \dfrac{a^2+b^2}{2}$ for any $a,b\in\mathbb R$ ? For $a$ and $b$ are both positive or both negative,I proved this. But I am not able to ...
1
vote
1answer
23 views

Suppose $f\in L_2(\mathbb R)$ and $f$ is a continuous function on $\mathbb R$. Is $\displaystyle\sum_{k\in\mathbb{Z}}|f(k)|^2<\infty $?

Suppose $f\in L_2(\mathbb R)$ and $f$ is a continuous function on $\mathbb R$. Is $\displaystyle\sum_{k\in\mathbb{Z}}|f(k)|^2<\infty $ ? I am trying to prove the relation $\displaystyle\sum_{k\in\...
0
votes
1answer
29 views

Let $f\in L_2(\mathbb R)$ be a function. Is $\displaystyle\sum_{k\in\mathbb{Z}}|f(k)|^2<\infty $?

Let $f\in L_2(\mathbb R)$ be a function. Is $\displaystyle\sum_{k\in\mathbb{Z}}|f(k)|^2<\infty $ ? I am trying to prove the relation $\displaystyle\sum_{k\in\mathbb{Z}}|f(k)|^2<\int_{\mathbb R}|...
0
votes
1answer
22 views

If $f(x)=e^{-x-y}-x-y-1,\forall\;(x,y)\in\Bbb{R}^2$, then $\exists\;\alpha>0$ and $\varphi:\,]-\alpha,\alpha[\to\Bbb{R}$ of class $\Bbb{C}^{\infty}$

Let $$f:\Bbb{R}^2\to\Bbb{R}$$ $$x\mapsto f(x)=e^{-x-y}-x-y-1,\forall\;(x,y)\in\Bbb{R}^2$$ $i.$ Show that there exists $\alpha>0$ and $\varphi:\,]-\alpha,\alpha[\to\Bbb{R}$ of class $\Bbb{C}^{\...
0
votes
1answer
30 views

If $f(x)=\langle x,a\rangle e^{-\langle x, x\rangle} \ \forall x, a \in \Bbb{R}^{n}$, then prove that $f$ is of class $C^{1}$ and compute $f'(x)$

Let $f:\Bbb{R}^n\to\Bbb{R}$ be a function defined by $$f(x) = \langle x, a \rangle e^{-\langle x, x\rangle}$$ $\forall x \in \Bbb{R}^n, \ a \in \Bbb{R}^{n}$. Questions: I want to Prove that $f$ ...
0
votes
0answers
13 views

Showing boundedness below by sobolev norm

Suppose I know that $h(f)$ is bounded by $h(f) \geq (1 -2ab)\Vert f' \Vert^2_{L^2} - \frac{4a}{b} \Vert f \Vert^2_{L^2}$ with $a,b > 0$ and $f \in H^1$. Then for any $b \leq \frac{1}{2a}$ we can ...
1
vote
1answer
44 views

Correct approximation estimate

Let $x$ be the exact value of "something" and $y$ be an approximate value of $x$. If a question asks to find such a $y$ correct to the nearest thousandth of a unit what should be the correct ...
0
votes
2answers
29 views

How is the set $S = \{ z \in \mathbb{C} \: |z − 2| \lt 3|z|\}$ simply connected?

I have sketched the set $ S = \{ z \in \mathbb{C} \: |z − 2| \lt 3|z|\} $ and to me it appears to be the complement set to the closed ball $B(\frac14, \frac34)$. As a result I can think can come up ...
0
votes
1answer
15 views

Convergence of power series by comparison test with geometric series.

Let $\sum_{n \in \mathbb{N}} a_nz^n$ be a power series, such that $\sum_{n \in \mathbb{N}} a_nz_0^n < \infty$. Thus $a_nz_0^n \rightarrow 0$. (As I understand) the author caps $\left|\sum_{n \in \...
0
votes
1answer
25 views

Prove that the function can be continued into a larger domain

Prove that the function $f(z)=\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}\frac{z^n}{n}$ can be continued into a larger domain by means of the series $$\ln2-\frac{1-z}{2}-\frac{(1-z)^2}{2\cdot 2^2}-\...
0
votes
0answers
14 views

Composition of absolutely continuous function

if I have a function $x:[0,\infty)\rightarrow H$ (H is an hilbert space) that is absolutely continuous (AC), why should the function $\|x(t)-y\|^2$ be AC, for $y\in H$? Yet again, why should $\|x_{\...
5
votes
2answers
88 views

Why $\sum_{k=1}^m \left(\cos{\frac{2\pi k}{2m+1}}\right)^{(2m+1)^2}$ converges to $\sum_{k=1}^\infty e^{-2\pi^2k^2}$?

While I was considering random walk on $S^1$, I needed to compute the following. $$\lim_{m\rightarrow\infty}\sum_{k=1}^m \left(\cos{\frac{2\pi k}{2m+1}}\right)^{(2m+1)^2}$$ I guessed it should ...
1
vote
1answer
46 views

Structure theorem for closed set in R

Any Closed set in $ \mathbb R$ can be given by intersection of countable open set. My attempt: Let $E$ be closed set in $\mathbb R$. Case 1 : $E=\emptyset$. $$E=\bigcap_{n\in \mathbb N} \left(0,\...
1
vote
0answers
43 views

Trouble with integration

I am having trouble calculating this integral. Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ defined by \begin{equation} f(x,y) := \begin{cases} 1 & \text{if} \ x\geq0, \ x\leq y \leq 1+...
4
votes
3answers
39 views

MVT inequality problem

I just sat a real analysis exam and this was a question in it that I couldn't answer... Prove that $\left|e^\frac{-x^2}{2t}-e^\frac{-y^2}{2t}\right| \leq \frac{|x-y|}{t}$ for $x,y \in [-1,1] ,t>0$ ...
1
vote
0answers
23 views

Reason for choosing a particular analytical continuation of the factorial

From this answer I know the choice of continuous extention $\ \Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}dt\ $ is not unqiue. But is that particular extension the unique best choice in some sense? E.g....
0
votes
2answers
33 views

Let $a_n \neq 0 \;\; \forall n \in \mathbb{N}$ and $\exists L = \lim |\frac{a_{n+1}}{a_n}|$. Show that, if $L<1$, then $\lim a_n = 0$ [duplicate]

Let $a_n \neq 0 \;\; \forall n \in \mathbb{N}$ and $\exists L = \lim |\frac{a_{n+1}}{a_n}|$. Show that, if $L<1$, then $\lim a_n = 0$ Please verify if my attempt makes sense and how could I fix it....
0
votes
1answer
45 views

On the Definition of the Derivative

How can I show that if a function $f$ from an open interval of the real numbers to an euclidean space is differentiable at a point $x_0$ in its domain, that is, the limit $\lim_{\epsilon \to 0;\ \...
2
votes
2answers
68 views

Lagrange Theorem with $\mathbb{tan}(x)$

I take $\mathbb{tan}(x)$ in $[30°, 45°]$ and I want to find $f'(c)$. The hypothesis are satisfied. I compute: $$f'(c)=\frac{\mathbb{tan}(45°)-\mathbb{tan}(30°)}{45°-30°}=\frac{1-0.577..}{15°}\simeq0....
2
votes
0answers
25 views

Extending laws for Riemann integral to Riemann-Stieltjes integral

I was reading Terence Tao's notes on Analysis. He says Theorem 13(g) cannot be extended from Riemann integral to Riemann-Stieltjes integral: Most (but not all) of the remaining theory from Week 9 ...
0
votes
1answer
44 views

Bounded operators versus bounded sequences

Let $X$ be a Banach space, $Y$ normed, $A_n\in B(X,Y)$. Prove that $(\|A_n\|)$ is bounded if and only if for all $x\in X$ and for all $f\in Y^*$: $(|f(A_n(x))|)$ is bounded. Any hints on that? $A_n$ ...
1
vote
1answer
58 views

Proof by induction for $f^{n+1}(x)=\frac{x}{\sqrt{1+(n+1)x^2}}$

Look at the following function f: $\mathbb{R} \to \mathbb{R}: x \mapsto \frac{x}{\sqrt{1+x^2}}.$ Show with the complete induction that the recursive ( given by $f^1:=f$ and $f^{n+1}:=f\circ f^n$) ...
1
vote
1answer
23 views

Sign change for a continuous Lipschitz function

Suppose I have a Lipschitz function $f$ defined on $[a,b]$. Suppose there is a measure zero set on which $f$ is equal to $0$. Can the following happen? For any $\epsilon > 0$, if $f(x) = 0$ for ...
1
vote
1answer
23 views

Inclusion of limit point in Collection set

Let $F$ be collection of set in $R^n$ and let $S=\cup_{A\in F}A$ and $T=\cap_{A\in F}A$. Then prove or disprove following facts. 1) If x is limit point of T,then x is limit point of each of $A\in ...
1
vote
2answers
69 views

Let $f:\Bbb{R}^n\to \Bbb{R} $ be differentiable. Prove that $f$ is linear and show that $f(0)=0$

Let $f:\Bbb{R}^n\to \Bbb{R} $ be differentiable such that $$f(\lambda x)=\lambda f(x),\;\forall\;\lambda\in\Bbb{R},\;\forall\;x\in\Bbb{R}^n.$$ Prove that $f(0)=0.$ Prove that $f$ is linear. Here's ...
0
votes
2answers
34 views

The domain of a continuous real function must be connected?

The domain of a continuous real function must be connected? For instance, the function $$ f(x) = \begin{cases} \ \ \, 1 & \text{if $x \in (1,2)$} \\ -1& \text{if $x \in (2,3)$} \end{cases} $$ ...
3
votes
1answer
60 views

$\forall x\in [0,1]$, if $f(t)f(t-x)$ is integrable, then $f(t)$ is integrable?

The original problem (from here problem 8) is: Let $\alpha>\frac{1}{2}$ be a real number. Prove that it is impossible to find a real function $f$ such that $$f(x)=1+\alpha\int_{x}^1f(t)f(t-x)dt$$ ...
1
vote
1answer
33 views

Differential of scalar function is a 1-Form. What for differential of vector-valued functions?

Let $U\subset\mathbb{R}^n$ be open and $f\colon U\to\mathbb{R}$ a continuously differentiable function. The the total differential $df$ is a differential $1$-form, i.e. $df(p)$ is a cotangent vector $...
1
vote
1answer
37 views

Is it possible to define a euclidean structure on infinite dimensional vector space

It is known that for every infinite dimensional vector space $V$, there exists a Hamel basis $B$, such that each $v\in V$ can be repsentated as $$v=\sum_{i=1}^{n(v)}a_ie_i$$ with $e_i \in B$. Now my ...
1
vote
2answers
36 views

Prove that $\Phi$ is differentiable on $E$ and find $\Phi'(x)$

Good day all, here is a problem on differentiabilty on $\Bbb{R}^n$ that I find difficult. Let $E$ be a normed-vector space over $\Bbb{R}$ such that $\dim E<\infty$ and $F$ also, be a normed-...
1
vote
4answers
51 views

$n$-th partial sum and convergence $\sum_{k=1}^{\infty}\frac{1}{k(k+2)}$

Having trouble with finding the $n$-th partial sum, and seeing if it diverges or not of, $$\sum_{k=1}^{\infty}\frac{1}{k(k+2)}$$ I know that it is a telescoping series, and I can solve $\...
1
vote
3answers
39 views

Determine whether a sequence is bounded above

If I want to determine whether a sequence, ${a_n}$, is bounded above $\forall n \in \Bbb{N} $, is it enough to find a sequence that is larger than $a_n$, and show that it converges and is therefore ...
1
vote
0answers
37 views

Pole in limit expression

I would like to compute $$ f(E,q):=\lim_{\epsilon\to 0}\text{Im}\left[ \frac{(E^2-q^2)^2-i\epsilon}{(E^2-q^2)^4+\epsilon^2}\left( E^2 + \alpha(m+in) \right) \right] $$ with an infinitesimal ...
0
votes
1answer
20 views

Let $f:E\to \Bbb{R},\;p\mapsto f(p)=\int_{0}^{1}p^3{(t)}dt$. Then $f$ is differentiable and we can compute $f'(u)$

I have been faced with this problem. Let $$f:E\to \Bbb{R}$$ $$p\mapsto f(p)=\int_{0}^{1}p^3{(t)}dt.$$ I want to prove that $f$ is differentiable and also compute $f'(u).$ $E=R_n[x]$ is provided with ...
0
votes
0answers
32 views

The relation between Frechet derivative and differentiation in $\mathbb{R}^2$

In the book of Complex Made Simple by Ullrich, at page 4, it is given that However, if we are considering $\mathbb{C} = \mathbb{R}^2$, then $Df$ is a $2\times 2$ matrix, and not a complex number, but ...
2
votes
0answers
36 views

How to Recursively Formulate a Speed vs Quality Problem

I'm studying the trade-off between speed and quality. We need to finish a fixed number of jobs, let's say $n$ jobs. We receive a reward for finishing a job and this reward is decreasing and convex in ...
1
vote
1answer
35 views

Difference of convex functions over a convex function

If $f(x)$ is a decreasing and convex in $x$, $f(x)>0$, $x\geq 0$, and $$g(x)=\frac{f(x+y)-f(x+z)}{f(x)}$$ for any $z\geq y \geq 0$. Can we make some assumptions or add some constraints to $f(x)$ so ...
1
vote
1answer
34 views

Let $f:L(E)\to L(E),$ $u\mapsto f(u)=u^3=u\circ u\circ u.$ Then, $f$ is differentiable and compute $f'(u)$

Let $$f:L(E)\to L(E)$$ $$u\mapsto f(u)=u^3=u\circ u\circ u.$$ I'm interested in proving that $f$ is differentiable and computing $f'(u).$ Here is what I've done: $$f(u+h)-f(u)=(u+h)^3-u^3$$ $$=3 u^2 ...
0
votes
1answer
15 views

Prove decreasing convex function has decreasing differences

Let $f: \mathbb R \rightarrow \mathbb R$ be a decreasing convex function. For any $t>0$, I want to show that for all $x_{1} \leq x_{2}$ we have: $$f(x_{1}) - f(x_{1}+t) \geq f(x_{2} ) - f(x_{2}+t)...
0
votes
1answer
26 views

Problem at proof of Cartan's theorem about the relation between metric and curvature in do Carmo's book

I'm reading DoCarmo's book, Riemannian Geometry and i don't understand something. At page 157, Cartan's theorem. My question is, why can we take a Jacobi field $J$ in such a way that $J(0)=0$ and $J(...
4
votes
2answers
87 views

inverse Fourier transform of $\frac{x^3y}{(x^2+y^2)^2}$

Let $f(x,y)=\frac{x^3y}{(x^2+y^2)^2}$. What is the inverse Fourier transform of $f$? Since $f$ is neither a $L^1$-function nor a Schwartz-function I am looking for an inverse Fourier transform in the ...
-1
votes
1answer
17 views

Vitali covering and find disjoint family balls

There is exersice in Folland's book which show that every family $(A_{n})_{n=0}^{\infty}$in $\sigma$-Algebra,there exists a disjoint famly $(B_{n})_{n=0}^{\infty}$ such that $U_{n=0}^{\infty}A_{n}=U_{...
4
votes
1answer
91 views

Inequality involving $f,f',f''$

Let $f:\mathbb{R}\to [0,+\infty)$ is strictly convex ($f''(x)\geq 0$) and is $\mathrm{C}^2$ with $\min f(x) = f(0) = 0$. If $f''(0) > 0$ then by Taylor's expansion around $0$, obviously $$ \...
-4
votes
0answers
43 views

What is smallest natural number $m$ is $a_m =0$ [closed]

$$a_m=\lim_{n\rightarrow\infty}\sum_{k=0}^n \frac{1}{\sqrt{n^2 +k^m}}.$$ Then what is smallest natural number $m$ for which that sequence is $0$? I can easily find when $m=1,2$ but I'm stuck on $m=...