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).

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2
votes
0answers
30 views

How to prove that $\|x+y\|_p<\|x\|_p+\|y\|_p$?

Let $l_p=\{(x_n)\in\mathbb{R}^\mathbb{N}: \sum_{n=0}^\infty |x_n|^p<\infty\}$ and consider the following norm in $l_p$: $$\|x\|_p=\left(\sum_{n=0}^\infty|x_n|^p\right)^{1/p}$$ for ...
3
votes
1answer
25 views

Hessian-Matrix positive definite $\iff$ $a$ local minimum?

It is commonly known that if $f$ is twice differentiable, $\nabla f(a) = 0$ and $H_f(a)$ positive definite, $a$ is a local minimum. So, in short: $H_f(a)$ positive definite $ \implies $ $a$ local ...
1
vote
0answers
15 views

No compact embedding

I have the two spaces $W_0^{1,p}$ and $L^{p^*}_1=\{ u~\text{measurable}, \int_{\Omega} (|x| u(x)|)^{p^*} dx<\infty\}$ equiped with the norm $||u||_{L^{p^*}_1}^{p^*}=\int_{\Omega} (|x||u(x)|)^{p^*} ...
0
votes
0answers
17 views

Second derivative and Hessian-Matrix

Suppose $f$ is twice differentiable. Why is $Df(a)[v] + \frac12 D^2f(a)[v,a] = \frac 12 \langle H_f(a) v, v \rangle $ ? $Df(a)[v]$ is the multidimensional derivative of $f$ at point $a$ in ...
4
votes
2answers
27 views

On a condition when bounded sets in $\mathbb R^n$ is convex ?

Is it true that a bounded set in $\mathbb R^n$ , $n>1$ , is convex iff every straight line through an arbitrary interior point of the set intersects the boundary of the set in exactly two points ? ...
2
votes
1answer
15 views

Connected-ness of the boundary of convex sets in $\mathbb R^n$ , $n>1$ , under additional assumptions of the convex set being compact or bounded

Is the boundary of every compact convex set in $\mathbb R^n$ , ($n>1 $ ) connected ? is it path connected ? What if we assume only that the convex set is bounded , is the boundary connected ( and ...
0
votes
0answers
10 views

About convergence in norm of the Fourier Transform

Duoandikoetxea's Fourier Analysis, on page 59 (Corollary 3.7) says that: \begin{equation} lim_{R \rightarrow \infty}||S_{R}f - f||_{p} = 0 \end{equation} for $1<p<\infty$, where $S_{R}f$ is ...
2
votes
1answer
17 views

Fourier transform of $L^1$ function square summable?

It is known that for a $L^1$ function $f: \mathbb{R} \rightarrow \mathbb{C}$ the Fourier transform vanishes at infinity and is continuous. Does this even mean that $(\hat{f}(n))_{n \in \mathbb{Z}}$ is ...
1
vote
1answer
24 views

Problem with the Definition of contractible set

I have this definition of contractible set: we say that $A\subset X$ is contractible in $X$ if there exists a continuous function $\eta:[0,1]\times A\rightarrow X$ such that $\eta(0,x)=x, \forall ...
1
vote
2answers
36 views

Finding a choice for Epsilon for open/closed set proofs

I'm studying the proofs for open/close sets by using the following definition: I'm having problems to understand the proofs. The proofs sounds pretty straightforward: just choose a value for ...
1
vote
0answers
16 views

Fourier Analysis Help - $\mathcal L^2$

Let $\{e_k | e_k(x)= e^{ikx}/\sqrt{2\pi\,}\}$ be the orthonormal basis in $\mathcal L^2$ per. I first have to use this basis define two infinite dimensional orthogonal subspaces of $\mathcal L^2$ per. ...
0
votes
1answer
22 views

If the boundary of a convex set in $\mathbb R^n$ ($n>1$) is connected , is it necessarily also path-connected ?

If the boundary of a convex set in $\mathbb R^n$ ( where $n>1$) is connected , is it necessarily also path-connected ?
5
votes
2answers
110 views

Infinite integrals$\int_0^{ + \infty } {\frac{1}{{\left( {x + 1} \right)\left( {{x^n} + 1} \right)}}dx} .$

How to calculate $$\int_0^{ + \infty } {\frac{1}{{\left( {x + 1} \right)\left( {{x^n} + 1} \right)}}dx} .$$
2
votes
1answer
41 views

Let $a_n>0$ for $n \geq 1$ and let series: $\sum_{n=1}^{\infty}a_n$ diverge. Let $S_n=a_1+a_2+…+a_n > 1$ for $n \geq 1$

Prove that the series: $$\sum_{n=1}^{\infty}\frac{a_{n+1}}{S_n \ln S_n}$$ diverges and the series : $$\sum_{n=1}^{\infty}\frac{a_{n}}{S_n \ln^2 S_n}$$ converges. (Using the famous criteria I ...
1
vote
0answers
12 views

FTC and Points of differentiability

Just having a little bit of difficulty understanding the solution to the following problem: Let $f$ be defined as follows, $$f(t) = \left\{\begin{array}{ll}0, & \quad t<0, \\ t, & \quad ...
4
votes
1answer
22 views

Help understanding an inequality on Rudin's construction of the Lebesgue measure

I am having trouble understanding an inequality in Theorem 2.20 from "Real and Complex Analysis." Rudin states that if $f\in\operatorname{C}_c(\mathbb{R}^k)$ , $f$ is real, $W$ is an open k-cell ...
1
vote
0answers
23 views

calculate complex integral $\int_{0}^ {\pi} \frac{(x \sin x)dx }{1-2a \cos x+a^2}, a>0$

I don't know how to calculate this complex integral: $$\int_{0}^ {\pi} \frac{(x \sin x)dx }{1-2a \cos x+a^2}, a>0$$
2
votes
2answers
27 views

How to find the length between 2 points given a pivot

I am not great at math but I have done the previous steps to my problem. This is the last step where I need to find out the distance between C,D. I am writing a program that will output this ...
0
votes
0answers
22 views

Help in finding a paper on nonlinear Schrodinger equations [on hold]

I'm looking for the following paper: Authors: Baillon, Jean-Bernard; Cazenave, Thierry; Figueira, Mario Title : Equation de Schrodinger non linéaire. (C. R. Acad. Sci. Paris Ser. A-B 284 ...
0
votes
1answer
20 views

Residue Calculus - Showing that the quotient of polynomials have integral $0$ in a simple closed contour in a special case.

I'm having difficulty understanding the solution to the following problem. In the solution below, I can't understand why since $b_m\neq 0$, the quotient of these polynomials is represented by a ...
2
votes
1answer
41 views

For which values of $x$ is the following series convergent: $\sum_0^\infty \frac{1}{n^x}\arctan\Bigl(\bigl(\frac{x-4}{x-1}\bigr)^n\Bigr)$

For which values of $x$ is the following series convergent? $$\sum_{n=1}^{\infty} \frac{1}{n^x}\arctan\Biggl(\biggl(\frac{x-4}{x-1}\biggr)^n\Biggr)$$
2
votes
1answer
44 views

Proving that a trigonometric sum is in $L^2$

how can I use Perseval's identity to prove that $$f(x)=\sum_{k=1}^\infty \frac{\sin(kx)}{1+k}$$ is in $L^2(0,\pi)$? thank you!
-1
votes
1answer
26 views

Antiholomorphic function

Let f be an antiholomorphic function in C. $z_0 \in C - C(0,1). $ Show that $\frac{1}{2 \pi i}\oint \frac {f(z)}{z-z_0} = \begin{cases}f(0) &\text{for } |z_0| < 1\\f(0) - f(\frac{1}{z_0}) ...
2
votes
1answer
34 views

Convergence of complex power series $z^{n!}$ at boundary

I'm revising for an exam at the moment and I'm struggling with part of a question. I'm asked to find the radius of convergence of the series $\sum_{n=0}^{\infty }z^{n!}$ and then find where it ...
0
votes
1answer
32 views

With the aid of series, show that if $f(z)=\frac{\operatorname{cos}z}{z^2-(\pi/2)^2}$, then $f$ is an entire function.

Prove that if $$f(z)= \begin{cases} \frac{\operatorname{cos}z}{z^2-(\pi/2)^2}, & \text{when} \; z\neq \pm \pi/2, \\ -\frac{1}{\pi}, & \text{when} \; z=\pm \pi/2, \end{cases} $$ then $f$ is ...
0
votes
0answers
27 views

Prove that the function $\xi\in R \mapsto {e^{i\cdot \xi\cdot λ}-1\over i\cdot \xi}-λ$ is $C^{\infty}$

Prove that the following function is $C^\infty$ in the point $\xi=0$: $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ Any ideas how to prove this? I am trying to think ...
-3
votes
1answer
22 views

Show, directly from the definition, that the following series is convergent. [on hold]

Using the definition of a convergent series, how do you show that the series $\sum_{n=1}^{\infty} (\frac{-2}3)^n $ converges.
0
votes
0answers
18 views

Pointwise convergence of Bernstein polynomials for piecewise continuous functions

I know that $B_nf \to f$ uniformly if $f:[0,1] \to \mathbb R$ is continuous. But can anybody explain to me, why $B_nf \to f$ pointwise if $f:[0,1]\to \mathbb R$ is only piecewise continuous (but ...
0
votes
0answers
34 views

Exercise to Heat equation

I am dealing with the following exercise on a generalized heat equation Let $U$ be a open bounded subset of $\mathbb{R}^n$ and $T \in (0, \infty).$ Now, let $u$ be a solution to $$(\partial_t - ...
0
votes
2answers
24 views

Recurrence sequences with two initial condition: how do I calculate the limit?

I've done some exercises with recurrence sequences with one initial condition. So, now that I'm attempting one exercise with two initial conditions I'm confused. Could you show me what to do? Let ...
0
votes
1answer
38 views

Bring a proof for the fundamental theorem of calculus.

If $f\in \mathscr{R}$ on $[a,b]$ and if there is a differentiable function $F$ on $[a,b]$ such that $F'=f$, then $$\int_a^b f(x)\ \ d(x)=F(b)-F(a)$$
0
votes
1answer
43 views

Why does the functional have a local minimum at $0$?

Definition: Let $J: A \to \mathbb{R}$ be a functional , where $A \subset V$ and $(V, ||\cdot||)$ a linear space with norm. Let $y_0 \in A$ and $h \in V$ such that $y_0+ \epsilon h \in A $ for ...
2
votes
1answer
71 views

A necessary condition to $F'(x)=f(x)$ for a continuous function $f$

Theorem: Consider , $$F(x)=\int_a^xf(t)\,dt$$ If the function $f:[a,b]\to \mathbb R$ is continuous then $F(x)$ is differentiable and $F'(x)=f(x).$ I know that the continuity condition ...
0
votes
1answer
16 views

Let $A$ be a convex set in $\mathbb R^n $ , where $n>1$ ; then is it true that $\bar A \setminus A^{\circ}$ i.e. the boundary of $A$ is connected ?

Let $A$ be a convex set in $\mathbb R^n $ , where $n>1$ ; then is it true that $\bar A \setminus A^{\circ}$ i.e. the boundary of $A$ is connected ? if yes , then is it also path connected ?
-3
votes
0answers
30 views

How can the elements $a_1, a_2, a_3\ldots, a_n$ be distinct in Theorem 2.13 of Rudin?

In Theorem 2.13 of Rudin, how could the elements $a_1,a_2,\ldots, a_n$ be distinct like he says they can? $A$ is a countable set (or just a set) and, therefore, all elements must be distinct. Perhaps ...
0
votes
1answer
63 views

How to prove this limit of derivative to zero [on hold]

This is a test question in real analysis and I need help to prove it. Let $f:\mathbb{R}\to\mathbb{R} \in C^{\infty}$ be periodic of period $1$ and nonnegative. Show that $$ ...
0
votes
2answers
44 views

If $f$ is a convex function in $(a,b)$ then $|f|$ is convex function in $(a,b)$.

TRUE or FALSE: If $f$ is a convex function in $(a,b)$ then $|f|$ is convex function in $(a,b)$. My Proof: Since $f$ is convex function so, $f(tx+(1-t)y)\le tf(x)+(1-t)f(y)$ , for all ...
5
votes
1answer
79 views

Prove $ \lim\limits_{n\to\infty}\int_0^1 f(x)g(nx)\,dx=\int_0^1 f(x)\,dx\int_0^1 g(x) \, dx $ [duplicate]

Let $f$ and $g$ be a real valued continuous functions on $\mathbb{R}$ such that $f(x+1)=f(x)$ and $g(x+1)=g(x)$ for all $x\in \mathbb{R}$. Prove that $$ \lim_{n\to\infty}\int_0^1 ...
0
votes
2answers
42 views

Calculate: $\lim_{n \to \infty} \frac{2^{\sqrt{(\ln n)^2 + \ln n^2}}}{n^2+1}$ and $\lim_{n \to \infty} \frac{10^{\sqrt{(\ln n)^2 + \ln n^2}}}{n^2+1}$

I have to evaluate the following limits (which are similar). However, I don't know how to evaluate them. Could you give me a hand? $$\lim_{n \to \infty} \frac{2^{\sqrt{(\ln n)^2 + \ln ...
2
votes
1answer
46 views

Verify solution to ODE

I am given the ODE $$\left(f''(x)+\frac{f'(x)}{x} \right) \left(1+f'(x)^2 \right) = f'(x)^2f''(x)$$ and I already know that the solution to this ODE is given by $$f(x)= c \cdot arcosh \left( ...
0
votes
1answer
15 views

Find limit inferior and limit superior of $[1+\sin n]$ and $n - [\sqrt n]$

I have to find the limit inferior and limit superior of the following sequences: $$[1+\sin n]$$ and $$n - [\sqrt n].$$ I have done similar exercises before, but never with the integer part function ...
2
votes
1answer
27 views

derivative of $f(r\cos\phi,r\sin\phi)=r^a\cos(a\phi)$

Let $$ f(r\cos\phi,r\sin\phi)=r^a\cos(a\phi) $$for some $r\in(0,\sigma)\subset\mathbb R$ and $\phi\in (0,\rho)\subset(0;2\pi]$. How do you calculate $Df=(\partial_1 f,\partial_2 f)$ ? I thought ...
0
votes
1answer
15 views

is it possible to decompose nonperiodic sinusoidal signal?

Using Fourier series we can decompose any any signal into it's elementary signals but condition is that signal should be periodic and sinusoidal one. Now, is it possible to decompose nonperiodic ...
-2
votes
0answers
28 views

A problem of Taylor series [on hold]

I need a step-by-step solution to the following problem. Sorry, I have nothing done because I don't know how to approach the problem. Determine $\alpha \in \mathbb{R}$ such that $$ \arctan^2 (2x) - ...
0
votes
0answers
5 views

Bipartite graph matching partitioning using clustering algorithm

I am identifying information from a document using bipartite graph model now I have to extract that information which are closely matched. hence I want to use clustering technique to group the data ...
0
votes
2answers
28 views

The Fourier transform of functions with compact support is differentiable.

1) How can I prove that if $f(x)$ is a continous function with compact support (let's say $f(x)=0$ $\forall x\in B(0,R)^c$), then its Fourier transform $\hat{f}(\xi)$ is differentiable? 2) Is there ...
4
votes
0answers
43 views

How to prove this integral [duplicate]

Let $f(x)$ be a real valued continuous function on $[0,1]$. Show that $$ \lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1) $$
1
vote
2answers
154 views

Any idea on this problem $\lim \limits_{x\to\infty}f(x)=0$ [on hold]

This is a question in real analysis. I think it needs ODE to prove, but not sure. Any idea is welcome. Let $f$ be a real valued continuous function on $[0,\infty]$ such that $$ \lim ...
2
votes
1answer
33 views

Relation between runge domain and polynomial convexity

Are these concepts the same? Just to state the definitions Definition 1 A domain $\Omega \in \mathbb{C}^n$ is a Runge domain if every function $f \in H(\Omega)$ can be approximated, uniformly on ...
1
vote
1answer
27 views

Limit vs interior definition of continuity

Suppose I have two topological spaces $X$ and $Y$ whose topologies are defined by interior operators $\text{int}_X$ and $\text{int}_Y$ respectively, as well as a function $f$ with domain $I$ (for ...