# Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

2,650 questions

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### Sub-Gaussian decay of the measure of Euclidean balls

Let $X$ be a random vector in $\mathbb{R}^d$ satisfying the following property: there exists $C_1,C_2>0$ such that
$$\int_0^{+\infty}\mathbb{P}(\|X-\mu_0\|\leq\sqrt{t})\exp(-t)dt\leq C_1\exp(-C_2\|\...

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**1**answer

60 views

### Relation between the measures of two sets defined via Lebesgue integration

I posted this question on StackExchange, people have upvoted it but I have not received any response. I read up here that it is okay to post unanswered StackExchange questions on Mathoverflow. So, ...

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31 views

### Energy-minimizing set of discrete points in a bounded domain

Let $\Omega \subset \mathbb{R}^3$ be a smooth, bounded domain.
Let $x_1,\ldots,x_n \in \overline{\Omega}$ be chosen so as to minimize
$$
\sum_{1\leq i<j\leq n} \frac{1}{|y_i - y_j|}
$$
over all ...

**1**

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**1**answer

46 views

### Subadditive function with special growth

Related to one of my previous question (for which I have received an answer, thanks) I have the following new one. Maybe I am describing the empty set but not being a specialist at all of the domain I ...

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**1**answer

176 views

### Is this sequence convergent? [on hold]

suppose $\exists S \subset \mathbb{R}$ and a function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x_0 \in S $ the sequence $x_{n+1} = f(x_n)$ converge to $x \in S$
now, let $\alpha \...

**2**

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48 views

### Can we approximate this matrix field with an invertible matrix field?

Let $\mathbb{D}^2=\{ x \in \mathbb{R}^2 \, | \, |x| \le 1\}$ be the closed unit disk, and set
$$\begin{equation*}
A(x,y)=\left(
\begin{array}{cc}
x & -y \\
y & x
\end{array} \right)
\end{...

**5**

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136 views

### Is $π:\mathcal{C}^∞(M,N)→\mathcal{C}^∞(S,N)$, $π(f)=f|_S$ a quotient map in the $\mathcal{C}^1$ topology?

This question was previously posted on MSE.
Let $M, N$ be smooth connected manifolds (without boundary), where $M$ is a compact manifold, so we can put a topology in the space $\mathcal C^\infty(M, N)...

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185 views

### An implication of the Zagier et al result on the hyperbolicity of Jensen polynomials for the Riemann zeta function?

In their paper recently published in the PNAS, Zagier et al demonstrated that
The Jensen polynomials $J_{\alpha}^{d,n}(X)$ of the Riemann zeta function of degree $d$ and shift $n$ are hyperbolic for ...

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57 views

### Find limit of sequence defined by sum of previous terms and harmonics

I came across this sequence as part of my work. Could someone indicate me the methodology I should follow to solve it? I guess it involves harmonic numbers and/or the digamma function?
I tried to ...

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**1**answer

178 views

### Bounding higher derivatives of $f(x) = 1/(1+x^2)^r$

Let $r\in \lbrack 0,\infty)$. Define $f(x) = 1/(1+x^2)^r$. It would seem to be the case that $$|f^{(k)}(x)|\leq \frac{2r \cdot (2r+1) \dotsb (2r + k-1)}{(1+x^2)^{r + k/2}}$$ for all even $k\geq 0$. ...

**10**

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**2**answers

297 views

### Is the composition of two nowhere differentiable functions still nowhere differentiable?

Let $f,g:\mathbb R\to\mathbb R$ be two continuous but nowhere differentiable functions. By the Denjoy–Young–Saks theorem for almost every point $x_0\in\mathbb R$ one has
$$
\limsup\limits_{x\to x_0}\...

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29 views

### Convergence acceleration of a series by using optimal parameters

One of the ways of accelerating the convergence of a series is by transforming into a faster series using optimal parameters. Examples of this approach can be found in this paper. I obtained a ...

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**1**answer

333 views

### A maximization problem

Consider the minimization problem described this paper. Let $f_{\lambda}$ be the minimizer. As a part of extending my work, I am able to show the following facts
$$\lim_\limits{\lambda \to 0}\|f_{\...

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44 views

### Differential Eq [closed]

Ok my first try of explaining my problem was not that understandable, I'll give it another try
Suppose you have n persons sitting in a circle, next to each other.
At time t0 everyone starts clapping ...

**-2**

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**1**answer

91 views

### Continuity of the Restriction Map Between Function Spaces [on hold]

Let $X,Y,Z$ be Hausdorff spaces and suppose that $Z\subset X$. Endow $C(X,Y)$ and $C(Z,Y)$ with the compact-open topologies and define the map $\rho$ as
\begin{align}
\rho:&C(X,Y)\rightarrow C(Z,...