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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63 views

Measures whose projections are absolutely continuous

Let $\mu$ be a finite Borel measure on the plane. Does there exist a characterization of the property that almost all (wrt rotations) projections of $\mu$ to lines on the plane are absolutely ...
2
votes
4answers
855 views

Continuous function over the reals which maps closed sets to a not closed set

My main question here may be notation but here is the actual problem: "Give an example of a function $f : \mathbb{R} \rightarrow \mathbb{R}$ which is continuous and a set $E \subset \mathbb{R}$ ...
3
votes
0answers
38 views

Borel lemma on rationality

I am looking for an enlightening proof of the following fact: Let $F(t)\in \mathbb{Z}[[t]]$ and suppose $S$ is a finite set of places on $\mathbb Q$ containing $\infty$. If for every $v\in S$ $F(t)$ ...
5
votes
0answers
103 views

Existence of smooth extension of a function defined on a closed interval

Suppose $ f: [0,1] \rightarrow \mathbb{R}$ is a function such that derivatives of all orders exist ( at the end points of the interval the appropriate one-sided derivatives exist) and are continuous $ ...
0
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1answer
130 views

Is $C_{0}^{\infty}(\mathbb{R})$ a subset of $L^{2}(\mathbb{R})$

I came across this while reading something but I cannot seem to prove it. Any ideas? Prove that $C_{0}^{\infty}(\mathbb{R}) \subseteq L^{2}(\mathbb{R})$ This is what I tried: Let $f \in ...
3
votes
4answers
89 views

proof by induction : $n^n \ge 2^{n-1} n!$

I am trying to show that $$\begin{matrix} n^n \ge 2^{n-1} n! & \text{(1)} \end{matrix}$$ I tried to solve it for n=n+1 $$(n+1)^{n+1}=(n+1)^n(n+1) \ge n^n(n+1) \ge 2^{n-1}n!(n+1)= ...
1
vote
1answer
36 views

Help with limit $\displaystyle\lim_{t\to 0^{+ }}\sup_{x\in[0, \infty)} |e^{-t^2-2tx}f(x+t)-f(x)|=0$..

can anyone help me showing $$\displaystyle\lim_{t\to 0^{+ }}\sup_{x\in[0, \infty)} |e^{-t^2-2tx}f(x+t)-f(x)|=0,$$ where $\displaystyle f\in C_0([0, \infty))=\{f\in C([0, \infty)): \lim_{x\to \infty} ...
4
votes
1answer
169 views

Properties of special rectangle (measure)

Let $I$ be a special rectangle in $\mathbb{R}^n$, and denote $\lambda(A)$ the measure of $A$. Prove that the following conditions are equivalent: a) $\lambda(I)=0$ b) $I^{\circ}=\emptyset$ (i.e., ...
5
votes
3answers
222 views

Can rearranging a SEQUENCE (not a series) change the limit?

I have this question on a homework assignment. I sat down with two other people for a long time and we derived the alternating harmonic series example, but I don't think that's valid because the ...
2
votes
2answers
199 views

A non-exponentially bounded analytic function?

A function $f:\mathbb R\to\mathbb R$ is said to be exponentially bounded if there is an $n$ such that for sufficiently large $x\in\mathbb R$, $\exp(\exp(\cdots \exp(x)))>f(x)$ (where the $\exp$ is ...
0
votes
2answers
107 views

What is the intersection of a countable set of intervals $(n, \infty)$?

If I have a countable set of intervals $\{ A_n \}^{\infty}_{n=1}$ where $A_n = (n, \infty)$, and take the intersection $ \cap_{n=1}^{\infty} A_n $ my assumption is that this set would be ...
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vote
3answers
844 views

What does directional derivative zeros imply when directional vector is not zero?

This question might sound stupid but I want to confirm an answer from it. I saw somewhere online that it means that when the directional derivative of function $f$ along the none zero vector $v$ at ...
1
vote
1answer
92 views

Piecewise $C^1$ function is element of $W^{1,\infty}$

Hey I'm confused about the following (apparantly) fact: Let $u:[a,b]\to\mathbb{R}$ a piecewise $C^1$ function, i.e. there exists $a=t_1<t_2<\cdots < t_n = b$ such that $u|_{[t_i,t_{i+1}]} ...
0
votes
1answer
62 views

Inequality, triangle, Law of cosines, integer

Prove $$|a^2+1-2a\cos{\theta}|^{\frac{1}{n}}\ge| a^{\frac{2}{n}}+1-2a^{\frac{1}{n}}\cos{\frac{\theta}{n}}|$$ where $a>0$ , $0<\theta<\pi$ and $n\ge2$ and $n\in N^+$.
1
vote
1answer
118 views

Proof of the completion of a metric space using cantors diagonal argument and showing a diagonal sequence is cauchy

I am studying applied functional analysis out of Applied Analysis by John Hunter. In chp. 1 of the text it gives a proof for the completion of a metric space. I am having trouble with understanding ...
1
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1answer
158 views

Ratio of convex functions with dominating derivatives is convex?

Let $f,g:\mathbb [0,\infty)\rightarrow (0,\infty)$ satisfy $f^{(n)}(x)\geq g^{(n)}(x)>0$ for all $n=0,1,2,\ldots$ and $x\in [0,\infty)$. In particular, $f\geq g> 0$ are increasing and convex ...
0
votes
1answer
104 views

Bound of $\det$ of positive definite matrices

I need to know that if the following holds for complex vectors $x=[a\cdot A \mid b\cdot B]u$, and $y= [A \mid B]u$ $$\det(I+d \frac{yy^*}{rI})\leq\det(I+\frac{xx^*}{rI})\leq \det(I+c ...
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vote
0answers
32 views

On the controllability function (minimising a functional)

Consider a system of ODEs $$\dot{x}(t)=f(x(t))+g(x(t))u(t),$$ where $f:\mathbb{R}^n\to\mathbb{R}^n$ and $g:\mathbb{R}^n\to\mathbb{R}^{n\times m}$ are smooth. Let $L:\mathbb{R}^n\times ...
2
votes
0answers
84 views

Meaning of quasiperiodicity in classical KAM

I'm learning about the classical KAM theorem, and I can't quite infer precisely what the term "quasi-periodic solution" means in the theorem's statement. I'm reading the following introductory note: ...
0
votes
1answer
61 views

$L^2$ dot product on surface of a sphere

If you have function $M,N:\mathbb{R}^3\rightarrow \mathbb{R}^3$ that are $M,N \in C^{\infty}$. Can we infer from this that on every surface of a sphere $B(0,R)$ this is a dot product: $$\langle f, ...
0
votes
1answer
62 views

Prove that $\bar A=\bar B \iff d(x,A)=d(x,B)$ for every $x\in\mathbb{R}^n$.

Prove that $\bar A = \bar B \iff d(x,A)=d(x,B)$ for every $x\in\mathbb{R}^n$. There is a lemma which states that $\exists x_0\in \bar A$ such that $d(x,A)=d(x,x_0)$. So for the forward direction (to ...
0
votes
1answer
276 views

If $F_1$ and $F_2$ are disjoint closed sets then there exist disjoint open sets $G_1$ and $G_2$.

Use an Urysohn function to give a solution of this problem: Prove that if $F_1$ and $F_2$ are disjoint closed sets in $\mathbb{R}^n$, then there exist disjoint open sets $G_1$ and $G_2$ such that ...
0
votes
2answers
77 views

If $U\subseteq X$ is open in $X$, then $U\cap Y$ is open in $Y$

If $X$ is a metric space, and $Y$ is a metric subspace of $X$ the show that if $U\subseteq X$ is open in $X$, then $U\cap Y$ is open in $Y$. So we have two cases: if $U\cap Y=\varnothing$ and $U\cap ...
3
votes
0answers
36 views

existence theorem of eliptic equation

Consider $\Omega \subset R^n$ a bounded and open set with $\partial \Omega$ smooth . Consider the problem: $$ - \Delta u + au = f \text{ in } \Omega $$ $$ u = 0 \text{ in } \partial \Omega $$ ...
2
votes
1answer
64 views

Analysis, Density of Rational Numbers

Suppose p/q and k/l are rational numbers with abs(p/q - k/l) < 1/ql. Prove p/q = k/l. Similarly, let p/q be a fixed rational number and suppose k/l is a rational number with 0 < abs(p/q - k/l) ...
0
votes
1answer
61 views

Method of steepest descent

Generally, the method of steepest descent describes the asymptotic behavior of integrals of the form $$\int_{-\infty}^\infty h_t(x)\exp(-tg(x)) \,dx$$ in terms of $t$. As long as $h_t(x)$ is ...
2
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0answers
67 views

Finding the limit of a quotient of sums (of functions)

I encountered the following basic problem in my research. Given $\alpha_{i} \geq \beta_{i} > 0$, $\lim_{t \rightarrow \infty} r_{i} (t) = 0$, where $r_{i}(t)$ is a continuous function on ...
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vote
0answers
32 views

Is it possible to extend a given function to be real analytic if its analytic wave front set consists of finitely many covectors at each point?

Specifically, suppose you have a function $f:\mathbb{R}^{2} \to \mathbb{R}$ and you assume that its analytic wave front set $\mathrm{WF}_{A}(f)$ contains at most finitely many covectors $\{(x, ...
0
votes
1answer
70 views

Derivative with respect to the Frenet frame

Given a regular smooth curve $\alpha: I \to \mathbb{R}^3$, we have that the Frenet Frame $\vec{t},\vec{n},\vec{b}$ is an orthonormal basis of $\mathbb{R}^3$ at each point $s \in I$, and hence we can ...
2
votes
1answer
60 views

A postive, decreasing function $f$ such that $\lim_{n \rightarrow \infty} \ln f(n) / \ln n$ neither diverges nor converges to $-\infty$.

Is there a positive, decreasing function $f$ such that $$ \lim_{n \rightarrow \infty} \frac{\ln f(n)}{\ln n} $$ neither converges nor diverges to $-\infty$? One candidate is a function like $$ f(n) ...
2
votes
0answers
112 views

Are non-constant, analytic functions topologically conjugate to $z\rightarrow z^m$ when $f'(0)=0$?

Given a non-constant, analytic function $f$ of the form $f(z) = z^m + a_{m+1} z^{m+1} + a_{m+2} z^{m+2} + \cdots$ one can show that $f$ can be written in the form $f(z) = f_1(z)^m$ where $f_1$ is ...
2
votes
0answers
43 views

sequence of p-harmonic functions

Consider $\Omega$ a bounded open set in $R^n$ with $ \partial \Omega$ smooth and $u_n \in C^{1,\alpha}(\Omega)$ a bounded sequence in $C^{1,\alpha}(\Omega)$. Suppose that each $u_n$ is p - harmonic ...
0
votes
1answer
63 views

What about the convergence of these series?

$$ \sum_{n=1}^{\infty} \arctan \frac{1}{2n + 1} $$ $$ \sum_{n=1}^{\infty} (\frac{\pi}{2} - \arctan ( \log n ) ) $$ $$ \sum_{n=1}^{\infty} \sin ( n \pi + \frac{1}{\log n } ) $$
1
vote
1answer
150 views

Fractional calculus

I have this exercise : "Consider the Cauchy problem's : $$ ^C D^{\alpha}y(t)=f(t,y(t),y'(t)), t\in [0,T] ....(1) $$ $$ y(0)=y_0, y'(0)=y_1 .... (2) $$ Where ...
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vote
0answers
62 views

function defined as an integral involving Bessel functions

i need to analyze a function of the form $$F(x,y) = \int_0^{1} e^{-(1+s)\alpha x}\sinh((1-s)\beta y) I_0(\sqrt{(x^2-y^2)s}) ds $$ Where $I_0$ is the modified Bessel function. $x>y$ always. ...
1
vote
1answer
131 views

Lower bound of Fourier transform

We know the Fourier transform of the Gauss-function: $\displaystyle\int_{\xi\in\mathbb{R}^d}e^{-\pi\, C\,|\xi|^2}e^{2\pi i \xi\cdot X}d\xi=C^{-d/2}e^{-\, \pi\, |X|^2/2}$ for any $C>0$. Then ...
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vote
2answers
83 views

$\det (A^{-1})$ from eigenvalues of $A$

Suppose I have invertible square matrix $A$ in the complex field and I know all of its eigenvalues and they may be assumed to be non zero. Is there a way to write $\det(A)$ and $\det (A^{-1})$? PS. ...
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vote
2answers
116 views

Eigenvalues of $(A+B)^{-1}$

Suppose I know the eigenvalues of $A$ and $B$. Is there a way to write eigenvalues of the following? (1). $(A+B)$ (2). $(I+A)$ (3). $(I+A)^{-1}$ (4). $(A+B)^{-1}$ where $A, B$ are matrices in ...
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votes
0answers
39 views

How to solve $ \tfrac {1}{2 \pi } \sum_{m=-n}^{n} \left [ \int_{0}^{2 \pi} f(t) e^{-imt} dt f_m \right ]$

I'm trying to solve the following question: Let $f$ a complex function defined on $[0,2 \pi]$. Let $f_m:[a,b] → ℂ$ be the function with $f_m(x) = e^{imx}$. Solve $$ \tfrac {1}{2 \pi } ...
3
votes
1answer
546 views

Fourier transform of the indicator of the unit ball

What is the Fourier transform of the indicator of the unit ball in $\mathbb R^n$? I think it is known as one of special functions, so I would be happy to know which one.
2
votes
2answers
157 views
1
vote
1answer
91 views

Looking for an analytical expression of this horror-integral

I have given a function $$a(m,n,\mu,\nu,p):=\frac{2p+1}{2}\frac{(p-m-\mu)!}{(p+m+\mu)!}\int_{-1}^1 P^m_n(x)P^\mu_{\nu}(x)P^{m+\mu}_p(x)dx.$$ (of course all parameters are appropriate integers, so that ...
2
votes
2answers
192 views

Double sequence $z_{mn}$ Converges but it doesn't imply $z_{mn}$ is bounded

I have noticed an interesting thing in double sequence $z_{mn}$ and I can't see why such thing happens. Definition: Double sequence $z_{mn}$ is a mapping from $\mathbb{N}\times\mathbb{N}\rightarrow ...
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vote
2answers
342 views

If x and y are distinct real numbers prove there's a neighborhood P of x and neighborhood Q of Y such that $P\cap Q = \emptyset$

So here is the proof that I have: By definition, $\exists \varepsilon > 0$ so that $(x-\varepsilon, x+\varepsilon) \subset P$ and $(y-\varepsilon, y+\varepsilon) \subset Q$ Choose $\varepsilon = ...
2
votes
2answers
247 views

Clarification of sequential compactness theorem with example

A set of real numbers $S$ is said to be sequentially compact provided that every sequence $\{a_n\}$ in $S$ has a subsequence that converges to a point that belongs in $S$. For clarification, does ...
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1answer
214 views

Monotonic log det function?

I want to claim that the follwoing function is monotonically increasing in $d_j$. ...
0
votes
1answer
84 views

Bound of $\log \det$

I want to find a bound to the function $$R(d_i, ...
1
vote
1answer
30 views

How to show $\|Pf\|_{L^1(\mathbb T^n)}\leq \|f\|_{L^1(\mathbb T^n)}$:

I need some help with the following problem: Let $P:S(\mathbb R^n)\rightarrow C^\infty(\mathbb T^n)$ be the operator given by $f\mapsto Pf$ where, $$Pf(x)=\sum_{k\in\mathbb Z^n} f(x+k).$$ How can I ...
1
vote
4answers
100 views

If $x,y,a,b > 0$ and $\frac{x}{y}$ < $\frac{a}{b}$ than prove $\frac{x}{y}$ < $\frac{x+a}{y+b}$ < $\frac{a}{b}$

So I am not really sure where to start. I understand what I have to do for the proof but Im thinking that there is someway to use the fact $\frac{x}{y}$ < $\frac{a}{b}$ to deduce which variables ...
0
votes
1answer
128 views

The Topologies generated by Borel sets.

If I have a topological space $(X, \tau)$, then we can consider the Borel hierarchy on it. Now the sets from a Borel class itself could be taken as generating a topology. Is something known or what ...