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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The question about the support of Fourier transform of $|f|^p$

Suppose $f$ is a smooth function with $\mathbb{supp}{(\mathcal{F}{f})} \subset B(0,1)$. In addition, assume $f$ is positive. We can observe that $|f|^2$ has some nice property : ...
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0answers
12 views

How to find the domain of the support function

First the definition of the support function of set $S$: Let $S$ be a nonempty convex set. The support function $h$ of $S$ is the real-valued function defined by $$h(x)=\sup_{s\in S}\langle ...
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1answer
26 views

Show that $\|f\|^{2}$ attains a minimum value on the interior of $B$

I am looking for any help, hints, or suggestions in how to go about this problem from a previous qualifying exam. We are given a smooth mapping $f: U \rightarrow \mathbb{R}^{n}$ whose differential ...
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2answers
43 views

History of convolution

Let $f, g\in L^{1}(\mathbb R),$ we may define the convolution of $f$ and $g$ as follows: $f\ast g(x)= \int_{\mathbb R} f(x-y)g(y) dy, (x\in \mathbb R).$ It is well known that it can be defined on ...
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1answer
33 views

What is the meaning of $\lim_{\Delta(P) \to 0} F(P) = L$ for partitions

Let $[a,b]$ be an interval, and denote by $\mathcal P[a,b]$ the family of all partitions of $[a,b]$, i.e. sets $P = \{ a = x_0 < x_1 < \ldots < x_n = b \}$. For some $P \in \mathcal P[a,b]$ ...
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1answer
24 views

Is the integral with respect to increasing continuous functions the limit of integrals with respect to $C^1$ functions?

if $\xi$ is continuous increasing can we find $\xi^n\in C^1$ such that $$\int_0^t f(u)\, d\xi = \lim_n\int_0^t f(u)\, d\xi^n$$ for every continuous $f$?
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1answer
27 views

Unit radial vector field

Lee's book defines the unit radial vector field in normal coordinates as $$ \partial_r:= \frac{x^i}{r(x)} \partial_i$$ and $r(x):=\sqrt{\sum_i (x^i)^2}$ Now this is a unit vector field iff ...
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1answer
24 views

Complement of the union of finitely many , mutually disjoint , closed balls in $\mathbb R^n$ , for $n>1$ , path connected? [on hold]

Let $D_1,D_2,...,D_k$ be finitely many , mutually disjoint , closed balls in $\mathbb R^n$ , where $ n \ge 2$ . Then is $\mathbb R^n \setminus \cup_{i=1}^k D_i$ path connected ?
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2answers
42 views

set of all accumulation points of A is countable

How do I approach this question stating: Construct a compact set A of R such that the set of all accumulation points of A is countable. F compact means closed and bounded. Let $x_k$ element in it. ...
4
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1answer
37 views

Prove that $\{ar+b:a,b\in Z\}$, where $r$ is an irrational number is dense in $R$ by using the following lemma

Prove that $\{ar+b:a,b\in Z\}$, where $r$ is an irrational number is dense in $R$ by using the following lemma: If $x$ is irrational, there are infinitely many rational numbers $h/k$ with $k\gt 0$ ...
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2answers
45 views

Is this sequence is dense?

Define $S _m, _n = $ n th smallest square number which is bigger or same than $10^ {m-1}$and smaller than $10^m$ Then is the sequence $ \frac{S_m,_n} {10^m}$ is dense in (0,1) or arbitary ...
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3answers
35 views

Convergence of improper integral?

Consider an improper integral such that: $$I = \int_0^{+\infty} \frac{f(x)}{x}dx.$$ If $\int_0^{+\infty}f(x)dx < + \infty$, Can we conclude that the integral I converges? Thanks for any answer or ...
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1answer
49 views

Does $\sum_{n=1}^{\infty}a_k^2<\infty$ imply $\sup_{n\in \mathbb{N}}na_n<\infty$? [on hold]

Let $a_k>0, k\in \mathbb{N}$. Suppose that $\sum_{n=1}^{\infty}a_n^2<\infty$. Does it implies $\sup_{n\in \mathbb{N}}na_n<\infty$? Thanks.
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0answers
42 views

Application of Arzela Ascoli theorem?

Let $\Omega$ an open bounded domain in $\mathbb R^n$ and consider $u_k: \Omega \to \mathbb R , k=1,2,...$ a sequence of functions. Suppose that $|u_k(x)| \leq 1$ for all $x \in \Omega$. Suppose that ...
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1answer
34 views

Continuously differentiable operator

I consider an operator $A:H^1_0\to H^1_0$ defined by $$Au(t)=\int_0^1 G(t,s) f(s,u(s))ds$$ where $$ G(t,s)=\begin{cases} t(1-s), &t\leq s\\s(1-t), &s\leq t.\end{cases}$$ I want to know what ...
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0answers
36 views

“Triangle” inequality for integrals

I have got two questions: 1) Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ be any continuous function. Let $\Gamma$ be a piecewise smooth curve on $\mathbb{R}^2$. The following inequality holds: ...
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0answers
33 views

Fundamental Lemma of the Calculus of Variations with higher derivatives

The fundamental lemma of the calculus of variations is often presented as: If $M(x) \in C[a,b]$ such that $\int_{a}^{b}{M(x)\eta(x)} = 0 ~~\forall\eta\in C^1[a,b],\eta(a)=\eta(b)=0$, then $M(x)=0$ for ...
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1answer
36 views

Application of inverse function theorem?

I am not completely sure if this a direct consequence of the inverse function theorem. Assume that we have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ that we can write in terms of ...
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1answer
13 views

Jacobian of the Kelvin transform

The Kelvin transform of the circle in $\mathbb{R}^n$ with centre $\textbf{u}$ and radius $r$ is defined by $$\textbf{y} \mapsto \textbf{u} + r^2|\textbf{y} - \textbf{u}|^{-2}(\textbf{y}-\textbf{u}).$$ ...
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0answers
35 views

Matrix of $f^p:\Lambda^p(E)\rightarrow \Lambda^p(E)$

Let $\Lambda^p (E)$ be the set of $p$-covariant exterior(alternative) tensors on linear space $E$ over field $K$ (dim$E=n$ and $ 0\leq p\leq n $ , $\Lambda^0E:=K$). We define linear map ...
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2answers
41 views

If $\langle f'(x) \cdot v , v \rangle > 0$ then $f$ is injective

Question: Let $f: U \to \mathbb R^m$ differentiable at the convex set $U \subseteq \mathbb R^m$. If $$\langle f'(x) \cdot v , v \rangle > 0 , \,\,\, \forall\,\, x \in U, v \neq 0 \in \mathbb ...
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1answer
23 views

First Eigenfunction of Simple Equation

Consider the interval $[-a,a]$ and the following problem: $$\phi'' + \lambda\phi=0$$ $$ \phi(\pm a) = 0. $$ The obvious sequence of orthogonal eigenfunctions seems to be $\sin(\frac{\pi n}{a}x)$ ...
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0answers
39 views

Explicit expression of a sequence [on hold]

In my research, I am stack in finding the explicit expression of the sequence $(x_k)_{k \geq 1}$ defined as the following: Given a positive integer $r.$ The sequence is such that $x_k=\frac{1}{rk+1}$ ...
3
votes
1answer
30 views

Show the following are not connected in $\mathbb{R}^n$

Is the interior, boundary and closure of a connected set in $\mathbb{R}^n$ connected? I know the interior is not connected we can show it by a counterexample but I am not quite sure for the ...
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0answers
78 views

Distance between sets

Let $K \subset K_1 \subset U \subset \Bbb R^2$, such that $K$ and $K_1$ are compact sets, with $K \subset \mathring {K_1}$, and $U = \mathring U \subsetneq \Bbb R ^2$. If $w \in \partial K_1$ such ...
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0answers
16 views

Prove that $\frac{\langle f^2,g\rangle_{L^2}}{\left\|f\right\|_{L^2}^2}\ge-\left\|g\right\|_{L^\infty}$ for $f\in L^2$ and $g\in L^\infty$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $f\in L^2(\Omega)$ and $g\in L^\infty(\Omega)$. How can we show, that $$\frac{\langle ...
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1answer
34 views

Existence of such a function

I am supposed to construct a function $f \in C_c^1((-\frac{3R}{4},\frac{3R}{4}))$ such that $f|_{(-\frac{R}{2},\frac{R}{2})}=1$ and $|f'(x)| \le \frac{4}{R}$ for almost all $x \in (-R,R)?$ I ...
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0answers
48 views

Preparations to finals, validation needed

I have an exam in a few days from now and I'm very nervous. I tried to tackle this one with all I got, but I'm not sure if I'm correct. If anyone can direct me, and tell me if and where I'm doing ...
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2answers
19 views

limit points of subset of real numbers

Let $$A=\{ \frac{\sqrt{m} -\sqrt{n}}{\sqrt{m}+\sqrt{n}} | m,n\in \Bbb{N} \}$$ I think that we must find sequence of $A$ and find limit of sequence,let $a_m =\frac{\sqrt{ k ^2 m^2} -\sqrt{ ...
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1answer
20 views

Inverse of the complex exponential function, considered as a multivariable function

Consider the complex exponential function $g: \mathbb{C} \to \mathbb{C}, z \mapsto e^z$. When identifying $\mathbb{C}$ with $\mathbb{R}^2$ in the natural way, then $g$ can be considered as a ...
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0answers
48 views

Abstract definition of a differential operator

In Kolar, Michor, and Slovak, a differential operator is said to be a rule transforming sections of a fibred manifold $Y \to M$ into sections of another fibred manifold $Y' \to M'$. Is this is a ...
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0answers
62 views

Show that if $f$ is differentiable as to function $x\mapsto ||x||$ with $x\in R$,then $f'(0)=0$ [on hold]

Let $f\in C^{\infty}(Ω)$ for some open set $Ω \subset R^n$ that contains $0$. Show that if $f$ is differentiable as to function $x\mapsto ||x||$ with $x\in R$,then $f'(0)=0$. I found this problem in a ...
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0answers
32 views

How to get the potential and the gradient of this function?

How to calculate the potential $P_3:\mathbb{R}^3\rightarrow\mathbb{R}$ with the $\nabla P_3=f_3$ of the function $f_3:\mathbb{R}^3\rightarrow\mathbb{R}^3$ with $\large ...
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0answers
30 views

Functions linearly independent and linearly independent gradients? [on hold]

Let $F_1,...,F_n: \mathbb{R}^n \rightarrow \mathbb{R}$ be a set of $C^{1}$ functions. Is it true that they are linearly independent on a joint level set $\Omega:= \{ p \in \mathbb{R}^n; ...
3
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1answer
20 views

finding polynomials to approximate a multivariable function

Let $U := B_1(0) \subseteq \mathbb{R}^2$, with $B_1(0) := \{(x, y) \in \mathbb{R}^2,\space \|(x, y)\| _1 < 1\}$. Now consider the function: $$g: U \to \mathbb{R}^2, (x, y) \mapsto ...
2
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1answer
18 views

Convolution with standard mollifier

Let $\Omega \subset \mathbb{R}$ open and $f \in L^p(\Omega).$ Now, we define $$\eta(x):=\chi_{[-1,1]}(x) e^{\frac{-1}{1-x^2}}.$$ Then we define $$\eta_h(x):=\frac{1}{h} \eta( \frac{x}{h}).$$ This ...
3
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1answer
29 views

diffeomorphism inbetween two subsets of $\mathbb{R}^2$

Consider the function $$f: \mathbb{R}^2 \to \mathbb{R}^2, \space\space f(x, y) := \pmatrix{x(1-y) \cr x y}$$ Now first, why is $f$ continuously differentiable? Then, I want to prove that $f$ ...
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2answers
23 views

second derivative of the composition of two multivariable functions

Let $U \subset \mathbb{R}^n$ be open, and let $\gamma: \mathbb{R} \to U$ and $f: U \to \mathbb{R}$ be to functions that are differentiable at least twice. I want to show that $\frac{d^2}{dt^2}(f ...
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0answers
14 views

Hesse matrix is negatively semidefinite if a function has a local maximum

Let $U \subset \mathbb{R}^n$ be open, and let $f: U \to \mathbb{R}$ be at least twice continuously differentiable. Also, we assume $f$ has a local maximum $a \in U$. I now want to show that the Hesse ...
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7answers
2k views

After switching a lamp on and off infinitely many times in one minute, is it on or off? [duplicate]

So we have a lamp. It's switched on. let's represent its state of being switched on with associating it with $1$ and being off with $-1$. after half a minute passes, you turn it off, after another ...
2
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1answer
44 views

An open interval as a union of closed intervals

For $a<b, a,b\in\Bbb R$ $$(a,b)=\bigcup_{0<\delta<(b-a)/2} I_{\delta} \quad I_{\delta}:=[a+\delta,b-\delta] $$ Clearly the RHS is an (uncountable) infinite sum of closed intervals. I ...
2
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1answer
40 views

Prove that $Df(p)=f(p)T$ where $T(q)=\int_{0}^1q$

Let $E=\mathcal{C}[0,1]$ provide with $\|\cdot\|_\infty$ norm. Let $f:E\to \mathbb{R}$, given by $f(p)=e^{\int_0^1 p}$. I need to prove that $f$ is differentiable. My approach: Let ...
2
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0answers
59 views

A few questions on the later chapters in Principles of Mathematical Analysis by Walter Rudin (3rd Edition)

I am currently reading Principles of Mathematical Analysis by Walter Rudin (3rd Edition). I am enjoying the book and it's terseness, which isn't an issue for me. What I do have a problem with is that ...
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1answer
53 views

Particular $f \in C_c^1$

is there a way to construct a function $f \in C_c^1( B(0,\frac{3R}{4}))$ such that $f|_{B(0,\frac{R}{2})}=1, \quad f|_{B(0,\frac{3R}{4})^C}=0$ and $0\le f\le 1$ everywhere such that $|Df(x)| \le ...
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1answer
75 views

Limit of measurable functions is measurable?

Suppose $(\Omega, \cal F)$ is a measurable space and $(X, \mathcal B_X)$ is a topological space with its Borel sigma algebra. If $f_n: \Omega \to X$ is a sequence of $(\cal F , B$$_X)$-measurable ...
3
votes
1answer
35 views

Estimate for a weak solution to a PDE

Let $f \in L^2(B_R(0))$ and let $u \in W^{1,2}(B_R(0))$ be a weak solution of the equation $$Lu = - \sum_{i,j=1}^{n} D_i(a_{ij}D_ju)+ \sum_{i=1}^{n} b_i D_i u + cu =f.$$ There are constants $0 \le ...
1
vote
1answer
41 views

Finding Laurent series of a function $f(z)=\frac{3z^2-6z+1}{(z-1)(3z-1)}$ [on hold]

How do i transform this function into Laurent series $$f(z)=\frac{3z^2-6z+1}{(z-1)(3z-1)}$$ where $ \frac{1}{3} < |z| < 1 $.
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0answers
17 views

combine analysis and artificial intelligence

I'm sorry if I ask this question at the wrong place, but I don't know a better one. I am a Master's student and I am really interested in analysis, but I also want to get into AI. Does anyone know a ...
0
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0answers
42 views

Is the field of real algebraic numbers a complete field?

Let $\mathbb{R}_{alg}$ be the field of real algebraic numbers. Is there exist a metric $|\cdot|$ for which $(\mathbb{R}_{alg}, |\cdot|)$ is a complete field (i.e. any Cauchy sequence converges in ...
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1answer
26 views

Prob. 3, Sec. 4.2 in Erwin Kreyszig's Functional Analysis: How to show that $\lim\sup$ is sublinear?

Let's consider the real space $\ell^\infty$ of all bounded sequences of real numbers. Let $p \colon \ell^\infty \to \mathbb{R}$ be defined by $$p(x) \colon= \lim\sup_{n \to \infty} \xi_n \ \mbox{ for ...