The real-analysis tag has no wiki summary.

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### A special approximation of BV functions

Suppose $\Omega:=B(0,1)\subset \mathbb R^2$. Let $u\in BV(\Omega)$ be a radially symmetric, i.e., $u(x)=u(Rx)$ for all $R\in SO(2)$. In addition, suppose $w$ to be an affine function,. i.e., ...

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### function bounds and comparison [on hold]

ok so basically i need to understand how i can compare functions so that i can find big O big theta and big omega for algorithms of a program
my mathematics background is not very strong but i have ...

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80 views

### Proof for this expression for Dottie number [on hold]

Neither Wolfram's mathworld nor Wikipedia mentions any series expanion for the cosine's fixed point.
Therefore I am asking for a proof for this Kapteyn series representation of Dottie number:
...

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**1**answer

106 views

### Function and its Gradient with Prescribed Norms

I'm not sure if the following question is too elementary for Mathoverflow. I'm sorry if it is the case.
Question:
Let $n\in\mathbb{N}$ and let $1\leqslant p<\infty$. Let $\alpha,\beta>0$. ...

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**1**answer

67 views

### Boundedness of a singular integral operator on $L^p(\mathbb{R})$, $1<p<\infty$

My singular integral operator is defined by
\begin{align}
Sf(x)=-\int_{-\infty}^{\infty}f(t-x) \frac{dt}{2\sinh\frac{\pi}{2}t},
\end{align}
that is, a convolution $-\frac{1 }{2\sinh\frac{\pi}2x}\ast ...

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### Zeros of a Real Analytic Function [on hold]

Let $f:[0,1] \rightarrow \mathbb{R}$ be a non-zero real analytic function. Consider $Z(f) \subseteq [0,1]$ as the set where $f$ vanishes. What can we say about $Z(f)$? This is (i) finite, (ii) ...

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### Property of summations [on hold]

Suppose to have the following identity:
$$
\sum_{i=i_0}^{i_1}\sum_{j=j_0}^{j_1} f(i,j) = \sum_{i=i_0}^{i_1}\sum_{j=j_0}^{j_1} f(i,j)g(i,j),
$$
for 'good' indexes $i,j$ and some functions $f,g$.
What ...

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44 views

### Graph of bounded continous functions with distance 1

Let $V = \{f:[0,1]\to \mathbb{R}: f \text{ is continuous}\}$ and consider the metric that is defined for $f,g\in V$ by $$d(f,g) = \max\{|f(t)-g(t)|: t\in [0,1]\}.$$
We set $E = \{\{f,g\}: f,g \in ...

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### On an example of an eventually oscillating function

For $x\in(0,1)$, put
$$f(x):=\sum_{n=0}^{\infty}(-1)^{n}x^{2^{n}}.$$
This function possesses interesting properties. It grows monotonically from $0$ up to certain point. Then it starts to oscillate ...

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### Interchange summation and differentiation [migrated]

I asked this question already on math.stackexchange, but did not receive any answers
see here
Let $f = \sum_{n=0}^{\infty} a_n e_n $ where $e_n$ are an ONB of $L^2[0,1].$
Now assume we have that ...

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52 views

### Good upper bound for an alternanting series [closed]

Someone know a good upper bound for the partial sums of $S=\sum(-1)^{n+1}\sqrt{n}$?
I mean how fast is the growth of this sum?

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### Deriving inequalities from a polynomially-bounded derivative

In this paper (p. 2, definition/remark) the following notion of ‘polynomial growth’ is defined for a non-negative real function $g(x)$ and a real constant $b\in(0;1)$:
There exist positive ...

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**1**answer

57 views

### Hadwiger-Nelson problem in higher dimensions

Given a positive integer $n\in \mathbb{N}$ we define the Hadwiger-Nelson graph $\text{HN}_n$ by
$V(\text{HN}_n) = \mathbb{R}^n$;
$E(\text{HN}_n) = \{\{v_1, v_2\}: v_1, v_2 \in \mathbb{R}^n \text{ ...

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**1**answer

65 views

### About preserving real-rootedness of multivariable polynomials

Say $f_i(z_1,z_2,..,z_m)$ are polynomials real rooted in the $z$s for a bunch of polynomials indexed by $i$. When can one say that $\sum_{i} p_i f_i(z_1,z_2,..,z_m)$ is also real rooted?
If ...

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**1**answer

89 views

### Inequality of the norm of the convolution in $L^p(\mathbb{R}^n)$ with symmetric decreasing rearrangement?

Is it true that
$$
||f*g||_p \le ||\,|f|^* * |g|^*||_p\quad ?
$$
where $|f|^*$ and $|g|^*$ are the symmetric decreasing rearrangements of the functions $|f|$ and $|g|$. Under what conditions on $f$ ...

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**1**answer

65 views

### “Schwarz symmetrization” on annulus

If $\Omega=\{x\in \mathbb R^n| 0<r_0<|x|<r_1\}$ is an annulus on $\mathbb R^n$, I am looking for a symmetrization result on $\Omega$. To be precise, for any $u \in W_0^{1,2}(\Omega)$, can we ...

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146 views

### Asymptotic behaviour of eigenvalues

If you look at $-\Delta + q$ on the sphere in $\mathbb{R}^3$ for example and $||q|| < \infty,$ is there a way to asymptotically describe the behaviour of the eigenvalues? Probably they behave ...

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**1**answer

118 views

### Proving a complicated inequality with powers of logarithms

I am currently formalising some results from complexity theory with a theorem prover. For that, I have to prove the following statement:
Let $p, b, \varepsilon \in \mathbb R$ with $\varepsilon>0$ ...

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81 views

### Summing a function at integer points

For a real function $f$ defined on the reals, and real $y$, define the 2-way infinite sum
$$ F_f(y) = \sum_{i\in\mathbb Z} f(y+i). $$
If $F_f(y)$ is defined for all $y$, it is periodic of period 1.
...

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71 views

### Extension of Sobolev Functions

Let $\,D\subseteq\mathbb{R}^{n-1}$ be a convex bounded domain. Let$A:D\to(0,\infty)$
be a Lipschitz continuous function. Let $\,\Omega\,$ be bounded domain in $\,\mathbb{R}^{n}\,$
of the form
...

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**1**answer

112 views

### Does $\int \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \to \Phi(u)$ imply that $f_t \to \delta_1$?

I'm looking at a family $(f_t)$ of densities of some continuous random variables and know that
$$\int_{-\infty}^{\infty} \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \xrightarrow{t \to ...

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78 views

### Number of critical points

Let $f:[0,2\pi]\rightarrow R^2$ be a smooth function such that $f([0,2\pi])$ is a smooth closed simple curve $C$. Suppose $(0,0)$ lies inside the the bounded open region enclosed by $C$ and ...

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74 views

### Set nor its compliment contain an uncountable closed set [closed]

Does there exist a set $X$ subset of the real numbers such that no uncountable closed set is contained in $X$ or $X^c$?

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65 views

### What is the fractional derivative smoothness of functions from the Zygmund class?

Let $\Lambda([0,1])$ be the Zygmund class of continuous on $[0,1]$ functions for which $$\sup h^{-1}|f(x+2h)-2f(x+h)+f(x)|<\infty.$$ What would be the exact smoothness class for the fractional ...

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61 views

### On Rényi entropy/divergence

The Rényi entropy for a probability density function $f$ with dominating measure $\mu$ of order $\alpha>0$ is defined as
$$H_\alpha(f)={1 \over {\alpha-1}}\log\int f^\alpha d\mu.$$
If $f$ is ...

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35 views

### minimal entropy approximation of a truncated discrete measure

Consider a measure $\mu$ on $\mathbb{N}$ given by the sequence $(\mu(n))_{n \geq 0}$ with $\mu(0)>0$. For example $\mu(n)=n^2+1$ on the figure below.
For each $n$, let $X_n \sim \mu(\cdot \mid ...

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61 views

### Must the Lebesgue measure of a $\rho$ - neighbourhood of an $(n-2)$ - dimensional set be at least $c\rho^2$?

The Lebesgue measure of a $\rho$-neighbourhood of a point in $\mathbb{R}^2$ is of course equal to $c\rho^2$. Similar such considerations in higher dimensions lead me to the following question:
Given ...

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123 views

### Infinite sum of asymptotic expansions

I have a question about an infinite sum of asymptotic expansions:
Assume that $f_k(x)\sim a_{0k}+\dfrac{a_{1k}}{x}+\dfrac{a_{2k}}{x^2}+\cdots$
with $a_{0k}\leq \dfrac{1}{k^2}$, $a_{1k}\leq ...

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118 views

### Orlicz Norm and A result on expectation

I am reading paper which is mainly about Dobrushin's contraction coefficient and its generalization. In page 27, the following is defined:
Consider arbitrary, non-negative, convex function ...

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### Extension of Premeasure to a Measure [migrated]

Question: Show that a set function on a σ-algebra is a measure if and only if it is a premeasure.
Sketch of proof:
A set on σ-algebra is a measure if it is an algebra and given all subsets of the ...

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480 views

### Inserting an open and simply-connected set between a compact set and an open set

In a paper I am reading, the following is considered obvious:
Let $K$ be a compact and connected subset of $\,\mathbb R^2$, with $\mathbb R^2\smallsetminus K$ also connected, and $U\subset \mathbb ...

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356 views

### Integrals of pullbacks and the Inverse function theorem(s?)

The usual story goes like this:
Smooth picture (?):
For a smooth bijection $\phi: M \to N$ between $n$-manifolds the following
is true:
$\phi^{-1}$ is a local diffeomorphism a.e.
...

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**1**answer

202 views

### A result attributed to Whitney

One of the basic results of real analysis says that any closed subset of a smooth ($C^\infty$) manifold $M$ is the set of zeros of some map $\lambda\in C^\infty(M;[0,1])$. This result (or some ...

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### Uniform estimate of a function given by an integral

consider the function $f_{n}(x,a,t):=e^{-(ax+n+1/2)^2t}$ with $t,x,a > 0$. The claim is now that there exists a constant $C>0$ such that for all even natural numbers $n=2k$, $k\in\mathbb{N}$ one ...

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291 views

### An inequality for copulas

Suppose that $f$ from $[0,\infty]$ onto $[0,1]$ is completely monotonic on $(0,\infty)$, and let $g$ be the inverse of $f$. For $(u,v)$ in $[0,1]^{2}$, define $C(u,v) = f(g(u)+g(v))$, and let $a = ...

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79 views

### Approximation of subharmonic functions

Let $u$ be an (upper semi-continuous) locally bounded subharmonic function in a domain in $\mathbb{R}^n$. Let $\chi_\epsilon$ be a standard smoothing kernel, namely
...

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163 views

### Equation between the two branches of the lambert w function

My question: Is there an equation connecting the two branches $W_0(y)$ and $W_{-1}(y)$ of the Lambert W function for $y \in (-\tfrac 1e,0)$?
For example the two square roots $r_1(y)$ and $r_2(y)$ of ...

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### A difficult integral which the Risch algorithm shows is not elementary

For reasons which aren't conceptually related to the problem a few of my colleagues and I are in need of finding an expression for the following integral in terms of $a$ and $\delta$:
...

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### Perturbation of Laplacian via Kato-Rellich theorem

Let's consider a potential $V(x)\in L^3(\mathbb{R}^3)$. I want to know if the following Hamiltonian
$$-\Delta+V(x)$$
is self-adjoint on $H^2(\mathbb{R}^3)$.
My idea is to use Kato-Rellich theorem; ...

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### Extension of a smooth function to a small neighborhood of a cone

Let $C\subset\mathbb{R}^n$ be an open polysimplicial cone. Let $f$ be a smooth function on $C$ such that all its derivatives extend by continuity to $\overline{C}$ (the closure of $C$). Does this ...

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### Does this symmetrization operator have a name? Any theory?

Consider a function $f(x_1,\ldots,x_n)$ of $n$ complex variables. Define
$$f_{\mathrm{symm}}(x_1,\ldots,x_n) =
2^{-n}\sum_{\varepsilon_1,\ldots,\varepsilon_n=\pm 1}
...

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### Generating the sigma algebras on the set of probability measures

I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and ...

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321 views

### Find an integrable, positive, unbounded, analytic function

Is there a standard example of a function $f \in L^1( \mathbb R)$ which is analytic, positive, integrable but not bounded?
An example which comes immediately to mind is to take the series of narrower ...

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**1**answer

133 views

### Legendre transform and Lipschitz approximation

Let $(X,d)$ be a compact metric space and $f:X \to \mathbb{R}$ a real valued continuous function. Let us agree that a modulus of continuity means concave, nondecreasing, uniformly continuous function ...

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### Recent trends in effective analysis

The references listed at http://en.wikipedia.org/wiki/Computable_analysis have all been published 30-15 years ago. Are the approaches which these references expose still up-to-date and relevant to the ...

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130 views

### Sequence of smooth maps converging to the identity [closed]

Let $\{g_{n}:B_{1}(0) \rightarrow \mathbb{R}^{2}\}_{n\in \mathbb{N}}$ be a sequence of smooth maps where $B_{1}(0)=\{x\in \mathbb{R}^{2} \mid |x|<1\}$ is the unit ball in $\mathbb{R}^{2}$. Assume ...

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65 views

### Are Carnot groups (as Carnot Caratheodory metric spaces) doubling?

I need to use the Lebesgue differentiation theorem for doubling metric measure spaces and was wondering if Carnot groups are doubling. If yes, is there any reference you can point me to? Thank you.

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### Characterizations of an exotic measure on the open sets in the circle $S^{1}$

Suppose that $U\subseteq S^{1}$ is open where $S^{1}=\{z\in\mathbb{Z}:|z|=1\}$. Then define $\mu_{n}(U)=\max_{t\in S^{1}}\frac{1}{n}\cdot|\{k\in\{1,...,n\}|t\cdot e^{\frac{2\pi ik}{n}}\in U\}|$. ...

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### Discrete measures and discrete kernels

This is a cross-post from math.stack. Let $d\in\mathbb N$ and $\mu$ be the probability measure on $\mathbb R^d$ defined by $\mu=\sum_{k=1}^\infty 2^{-k}\delta_{x_k}$ for some sequence ...

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253 views

### A question on the Lebesgue differentiation theorem

In the paper [Jessen, B., Marcinkiewicz, J., and Zygmund, A. Note on the differentiability of multiple integrals. Fundamenta Mathematicae 25.1 (1935): 217-234] it is considered the limit
$$
...