Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.

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0answers
13 views

Does the series $\sum_{i=0}^{\infty}\exp\{{-(m!)!}\}(D^m\phi)(0)$ converge for every $\phi \in C^\infty$?

Does the series $$\sum_{m=0}^{\infty}\exp\{{-(m!)!}\}(D^m\phi)(0)$$ converge for every $\phi \in C^\infty$? For analytic function $\phi$, we can show that the series converges by using ...
0
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0answers
9 views

Proof that quantum relative entropy is $\leq$ 0 using Klein's inequality for positive semi-definite operators

I was asked to prove that $S(\rho) \leq - {\rm Tr} \left[ \rho \log \tau \right] $ where $\rho, \tau$ are density operators on a finite dimensional complex inner product space and $S(\rho)$ is the von ...
0
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0answers
16 views

Expand trigonometric expression

I am supposed to expand this expression $${\frac {\sin \left( x \right) b \left( 4\,b\cos \left( x \right) + \sqrt {16\,{b}^{2}+1}+5 \right) }{4\,b\cos \left( x \right) +\sqrt {16 \,{b}^{2}+1}+1}} $$ ...
1
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1answer
14 views

$|\int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t}| dt \leq G(x,w), G\in L^{1} ? $

Put $\lambda >0,$ and we define, $$F_{\lambda}(x, w)= \int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t} dt;(x,w) \in \mathbb R^{2}$$ we note that, $F_{\lambda} \in ...
1
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0answers
29 views

Does a function that is twice weakly differentiable have a version that is classically differentiable?

I have been wondering about the idea of functions that are weakly differentiable. My intuition tells me that the weak derivative allows one to differentiate functions that either have a removable ...
0
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2answers
60 views

a question about sequence and series. prove $ \lim_{n \to \infty}( n\ln n)a_{n}=0$? [on hold]

Suppose $a_{n}>0$. $na_{n}$ is monotonic, and it approaches 0 as n approaches infinity. $\sum_{n=1}^{\infty} a_{n}$ is convergent. please prove $$ \lim_{n \to \infty} n\ln(n)\,a_{n}=0$$ I totally ...
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0answers
17 views

an inequality derived from conformal automorphisms of unit disk

Let $f$ be a holomorphic function on $D(0,1)$ such that $|f(z)|<1$ for all $z\in D(0,1)$. I have obtained $$ \frac{|f(0)|-|z|}{1+|f(0)||z|}\leq |f(z)|\leq \frac{|f(0)|+|z|}{1-|f(0)||z|}. $$ Is it ...
1
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1answer
25 views

Constructing a set that contains at most one point on vertical and horizontal

I'm not sure how to answer this question: Construct a set $A$, which is a subset of $[0, 1] \times [0, 1]$, such that $A$ contains at most one point on the horizontal and vertical lines, but ...
4
votes
7answers
139 views

Convergence in a metric space

Is it possible to define a metric on $\mathbb R$ such that $(1,0,1,0,...)$ converges on $(\mathbb R, d)$? I believe it is impossible. But how to show analytically? Any hint would be appreciated.
-1
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0answers
25 views

Two measures having the same moments [duplicate]

Let $\mu_{1}$ and $\mu_{2}$ be two finite Borel measures supported on $[0, 1]$. Suppose $\int_{\mathbb{R}}x^{k}\, d\mu_{1}(x) = \int_{\mathbb{R}}x^{k}\, d\mu_{2}(x)$ for all $k = 0, 1, 2, \ldots$. ...
0
votes
0answers
18 views

Property of linear growth function [on hold]

Prove: Suppose function $f(X)$ has linear growth and there exists an $X^*$ such that $f(X)$ is monotonic for $|X|>X^*$. Then it is possible to define a function $g$ such that $g$ a rotation of $f$ ...
2
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1answer
29 views

Minimum of a potential function

I'm looking for extremes (minimum) of $$V = \frac{\alpha}{|\vec{r}_1-\vec{r}_2|} + \beta (\vec{r}_1 + \vec{r}_2)\cdot \vec{e}_z$$ where $\vec{r}_i = ...
0
votes
1answer
44 views

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$ I want to use dominated convergence theorem obviously. However, not sure how to dominate it. ...
2
votes
1answer
37 views

Differentiation under the integral if and only if we have an $L^1$ dominator

Let $f(x)\in L^2(\mathbb{R})$ and define $$g(t) = \int_\mathbb{R} f^2(x)\exp(-tx^2)dx$$ for $t\geq0$. We want to show that $g(t)$ is continuously differentiable if and only if $xf(x)\in ...
0
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1answer
19 views

Proving or disproving continuity of a function

Consider a function $f:\mathbb{R}^{n}\times \mathbb{R}^{+} \rightarrow \mathbb{R}$, with the property that for a fixed vector $a:=(a_1,a_2,\cdots,a_n) \in \mathbb{R}^{n}$, there exist a finite ...
1
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1answer
30 views

The functions $\{f_n(x) = n\}$ are analytic and each miss the points $-2, -3$. But, they are not a normal family. So what am I missing. Thanks.

Here is a theorem of Montel: Let $\mathcal{F}$ be a family of analytic functions defined on a domain $\Omega$ . If there are two fixed complex numbers $a$ and $b$ that are omitted from the range of ...
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0answers
39 views

Prove that $f\ast g$ is defined a.e., integrable, and such that $∥f\ast g∥_1 ≤ ∥f∥_1 · ∥g∥_1$

Let $f,g : \mathbb{R} → \mathbb{R}$ be $L_1$-functions. Set $h(x) = \int_\mathbb{R}f (x − y)g(y) \, dm(y).$ Prove that $h(x)$ is defined a.e., $h ∈ L_1(\mathbb{R})$ and $∥h∥_1 ≤ ∥f∥_1 · ∥g∥_1.$ So I ...
1
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1answer
48 views

Let $f(z) = \sum_{n = 0}^\infty a_nz^n$ be the Taylor series around $0$. Prove that lim $a_n/a_{n+1} = z_0.$ [duplicate]

Let $f(z) = \sum_{n = 0}^\infty a_nz^n$ be the Taylor series around $0$ of a function which is analytic in $\mathbb{C}$ \ ${z_0}$, $z_0\neq 0$ and has only a simple pole at $z_0.$ Prove that $lim_{n ...
1
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0answers
15 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f_{r}(t-y)- f_{r}(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $ r\to \infty $?

Let $f\in \mathcal{S}(\mathbb R)$ with $\hat{f}$ has a compact support. For $r>0,$ put $f_{r}(x)= r^{-1}f(x/r), (x\in \mathbb R).$ We note that, $\int_{\mathbb R} |f_{r}(x)| dx = r^{-1} ...
1
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1answer
81 views

Spectral theory

I have absolutely no idea about Spectral theory and want to ask some fundamental questions. 1.) What does it mean that the resolvent of an operator is Hilbert-Schmidt? (Cause I saw a theorem that was ...
3
votes
2answers
123 views

Evaluating $\int^b_a \frac{dx}{x}$ from the definition of the integral

I know that $$\int^b_a \frac{dx}{x}=\ln b-\ln a$$ I'm trying to evaluate this integral using the same method used in this answer: http://math.stackexchange.com/a/873507/42912 My attempt $\int^b_a ...
2
votes
2answers
78 views

Convergence of the series $\sum\limits_{n\geqslant1}(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n})$

If $x >0$, consider the series $\sum\limits_{n=1}^\infty a_n$, where $$a_{n}=(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n}).$$ Is it convergent? This question is in my textbook. I don't know how to ...
1
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1answer
27 views

variation of a function over countable intervals

Let $f$ be a function of bounded variation on $[0,1]$. Let $\{[a_n,b_n]\}_{n=1}^\infty$ such that $(a_n,b_n)$ are pairwise disjoint and $\cup_{n=1}^\infty [a_n,b_n]=[0,1]$. (for example, $[1/2, 1], ...
0
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0answers
21 views

Checking if the Hessian is the derivative of the gradient

Suppose f: R^n --> R. I have a code that computes the gradient of f. I have another code that computes the Hessian of f times a vector. Now I want to check if they are correct. Specifically, I am ...
0
votes
0answers
16 views

Minimization of an evaluation under the weak* topology

I'm self studying (for fun) the book "Functional Analysis, Calculus of Variations and Optimal Control" (by Clarke), and I'd appreciate some feedback for my proposed solution for an exercise from the ...
1
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1answer
19 views

Lower sum/Riemann Integral

Let $c>0$ and $f(x)=x$ for $x \in[0,c]$ . Let $P$={$x_0,x_1,...x_n$} be a partition of $[0,c]$ where $x_i=\frac{i}{n}c$ for $i=0,1,2,...n$ How do you find $L(P,f)$ and $\lim_{n \to \infty} ...
3
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0answers
27 views

Let $(X,μ)$ be a measure space. Find a necessary and sufficient condition on $(X,μ)$ that $L_q(E) ⊂ L_p(E)$ for all $1 ≤ p < q ≤ ∞.$

Let $(X,μ)$ be a measure space. Find a necessary and sufficient condition on $(X,μ)$ that $L_q(E) ⊂ L_p(E)$ for all $1 ≤ p < q ≤ ∞.$ I want to say that the condition is that $E$ is finite. This ...
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0answers
26 views

Rapidly Decreasing Functions

Can someone explain the notion of a rapidly decreasing function? Namely, a function in the Schwartz space: $$\mathscr{S}(\mathbb{R}^n):= \{ f \in C^{\infty} (\mathbb{R}^n) : ||f||_{\alpha, \beta} ...
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0answers
15 views

intuitive fact of a class of functions defined in $R^n$

I am reading an article and i have the following situation: Let $u: R^n \rightarrow R$ a continuous function in $R^n$. Supoose that u is nonnegative and that for all $t \geq 0$ the set $L_t = \{ x ...
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votes
2answers
28 views

Determine all values of $p \in R$ such that the sequence is in $l^2$. $\left\{\frac{1}{\sqrt{k}(\ln k)^p}\right\}_{k=2}^{\infty}$ [on hold]

Determine all values of $p \in R$ such that the sequence is in $l^2$ $$\left\{\frac{1}{\sqrt{k}(\ln k)^p}\right\}_{k=2}^{\infty}$$
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0answers
21 views

Determine all values of $p \in R$ such that the sequence is in $l^2$. $\left\{\frac{k^p}{p^k}\right\}_{k=1}^{\infty}$ [on hold]

Determine all values of $p \in \mathbb{R}$ such that the sequence is in $l^2$ $$\left\{\frac{k^p}{p^k}\right\}_{k=1}^{\infty}$$ I'm so lost!
0
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0answers
36 views

Proving pointwise convergence of Fourier series [on hold]

I'm trying to prove the pointwise convergence of a $2\pi$-periodic function $f$ to its Fourier expansion. The proof on my lecture notes stops at this formula: $$f(t)-P_{N,f}=\frac 1 {2 \pi} ...
1
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1answer
23 views

If $p$ and $q$ are positive real numbers, show that $\sum_{k=2}^\infty(-1)^k\frac{(lnk)^p}{k^q}$ converges [on hold]

If $p$ and $q$ are positive real numbers, show that $$\sum_{k=2}^\infty(-1)^k\frac{(ln k)^p}{k^q}$$ converges
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1answer
22 views

Test for absolute convergence $\sum_{k=1}^\infty \frac{(-1)^{k+1}k^k}{(k+1)^k}$

Test for absolute and conditional convergence. $$\sum_{k=1}^\infty \frac{(-1)^{k+1}k^k}{(k+1)^k}$$ $$\lim_{k\to\infty}|a_n| = \lim_{k\to\infty} \frac{k^k}{(k+1)^k}$$ I'm stuck on what to do next.
2
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0answers
35 views

Show that $\int_E f (x, y) dx$ is differentiable with respect to $y$ and $\frac{d}{dy}\int_E f(x,y)dx=\int_E \frac{d}{dy}f(x,y)dx.$

Assume that $f = f(x,y)$ is a function defined on $E × (a,b).$ For each fixed $y ∈ (a,b),$ $f$ is integrable with respect to $x$ on $E$, and for each fixed $x ∈ E$, $f$ is differentiable with respect ...
0
votes
3answers
57 views

Proof that the continuous image of a compact set is compact [duplicate]

Let $X\subset \mathbb R^{n}$ be a compact set, and $f :\mathbb R^{n}\to \mathbb R $ a continuous function. Then, $F(K)$ is a compact set. See, I know that this question may be a duplicate, but the ...
2
votes
1answer
31 views

Problem with finding a suitable partition of unity

Let $\{U_s\}_{s\in S}$ be a family of open subsets of $\mathbb R$ with union $W$ with the following property: for each $x\in U_s$ we have $x+1, x-1\in U_s$. Does there exist a sequence ...
1
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0answers
20 views

Showing a function is upper semicontinuous

Let $f: \mathbb{R} \rightarrow [0, B]$ and for every $\varepsilon > 0$, let $\varphi_{\varepsilon}(x) := \sup_{\{y: |x - y| < \varepsilon\}}f(y)$. Since for each fixed $x$, ...
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0answers
19 views

Interesting properties of functions and sets that depends on dimension of space.

For $n=1$ (or $m=1$), we have some basic properties of functions and sets that are not valid (or not necessarily valid) for $n\neq 1$ (or $m\neq 1$). For exemple: Calculus. Let $[a,b]$ be a closed ...
2
votes
0answers
28 views

Is it possible to calculate the limit $\displaystyle \lim_{n \to \infty} \frac{a_n}{n}$ of the conditions?

Is it possible to calculate the limit $\displaystyle \lim_{n \to \infty} \frac{a_n}{n}$ of the conditions? $ a_n>0$ for $ {n \geq 1}$ and $ a_{n-1} \leq \left(a_{n+2} - a_n \right)^n \leq ...
3
votes
1answer
117 views

Monotone Convergence Theorem for Riemann Integrable functions

I'm having a really hard time proving this statement (this is not homework): If $f_{n} : [0,1] \rightarrow \mathbb{R}$ is a Riemann integrable function for all $n \in \mathbb{N}$, and $0 \leq f_{n + ...
3
votes
0answers
43 views

Rational analysis

I found myself thinking about how much of real analysis that can also be developed within the rational numbers. Of course, $\Bbb Q$ is lacking what is perhaps the most important property of the real ...
2
votes
1answer
39 views

Are these two expression square integrable?

I have two expressions (let's call them functions $f,g$) on $[0,1]$, where I want to find out whether they are square-integrable or better: for which $m \in \mathbb{Z}$ they are square-integrable ( ...
0
votes
0answers
31 views

Convergence of norms

I have this space $H_{0,p}^1=\lbrace u\in AC([0,+\infty),\mathbb{R}),u(0)=u(+\infty)=0, \sqrt{p} u'\in L^2(0,+\infty)\rbrace $ endowed with the norm $||u||^2=\int_0^{+\infty} p(t) u'^2(t) dt$ ...
0
votes
0answers
11 views

Can we identify Fourier transform of continuous compacltly supported functions with finte complex Borel measure?

It is well-known that, $L^{1}(\mathbb R)$ can be embed into $M(\mathbb R)$ (= The space of complex Borel measure on $\mathbb R$); by identifying $f\in L^{1}(\mathbb R)$ with the measure $d\mu= f dm.$ ...
2
votes
2answers
30 views

Establish the absolute maximum of a function

We have this function:$$f(x)=\begin{cases} \sin(x) \cdot\ln(\sin2x), & \mbox{if }0<x<\pi/2 \\ 0, & \mbox{if }x=0,\mbox{or }x=\pi/2 \end{cases}$$ So, how to prove that it decreases and ...
2
votes
2answers
74 views

Pick a smart function

Our teacher wants us to find a function $f$ on $(0,\pi)$ such that $$\sqrt{\sin(x)} f(x)^{\frac{1}{4}} =k_1 + \cos(x)$$ and $$\sqrt{\sin(x)} f(x)^{-\frac{1}{4}} = k_2 + \cos(x).$$ The two constants ...
1
vote
2answers
11 views

Asymptotic Notations in limits

Can the asymptotic notations, like Big O, be defined using limits? example:- Lim x->infi (f(n)/g(n))=c for defining f(n)=O(g(n)) If not, why??
2
votes
1answer
18 views

Hadamard's Lemma in multidimensional real analysis

This is Hadamard's Lemma: Let $U \subset \Bbb R^n$ be an open set, let $a \in U$ and $f: U \to \Bbb R^p$. Then the following assertions are equivalent. The mapping $f$ is differentiable at $a$. ...
2
votes
1answer
23 views

convolution of compactly supported continuous function with schwartz class function is again a Schwartz class function?

Suppose $f$ is continuous function on $\mathbb R$ with compact support; and $g\in \mathcal{S}(\mathbb R),$ (Schwartz space) My Question is: Can we expect $f\ast g \in \mathcal{S(\mathbb R)}$ ? ...