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
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1answer
86 views

Why is $-\Delta+1$ an isomorphism between Sobolev space $W^{2,p}$ and $L^p\,$?

Consider the linear elliptic operator $L: W^{2,p}(\mathbb{R}^N)\to L^p(\mathbb{R}^N)$ defined by $Lu=-\Delta u + u$. Can we prove that $L$ is an isomorphism for all $p\geq 1$? It can be proved that it ...
2
votes
1answer
55 views

Product of Hölder and Sobolev functions

Here $C^{\kappa , \lambda} ( \overline{\Omega} ) = \left\{ h|_{\overline{\Omega}} :h \in C^{\kappa , \lambda} ( \mathbb{R}^{n} ) \text{ and } h \text{ has compact support} \right\}$ denotes $\kappa$ ...
2
votes
0answers
34 views

Proving this is Lebesgue integrable using radial functions

Show that $f:\Bbb R^n\to\Bbb R$, given by: $$ f(x) = \begin{cases}\sin\left(\frac{1}{\|x\|}\right)\|x\|^{-n-\arctan(\|x\|-1)} & x\not=0 \\ 0 & x=0 \\ \end{cases}$$ is Lebesgue ...
2
votes
0answers
46 views

Essential hypothesis of Fourier Inversion formula

Let $f\in L^{1}(\mathbb R)$ and we define its Fourier transform as follows: $\hat{f}(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i \xi\cdot x} dx, (\xi \in \mathbb R);$ and we define $f^{\vee}(x):=\hat{f}(-x)= ...
1
vote
1answer
47 views

Show uniform convergence of a series of complex function on every compact subset

Let $f:B(0,1)\rightarrow \mathbb{C}$ be an analytic function. Suppose $\sum^\infty_{n=0}f^{(n)}(0)$ converges absolutely. Show that there exists an entire function $g(z)$ such that $g(z)=f(z)$ for ...
0
votes
0answers
43 views

Convergence of eigenvectors/eigenvalues for infinite, non-negative, irreducible matrices with row sums less than $1$

I actually had two questions but decided to put them in separate posts for clarity. Suppose you have a matrix $M$ with infinitely many rows and columns, and a sequence of matrices $M_m$ with the ...
0
votes
1answer
83 views

Prove $\nabla f(\mathbf x) = \mathbf 0.$

Suppose that the function $f:\Bbb R^n \to \Bbb R $ has first-order partial derivatives and that the point $\mathbf x$ in $\Bbb R^n$ is a local minimizer for $f:\Bbb R^n \to \Bbb R $, meaning that ...
0
votes
1answer
29 views

showing that a sequence converges in the dual space of a normed vector space

Suppose that $S=\left\{s_\alpha: \alpha \in A\right\}$ is a set of points in a normed vector space $X$ such that $\overline{span}(S)=X$. If $\left\{f_n\right\}$ is a bounded sequence in $X^*$ and ...
0
votes
1answer
33 views

Is it possible to integrate this asymptotic expansion?

Suppose that some smooth function $f \in C^\infty\bigl(\mathbb R^n \times (0,+\infty)\bigr)$ possesses an asymptotic development $$ f(x,t) \sim t^{-\alpha} e^{ith(x)} \sum\limits_{k=0}^{+\infty} ...
0
votes
2answers
64 views

Understanding the Definition of $\int_\gamma f\ \overline{dz}$

Definitionally, we have that $$ \int_\gamma f\ \overline{dz} = \overline{\int_\gamma \overline{f}\ dz} $$ Now let $\int_\gamma f\ dz = w = x +yi$. Question 1: Is it not the case that $\int_\gamma ...
0
votes
1answer
18 views

Defining the Complex Line Integral w.r.t. $x$ and $y$

Ahlfors defines line integrals with respect $x$ as follows: $$ \int_\gamma f\ dx = {1 \over 2} \left( \int_\gamma f\ dz + \int_\gamma f\ \overline{dz} \right) $$ From this I take it as obvious that ...
0
votes
1answer
73 views

Find the rate of convergence?

Given is the iteration $x_{k+1}=\frac{1}{11}(1-\cos(x_{k}))$ with $x_{0}\in (-\frac{\pi }{2},\frac{\pi }{2})$ without $0$. Check if the sequence converges to $x^{*}=0$ and find its convergence rate. ...
1
vote
1answer
33 views

Fix point of a continuous function under some conditions [closed]

Prove that under each of the following conditions the continuous function $f:[a,b]\to\Bbb{R}$ has a fix point: $f([a,b])\subset [a,b]$ $f([a,b])\supset [a,b]$ When $f$ is bijective and ingective.
0
votes
2answers
34 views

Explore the convergence of a series with ln

How to explore the convergence of this series: $$ \sum_{n=1}^{\infty}\dfrac{1}{\ln^n(n+1)} $$ What would be better to use: De Lamber indication or feature comparison. And if comparison is a good ...
0
votes
1answer
30 views

Proving triangle inequality

A metric is usually said to be a function $d: S^2 \rightarrow\mathbb{R}$ (where $ S$ is some set), for which the following conditions hold: $$(1) \: d (x, y)=d (y, x)$$ $$(2)\: d(x, y)=0 ...
1
vote
1answer
27 views

Verify why this is not a metric $d(x,y)=\|x-y\|_p$ ( $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$).

$d(x,y)=\|x-y\|_p$ $p$ is prime and $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$, where $m, n$ are coprimes with $p$. This is not a metric because if $x=y=p^k\dfrac{m}{n}$, then $x-y=0=p^0\dfrac0n$. ...
1
vote
1answer
47 views

Proving that $\Gamma (x) = \int_{0}^{1} \left( \ln \left(\frac{1}{u} \right) \right)^{x-1} du$

I want to prove that $$ \Gamma (x) = \int_{0}^{1} \left( \ln \left(\frac{1}{u} \right) \right)^{x-1} du $$ I start with $$ \int_{0}^{1} \left( \ln \left(\frac{1}{u} \right) \right)^{x-1} du = $$ ...
0
votes
3answers
107 views

Determine whether this series converges or diverges:

I am taking a course in analysis and would just like to clarify my understanding with this motivating example? The series is: $$ \sum_{n=0}^\infty \frac{2^n - 1}{3^n + 1} $$ My textbook defines ...
0
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0answers
38 views

Approximating $C^2$ functions with compactly supported $C^2$ functions

Let $C^2$ be the space of twice-continuously differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$ and $C^2_K$ be the subset of functions in $C^2$ with compact support (that are zero outside ...
1
vote
1answer
129 views

Gaussian Quadrature -Deriving a Formula-

eThe following is an exercise in the problem section of the Gaussian Quadrature chapter. The theorem: Derive a formula of the form $$\int_{a}^{b} f(x)dx \approx w_0f(x_0) + w_1f(x_1) + w_2f'(x_2) ...
0
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0answers
35 views

Designing a state feedback law for a nonholonomic system

Consider the set \begin{equation*} A_r=\left\{(e_x,e_y,L)\in\mathbb{R}^3:e_x=e_y=0,L(t)=\sqrt{\dfrac{\mu}{p_0^3}}t,t\in\mathbb{R}_{\geq0}\right\} \end{equation*} I have been trying to design a state ...
2
votes
4answers
127 views

Show that $\,\,\, \lim_{n\to\infty} \sin\bigl(\pi\sqrt{n^2+1}\bigr)=0 $

Can anyone help me to solve this problem? Show that $$ \lim_{n\to\infty} \sin\bigl(\pi\sqrt{n^2+1}\bigr)=0 $$
1
vote
2answers
91 views

Show $\lim_{x \to0^+} \sum_{n=1}^{\infty} \frac{2x}{n^2x^2+1} = \pi$

Show that: $$\lim_{x \to0^+} \sum_{n=1}^{\infty} \frac{2x}{n^2x^2+1} = \pi$$
3
votes
0answers
68 views

Conservation of momentum for nonlinear Schrodinger equation

I am having trouble proving the following momentum conservation law. Given smooth compactly supported solution $u(x,t)\in \mathbb{C}$ where $(x,t)\in \mathbb{R}^n\times \mathbb{R}$ of $iu_t+\Delta u= ...
0
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0answers
26 views

Gaussian Quadrature Confusion

For the Gaussian Quadrature Thrm., why are we letting$ c_0$ and $c_1$ be $0$ in the example in the picture?
3
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0answers
56 views

The dual of the Annihilator

Let $X$ be a Banach space, and $I$ be a closed subspace. Then it's known that $(X/I)^*=I^{\perp}$. My question is what is the second dual of $X/I$? or what is the dual of $I^{\perp}$ ? If we know ...
10
votes
4answers
802 views

Why would you expand a square wave in a Fourier series?

The periodic square wave $$ f(x) = \cases{ 1 \text{ if } 0 \le x \le \pi \\ 0 \text { if } -\pi \le x < 0} $$ seems easy enough to work with. Why transform it into a series of sines and cosines? ...
5
votes
1answer
57 views

Reference for a Cantor set in the plane formed from series of roots of unity

This is a long shot, but I'm looking for a particular article that I once read, and I'm trying to find it again. It deals with a certain Cantor set in the plane. The set could be written as something ...
0
votes
0answers
39 views

finite-dimensional continuous vector bundle

Let $M$ a compact metric space and $\pi: F \rightarrow M$ a finite-dimensional continuos vector bundle over $M$, endowed with a continuous Riemannian metric. I was wondering if it will be true that: ...
0
votes
1answer
39 views

Convex Functions and derivatives

I am currently going back through all the "Challenge" questions in preparation for exams, and for this I do not know where or how to start, any hints would be appreciated.
2
votes
1answer
29 views

For $\mathbb{X}$ with order relation and field structure extended from $\mathbb{R}$, if it includes real line, then is it real line?

For a set $\mathbb{X}$ given order relation and field structure extended from those of $\mathbb{R}$, if $\mathbb{X} \supseteq \mathbb{R}$ then $\mathbb{X} = \mathbb{R}$ ? This question is derived ...
1
vote
0answers
25 views

Retanguloids and volumes

I am currently going back through all the "Challenge" questions in preparation for exams, and have come across this, any hints on how to approach?
1
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3answers
42 views

For a set $\mathbb{X}$ given order relation extended from that of $\mathbb{R}$, if $\mathbb{X} \supseteq \mathbb{R}$ then $\mathbb{X} = \mathbb{R}$?

For a set $\mathbb{X}$ given order relation extended from that of $\mathbb{R}$, if $\mathbb{X} \supseteq \mathbb{R}$ then $\mathbb{X} = \mathbb{R}$ ? Motivation for this question rose from an ...
1
vote
4answers
171 views

About $ \lim_{x\rightarrow 0}\frac {\sin x}{x} = 1$ [duplicate]

I do not understand how $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$ As if $$ x = 0, \frac{\sin (0)}{0} = \frac {0}{0} $$ So if someone could explain this I would appreciate it! Thanks!
1
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3answers
95 views

If Limit of function and derivative exist, then limit of derivative is 0 [duplicate]

Any hints for this question , My attempt; Say $f(x):0$$\rightarrow$$\mathbb{R}$ The by MVT, there exists a $c$$\in$$(0,\infty)$ , such that; $f'(c)=$$\frac{f(x)-f(0)}{x-0}$ but im not sure about this ...
1
vote
1answer
89 views

$v$ is Conjugate Harmonic to $u$ $\implies$ $f = u + iv$ is Analytic (Proof Verification from Ahlfors)

Hypothesis: Let $u$ and $v$ be two functions from $\mathbb{R}^2$ to $\mathbb{R}$ s.t. $$ \Delta u = {\partial^2 u \over \partial x^2} + {\partial^2 u \over \partial y^2} = 0 $$ and $$ \Delta v = ...
1
vote
0answers
215 views

The Cantor set is nowhere dense

I am considering the so called Cantor ternary set $C$ on $[0,1]$. I have just proved that its Lebesgue measure is $0$. To show that $C$ is nowhere dense, is it correct the following reasoning? By ...
6
votes
2answers
112 views

If $f^3=\rm id$ then it is identity function

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function such that $f^3(x)=f\circ f\circ f(x)=x$ for all $x$. How can I prove $f$ is the identity function?
1
vote
1answer
66 views

Distribution function and decreasing rearrangement

Let $(X,dx)$ a measure space and $f\in L^p(X,\mathbb{C})$; let's define its distribution function $$F(\alpha)=meas(\{x\in X||f(x)|>\alpha\})$$ and the decreasing rearrangement ...
1
vote
3answers
87 views

Non-decreasing functions and continuity

I have the following situation: $f\colon\mathbb{R}\to\mathbb{R}$ is a non-decreasing $g\colon\mathbb{R}\to\mathbb{R}$ is defined as $\ g(x):=\lim_{t\to x^+}f(t)$ I have proved that also $g$ is ...
1
vote
1answer
22 views

Connected Subset of Finite Topological Space

What I want to know is this: Am I correct in thinking that if we have a finite topological space (so no reals or anything here), that any connected subset contained in this space has only one ...
2
votes
4answers
105 views

A question about metrizability

In a lecture in Topology I had earlier this week, I was told (without proof) that not every topological space $(X,O)$ is metrizable, i.e, it is impossible to find some metric $d$ such that $O$ and ...
0
votes
1answer
49 views

Integrability of sums of Dirac deltas

this is my first post in the forum and I am an engineer, so I apologize in advance if my question is not clearly stated. Consider the function $f(x)=\sum_{i=1}^N a_i\delta(x-x_i)$ where ...
1
vote
1answer
30 views

Continuity of an application between function spaces.

I'm trying to prove the following statement... Let $f:[a,b] \times \mathbb{R} \to \mathbb{R}$ a bounded and continuous function, $t_{0} \in [a,b]$, $x_{0} \in \mathbb{R}$, $r>0$ and $$B= \{ x ...
3
votes
0answers
52 views

What extra assumption makes this transformation affine?

Let a vector space $V$ be given. Let $f:V\to V$ have the property that for all $x,y,a\in V$, $$ f(x+a)-f(y+a) = f(x) - f(y) \tag{$\star$} $$ Q1. I'd like to know how weak one can make additional ...
0
votes
3answers
50 views

Find min and maxima

Find local min and maxima of $ \sin(x^3)$ on the interval $]-2,2[$. I take the derivative and get: $$3x^2 \cdot \cos (x^3)$$ I set this equal to zero and get $$x^3 = \cos^{-1}(0)$$ $$ \Rightarrow ...
1
vote
0answers
35 views

Differentiation of Radon measures

Assume $\ (X,d)$ is a locally compact metric metric space and $\ \nu,\, \mu$ are Radon measures on $X$. Then, suppose that the following hypothesis hold: $\ w\in ...
0
votes
1answer
65 views

slope of curve represented by discrete points

I have data which are visualized in this chart: I need to compute slope of increasing / decreasing parts of the curve. I can't use any 2 points because of noise in data. Maybe numerical derivative ...
7
votes
2answers
312 views

Real roots of a polynomial

Let $p$ be an even degree polynomial with real coefficients such that the product of the constant term and the leading coefficient is negative. Show that $p$ has at least two real roots. Thanks!
3
votes
2answers
142 views

Smooth spectral decomposition of a matrix

Let $A : x \mapsto A(x)$ be a $C^\infty$ map from the half-plane $\left\{ (x_1,x_2,\cdots,x_n) \in \mathbb{R}^n,\ x_n>0\right\}$ to the space of symmetric matrices with real coefficients. Suppose ...