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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Problem with constructing a smooth function with given properties

I wish to construct a function $f:\mathbb R \rightarrow \mathbb R$ of class $C^\infty (\mathbb R)$ with the folowing properties: $f(x)=0$ for $|x|\leq 1$ $f(x)=x$ for $|x| \geq 2$, $|f(x)| \leq |x|$...
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1answer
38 views

Is the action on $L^2$ arising from a measure preserving action continuous?

Let $G$ be a locally compact topological group, $X, \mu$ a probability space, and $G\times X \rightarrow X$ a measurable group action which preserves $\mu$ (i.e. $\mu (gA)=\mu(A)$) . Does it follow ...
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0answers
35 views

Proving Equalities in Analysis

In measure theory, I saw that while proving some "equalities" - $``a=b"$ - (such as measure of any type of an interval is its length, ...), the argument goes as follows: We prove that $a\leq b$ ...
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1answer
54 views

Integral over compact boundary is finite in the context of potential function

Consider some bounded domain $\Omega\subset \mathbb{R}^n$, s.t. the boundary $\partial\Omega$ is a smooth submanifold of dimension $(n-1)$ and fix some point $x_0\in \partial\Omega$. Now I want to ...
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1answer
94 views

What is the necessary and sufficient condition of linear dependence of $n$ functions?

If $n$ functions are linear dependent, then the Wronskian determinent is zero, While that the Wronskian determinent is zero cannot imply $n$ functions are linear dependent. So what is the necessary ...
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1answer
108 views

Fubini's theorem applied to heaviside step functions

First of all, I should probably mention that I am a physicist, not a mathematician so I sincerely apologize for any lack of rigour in my explanation of my problem. Recently, I have been trying to ...
2
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1answer
40 views

Is set with this property is homeomorphic to Cantor set?

(1) $A$ is nonempty subset of $\mathbb{R}$. (2) For all $x<y \in A$ there is $z \notin A$ such that $x<z<y$. (3) $A$ is perfect. Then is there homeomorphism between $A$ and cantor set? ...
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1answer
205 views

Fourier transform of a compactly supported function.

Can someone help me the question below? Is there a positive-valued compactly supported function $f$ such that the Fourier transform ${{f}^{\operatorname{ft}}}\left( t \right)=\int_{-\infty }^{\infty }{...
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4answers
321 views

Closure of a subset of a metric space is closed

From definition, if $X$ is a metric space, if $E \subset X$, and if $E'$ denotes the set of all limit points of $E$ in $X$, then the closure of $E$ is the set $\overline{E}=E \cup E'$. I need to ...
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1answer
49 views

Partial derivatives of $xy^2/(x^2+y^2)$ at the origin

I noticed that this is a big black hole in my understanding of partial derivatives at the point. I don't know how to count it: $$ f(x,y) = \frac {xy^2}{x^2+y^2} $$ $$ \frac {df}{dx}(0,0)=\lim_{t\to ...
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37 views

Can we expect to find $r,$ large enough, so, $\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}\leq C (1+y^{2})^{s} $ for all $y\in \mathbb R$?

Fix $y\in \mathbb R$ and $s>1.$ Consider the series: $$I(y)=\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}.$$ My Question is: Can we expect to find $r$ large enough, so that $$I(y)\...
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1answer
62 views

Two Strictly Convex Functions with Contact of Order 1

Let $f,g: \mathbb{R}\rightarrow \mathbb{R}$ be two strictly convex functions, where $f$ is differentiable, $g$ is smooth, and $f\geq g$. Suppose that for some $x_0\in \mathbb{R}$: $f(x_0)=g(x_0)\...
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1answer
50 views

Differentiable Strictly Convex Function on Interval

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a differentiable, strictly convex function. Let $I\subset \mathbb{R}$ be a closed, bounded interval such that $f'(x) \neq 0$ on $I$. Is $f$ strongly ...
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1answer
67 views

Strictly Convex and Differentiable Implies

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be strictly convex and differentiable. Is $f$ strongly convex when restricted to a closed and bounded interval $[a,b]$? This is true if $f$ is smooth but am ...
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80 views

Is the “difference-integral” of an integrable function integrable?

assume $f$ is Lebesgue integrable (on $\mathbb{R}$) and $h >0$. Is it true that then the double integral $\int_{-\infty}^{\infty} \int_{x}^{x+h} f(u) \, du \, dx$ always exists? Intuitively I have ...
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1answer
90 views

Show that the set $\left\{\sin\frac{1}{2}x,\sin\frac{3}{2}x,\sin\frac{5}{2}x,\ldots\right\}$ is complete on $[0,\pi]$

Show that the set $$\left\{\sin\frac{1}{2}x,\sin\frac{3}{2}x,\sin\frac{5}{2}x,\ldots\right\}$$ is complete on $[0,\pi]$ I think I can change it to $\left\{\sin\left(\frac{2n-1}{2}x\right)\right\}_{n=...
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1answer
105 views

Disconnected Topoological Space with Intermediate Value Property

Does There exist a disconnected topological space with intermediate value property? Intermediate Value Property states that 'a topological space X is said to have intermediate value property if for ...
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3answers
130 views

Nullhomotopic map extended

I have troubles understanding this proof: Let $h:S^1 \rightarrow X$ be a continuous map, then we have that if $h$ is nullhomotopic, $h$ can be extended to a continuous map $k:B^2 \rightarrow X.$ ...
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1answer
310 views

Rudin's proof of the change of variable theorem

I am having trouble with Rudin's proof of the change of variable theorem for multiple integrals. The theorem is for 1-1 $\mathscr{C'}$ mappings from $R^k$ into $R^k$. In theorem 10.7 just before the ...
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1answer
66 views

Limit of the p-norm of a function on subdomains equals the p-norm of the function on the union domain

Let $\Omega$ be an open subset in $\mathbb{R}^n$. Given a measurable function $f$, define $$ ||f||_{p,\Omega}=\inf_{a\in\mathbb{R}}||f-a||_{L^p{(\Omega})}. $$ Let $\{\Omega_n\}$ be a sequence of open ...
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42 views

amazing boundedness problem from maximal function

Let $n\geq 2$. For any $M>1$, prove that there exists a constant $C_M>1$ such that for any ball $B$ in $\mathbb{R}^n$, if we denote $MB$ as the concentric ball of $B$ with $M$ times radius of $B$...
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3answers
43 views

Zero point when $f'(x)\gt c$

Suppose that the function $f:\mathbb R\to\mathbb R$ is continuously differentiable and that there is a positive number $c$ such that $f'(x)\ge c$ for all points $x$ in $\mathbb R$. Prove that there is ...
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4answers
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Can a set be infinite and bounded?

I don't understand a statement in my math book course, I was restudying the compact sets part of the chapter when at a certain moment there is a corollary saying : 'every infinite and bounded part of ...
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2answers
65 views

Separating a Complex Valued Function

Is there a formula (with mathematical reasoning) for separating a complex-valued function $f(z)=f(x+iy)$ into the form $ f(z)=u(x,y) + iv(x,y)$? Thank You, C.A
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2answers
107 views

Real and imaginary parts of a complex-valued function

How do you get a complex-valued function $ f(z) = f(x+iy) = \frac{z^{s-1}}{e^{-z}-1}, $ where $s$ is a constant complex number and $z$ is a complex variable, into the form: $ f(x+iy) = a(x,y) + ib(x,...
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1answer
151 views

Division algorithm for the natural numbers.

I am trying to prove the following statement from Tao's analysis book. Definition of multiplication $ab++=ab+b$. Definition of addition $(a++)+b=(a+b)++$. Let $n$ be a natural number, and let $q$ ...
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1answer
19 views

First order condition of $\frac{1}{2}\sum_{t=t_0}^{\infty}{y_t}^2$

$$\frac{1}{2}\sum_{t=t_0}^{\infty}{y_t}^2$$ where $t=t_0,t_1,\dots$ What is the first order condition? I'm a bit confused since if we differentiate this using $y_t$, the f.o.c is $$\sum_{t=t_0}^{...
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2answers
43 views

Space of bounded functions vs. bounded space of functions.

Suppose I have a bounded set of functions, say $B\subset C[0,1]$. What exactly does this mean? I.e. is a bounded set of continuous functions equivalent to a set of continuous bounded functions? For ...
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2answers
204 views

Scalar-by-matrix Derivative of Quadratic Product

I'd like to know $\frac{\partial f(\mathbf{U})}{\partial \mathbf{U}}$, i.e., the 'by-matrix derivative' of the following scalar function $f(\mathbf{U})$ w.r.t. $\mathbf{U}$. $$f(\mathbf{U}) = \vec{x}^...
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2answers
93 views

How to do this integral $\int_{-\pi}^{\pi} x^n \cos^m(x) dx$?

is there a way to explicitely evaluate this integral for natural numbers $n,m$: $$\int_{-\pi}^{\pi} x^n \cos^m(x) dx.$$ Apparently, if $n$ is odd, this integral is zero due to symmetry.
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0answers
49 views

Different Formulations of Riesz' lemma

Version I: Let $U$ be a closed subspace of the normed space $X$ with $U \ne X$. Also let $0 < \delta < 1$, then there exists $x_{\delta} \in X$ with $||x_{\delta}|| = 1$ and $$ || x_{\delta} -...
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1answer
116 views

What's $\sum{\frac{x^n}{n^3}}$?

What's $\displaystyle f(x)=\sum_{n=1}^\infty{\frac{x^n}{n^3}}$? Note its derivative: $$\displaystyle f'(x)=\sum_{n=1}^\infty{\frac{x^{n-1}}{n^2}}$$ and the next derivative: $$\displaystyle f''(x)=\...
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1answer
40 views

Fibre is open in covering space

I think I don't see the wood for the trees: In my notes I found the remark that if $p:E \rightarrow B$ is a covering map, then for each $b \in B$ we have that $p^{-1}(b)$ in $E$ has the discrete ...
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1answer
65 views

Topological properties of $(0,1)\times \{0\}$

I am having a real hard time solving simple proofs involving open sets. I am confronted with this one: Is $(0,1)\times \{0\}$ open? Is it compact? What is its interior? I know $(0,1)$ is open. ...
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2answers
70 views

Can we expect to find some constant $C$; so that, $\sum_{n\in \mathbb Z} \frac{1}{1+(n-y)^{2}} <C$ for all $y\in \mathbb R;$?

Fix $y\in \mathbb R;$ and consider the series: $$\sum_{n\in \mathbb Z}\frac{1}{1+(n-y)^{2}}.$$ My Question is: Can we expect to find some constant $C$; so that, $$\sum_{n\in \mathbb Z} \frac{1}{...
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1answer
71 views

If holomorphic $\{f_n\}\to f$ uniformly on compact subsets of $U$, then do $f_n$ and $f$ eventually have the same number of zeros?

Let $U$ be an open subset of $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $U$ such that $f_n\to f$ uniformly on any compact subset $K$ in $U$. Suppose $f$ is not constant, ...
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2answers
124 views

For holomorphic functions, if $\{f_n\}\to f$ uniformly on compact sets, then the same is true for the derivatives.

Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset $K\...
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1answer
44 views

$\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}\leq C$ for all $y\in \mathbb R$?

Fix $y\in \mathbb R$ and $s>1.$ Consider the series: $$I(y)=\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}.$$ My Question is: Can we choose $r$ large enough so that $I(y)< C$ for ...
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1answer
26 views

From the given measure $\mu,$ how to construct another measure $\mu^{\ast}$; so that $d\mu^{\ast}(y)= (1+y^{2})d\mu(y)$?

Put $\mu= \sum_{n\in \mathbb Z}c_{n}\delta_{n};$ where $\delta_{n}$ is the unit Dirac mass at $n.$ We note that, $\mu$ is a complex Borel measure on $\mathbb R$ and the total variation of $\mu,$ that ...
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1answer
48 views

$\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx<\infty $ for some large $n$?

Fix $y\in \mathbb R.$ Define, $$I(y)=\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx.$$ My Question is: Can we show that $I(y)<\infty$ for some large $n\in \mathbb N$ ? If yes, what is a value ...
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0answers
50 views

Can we expect, $h\ast \mu \in L^{2}(\mathbb R, (1+|x|^{2})^{s})$ for $h\in \mathcal{S}(\mathbb R), \mu\in M(\mathbb R)$ and $s>1/2$?

We put, $M(\mathbb R)=$ The space of complex bounded Borel measure on $\mathbb R$ [With each complex Borel measure $\mu$ on $\mathbb R$ there is associated a set function $|\mu|,$ the total variation ...
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4answers
127 views

a question how to prove:$\sum_{n=1}^{\infty}{{(-1)}^{n-1}{\cos(nx)}\over {n}}=\ln(2\cos(x/2))$

I found a complicated question in my textbook, I can't solve it? How to prove $$\sum_{n=1}^{\infty}{{(-1)}^{n-1}{\cos nx}\over {n}}=\ln(2\cos(x/2))$$ where $x\in(-\pi,\pi)$. My tried method: I tried ...
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1answer
219 views

Eigenvalues gone wild

I added some significant details to this problem, as it was apparently not clear to everyone what I want to know: This is a question about convergence of eigenvalues which essentially came up in ...
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1answer
19 views

Importance of specifying indexing sets

I was going through a rudimentary course in mathematical analysis covering Metric spaces and the book opens up with the idea of open sets. While mentioning the property of open sets it cites that if ...
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0answers
26 views

The variation of Calderon reproducing formula

I'm reading the book 'Classical and multilinear harmonic analysis, Muscalu'. I fail to understand the page 261. Actually, I doubt that the proof is right. Let $f \in BMO(\mathbb{R^d}$) have compact ...
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52 views

Is Sobolev space $H^{s}(\mathbb R),$ for $s>\frac{1}{2},$ closed under point wise multiplication? [duplicate]

We note that, $L^{2}(\mathbb R)$ is not closed under point wise multiplication. Let $s>\frac{1}{2};$ and we define Sobolev space, as follows: $H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\...
3
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1answer
98 views

Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces

Let's $(X_1,d_1), (X_2,d_2)$ be compact metric spaces such that for every finite subset of $X_1$ like $A$ (respectively any finite subset of $X_2$ like $B$ ) there exists a finite subset of $X_2$ like ...
5
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1answer
86 views

how to determine the existence of double limit?

Let $f(x,y)$ be a function of two variables. Are there any criterions to determine the existence of double limit $$ \lim_{(x,y)\to(x_0,y_0)} f(x,y)? $$ If for all $y\in(y_0-\delta,y_0+\delta)$, $\...
2
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1answer
71 views

direction limits and double limit

Let $f(x,y)$ be a function of two variables. What is the counterexample that there exists $A$ s.t. for all $\theta$, $$\lim_{r\to 0+}f(r\cos \theta,r\sin \theta)=A$$ but double limit $$ \lim_{(x,y)\...
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1answer
56 views

Are all singular functions of bounded variation?

Let $f$ be a function of bounded variation on $[a,b]$. Then there exist a unique pair (up to adding a constant) of absolute continuous function $g$ and singular function $h$ (i.e., $h'=0$ a.e.) such ...