Questions tagged [real-analysis]

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

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5
votes
0answers
322 views

Partitioning $\mathbb{R}^n$ into closed sets

Let $n$ be a positive integer. It is well-known that $\mathbb{R}^n$ cannot be non-trivially partitioned into open sets, since it is connected. Let $\frak P$ be a partition of $\mathbb{R}^n$ into ...
9
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0answers
245 views

Concerning Luzin-(N)-property

Definition: a function $f:\mathbb{R}\to \mathbb{R}$ has Luzin-(N)-Property if $f$ maps any null set to a null set. By https://www.encyclopediaofmath.org/index.php/Luzin-N-property, it is known that ...
1
vote
0answers
54 views

Regularity of a shrunken domain

I am encountering a geometrical question that intuitively seems obvious but I have a lack of argument to prove or disprove it in a rigorous manner. Let $\Omega\subset\Bbb R^d$ be an open bounded (...
5
votes
1answer
132 views

Ratio of integrals with increasing dimension over Euclidean balls

Let $f_n(x)\geq0$ be any sequence of nonnegative $L^1(\mathbb{R}^n)$ functions such that $\int_{\mathbb{R}^{n}}f_n(x)dx=1$ where $dx$ is the Lebesgue measure on $\mathbb{R}^n$. For any $a>1,\...
8
votes
2answers
242 views

Reference request: Extensions of Wiener's Tauberian Theorem

Wiener's Tauberian Theorem says that linear combinations of translations of a function $f$ are dense in $L^1(\mathbb{R})$ if and only if the zero set of the Fourier transform of $f$ is empty. This is ...
3
votes
2answers
294 views

Good upper bound for a certain sum

Given $\gamma \in [0, 1)$, an integer $N \ge 2$, and a decreasing null sequence of positive numbers $e_1,e_2,\ldots,e_t,\ldots$, I'm interested in estimating the sum $S_N := \sum_{t=1}^N\gamma^t e_{N-...
1
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0answers
59 views

Real root isolation for exponential polynomials

Suppose we are given an exponential polynomial $f:\mathbb{R}\mapsto\mathbb{R}$ $$ f(t)=\sum_{i=1}^n p_i(t)e^{\lambda_i t} $$ where $p_i(t)$s are polynomials with algebraic coefficients and $\lambda_i$...
2
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1answer
90 views

Control the oscillation of a function by its total variation

Is it possible to control the oscillation of a BV vector field $u:\mathbb R^N \to \mathbb R^N$ at a point $x_0$ by the total variation of $u$?
2
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0answers
129 views

Karamata's proof of Hardy-Littlewood Tauberian theorem

I understand Karamata's proof of the Hardy-Littlewood Tauberian theorem as in http://individual.utoronto.ca/jordanbell/notes/karamata.pdf, but what on earth is the motivation behind Lemma 4 - i.e, ...
6
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1answer
253 views

Sub-Gaussian decay of convolution of $L^1$ function with Gaussian kernel

I think it might be helpful to put the new statement at the beginning and put the original post at the end. This new statement is more mathematically elegant. Let $f\geq0$ be in $L^1(\mathbb{R}^d)$ ...
1
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1answer
78 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, ...
0
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0answers
56 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
vote
1answer
49 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 ...
-3
votes
1answer
211 views

Is this sequence convergent? [closed]

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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0answers
51 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
votes
1answer
148 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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0answers
232 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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0answers
59 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 ...
4
votes
1answer
204 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 $k\geq 0$. ...
11
votes
2answers
343 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}\...
1
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0answers
34 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 ...
1
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2answers
472 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_{\...
-2
votes
1answer
100 views

Continuity of the Restriction Map Between Function Spaces [closed]

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,...
5
votes
1answer
175 views

Simple proof of Prékopa's Theorem: log-concavity is preserved by marginalization

The following result is well-known: Suppose that $H(x,y)$ is a log-concave distribution for $(x,y) \in \mathbb R^{m \times n}$ so that by definition we have $$H \left( (1 - \lambda)(x_1,y_1) + \...
1
vote
1answer
119 views

Measure theory problem concerning convergence of integrals

Let $X$ be a measure space. Let $S_j$, $j \in \mathbb N$ be an increasing sequence of $\sigma$-algebras on $X$ such that $S := \bigcup_{j \geq 0} S_j$ is a $\sigma$-algebra. For every $j$, let $\mu_j$ ...
3
votes
2answers
207 views

A reduction problem from $\mathbb{R}^2$ to $\mathbb{R}$

Let $f,g \in L^1_\text{loc}(\mathbb{R})$, with $g \geq 0$, and such that for almost every $(x,y) \in \mathbb{R}^2$, at least one of the following equations is true : \begin{align*} f(x) + f(y) + g(...
4
votes
1answer
438 views

Integrate $1/(x_1x_2\cdots x_n)^k$ for $1\le x_i \le a$, where product of coordinates satisfies $ b\le x_1\cdots x_n\le c$

I need to integrate $$ \int\limits_{[1,a]^n} \frac {\chi(\{ b \le x_1 \cdots x_n \le c \})} {( x_1 x_2 \cdots x_n)^k} \,dx_1 \cdots dx_n, $$ where $\chi(E)$ is the characteristic function of a set $E$....
4
votes
0answers
101 views

Korovkin subset of $C(\mathbb{T})$

Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if for every sequence $(T_n)$ of positive linear operators on $C(K)$, the condition $\|T_n(f)- f\|_\...
23
votes
0answers
408 views

Three real polynomials

Theorem. Let $f,g$ be two real polynomials, and suppose that their Wronskian $W(f,g)=f'g-fg'$ has only real roots. Then on any interval $I\subset\mathbf{R}$ containing no roots of $W$ every non-...
2
votes
1answer
88 views

How do the balls maximizing the maximal function depend on their centers?

Let $\mu$ be a finite Borel measure on $\mathbb R^N$ and let $f\in L^1(\mu)$ be a non-negative function. Let $M_\mu f$ denote the maximal function of $f$ relative to $\mu$, i.e. $(M_\mu f)(x)=0$ if $\...
3
votes
0answers
106 views

Characterising functions of bounded variation by their modulus of continuity

Given a a.e. finite measurable function $ \mathbb R^n \to \mathbb R$, define the essential modulus of continuity, $M(f): \ \mathbb R^n \times \mathbb R+ \to \mathbb R$ by $$ M(f) (x, e)=\sup_{m(A) = 0}...
1
vote
1answer
76 views

Box counting dimension of a set and Lipschitz functions

If $f$ is Lipschitz, then the following holds for the Hausdorff dimension: $$\dim_H f(A) \le \dim_H A.$$ Is the same true for the box counting dimension?
7
votes
1answer
249 views

Taylor's polynomials and loss of real roots

Real-rootedness, log-concavity, and unimodality are intertwined properties. It's in this light that I was prompted to ask the question below. Suppose the roots of a polynomial $p(x)$ are all real and ...
6
votes
1answer
161 views

Differentiability of the distance function from a (variable) point to a (fixed) set

The distance of from a point $x$ to a set $A$ is defined by $$ d(x,S) = \inf\{d(x,a)\mid a\in A\}, $$ where you may choose the setting to be $\mathbb R^n$, a Banach space or a complete metric space. ...
8
votes
1answer
295 views

A Besicovitch-type Covering Theorem

In the book The Geometry of Domains in Spaces by Krantz and Parks, the authors proved the weak $(1,1)$-type estimate of the maximal function $M_\mu f$, where $\mu$ is a Radon measure, using their ...
0
votes
1answer
77 views

A measurable set such that its intersection and difference with every interval have the same measure [duplicate]

Let $\Omega = [0,1]$. I want a Lebesgue measurable set $S$ with the following property. $$ \ell(S \cap I) = \ell(I \backslash S)$$ for every subinterval $I$ of $[0,1]$, where $\ell(A)$ is the ...
0
votes
0answers
61 views

General Term Formula for Sequences

Let $k_1,k_2,\cdots,k_n,\cdots$ be a sequence of known positive numbers. Define $$ a_1:=k_1,\\ a_2:=C_2^2k_2+C_2^1k_1a_1,\\ a_3:=C_3^3k_3+C_3^2k_2a_1+C_3^1k_1a_2,\\ a_4:=C_4^4k_4+C_4^3k_3a_1+C_4^...
2
votes
0answers
55 views

Essentially anti-Cauchy functions

Call a function $f: \mathbb R+ \to \mathbb R$ essentially $C^\infty$ if there exists a sequence $f_n$ $(n \geq 0)$ such that each $f_n$ is differentiable a.e., $f_0 = f$ a.e., and $f_n’$ is equal to $...
4
votes
0answers
41 views

Commuting flows problem for non-Lipschitz vector fields

Let $X$ be a continuous vector field on a (say compact) manifold $M$, if $X$ has ODE uniqueness then we can define its associated flow $\mathcal F_X:\mathbb R\times M\to M$ uniquely given by $\mathcal ...
5
votes
0answers
76 views

Projection of a function $f\in L^1(\Omega)$ onto a finite dimensional subspace

Suppose $\Omega \subset \Bbb R^n$ be a domain such that $|\Omega|<\infty$, $f\in L^1(\Omega)$. Let $Y= \text{span}\{g_1,\dots, g_k\}$. Is there a characterization of the set of projections of $f$...
2
votes
0answers
174 views

Find real function $f(x)$ such that $f(f(x))=f'(x)$ [duplicate]

Absolutely there is a trivial solution $f(x)=0$. Actually, assuming $f(x)$ being smooth and expanding $f(x)$ into power series one can get $f(0)=0\to f(x)=0$. Also, in the complex field there are ...
0
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0answers
38 views

Existence of a solution for the Laplace equation with sub-linear non-linearity

At first, I do apologize if my question is silly. I know that by variational methods it is possible to prove the existence of a solution for $$ \begin{cases} -\Delta u = u^p & \Omega \subset \...
3
votes
1answer
67 views

Representation of finite differences of order k

We define recursively finite differences $ g_k (x) $ of order $ k $ of function $ f $ as follows: $g_0(x)=f(x)$, $g_n(x)=g_{n-1}(x+h_n)-g_{n-1}(x) (n\in\mathbb{N})$. It is known that all arguments of ...
1
vote
0answers
132 views

Is the normalized derivative of a holomorphic function Sobolev?

This question is a cross-post from MSE. it is also a special case of this question. Let $B=\{z\in \mathbb C \,|\,|z|\le 1\}$, and let $f:B \to \mathbb{C}$ be holomorphic on the interior $B^o$, and ...
2
votes
1answer
95 views

Proving that $\lim_{j \to i} Z_{ij} = [\ln(\frac{\Delta s_i}{2})-1]\Delta s_i$

If I have the following integral equation $$\phi(\vec{x})=\frac{1}{\pi}\int [\phi\frac{\partial (\ln r)}{\partial n} -\ln(r) \frac{\partial \phi}{\partial n}] ds$$ An approximate solution of $\phi$ ...
0
votes
1answer
89 views

Reference request : How to use Lagrange multiplier technique with infinite (infact uncountably) number of constraints?

I have a constrained maximization problem (maximizing a functional), with number of constraints being uncountable infinite. It looks something like this. I want to maximize the convex functional $C(f)...
1
vote
1answer
64 views

A question about positivity preserving property of semigroup of Laplacian

Sry this the second question from the following article, I am asking in this week. At page 6 (126), 3th line, of the following article. THE HEAT EQUATION WITH A SINGULAR POTENTIAL the authors say ...
3
votes
0answers
43 views

Random Two-Player Asymmetric Game

About half a year ago I asked a question on MSE about a random two player game. At the time, the question received some attention and some progress was made, but was not resolved completely. I have ...
4
votes
1answer
131 views

Compute $ \partial_t\int_{\{u(t,\cdot) >0\} } 1\, dx$ in the sense of distributions where $u$ solves a PDE

Let $u:\Omega\subset \mathbb R^N \to \mathbb R$ be bounded function that solves an evolution PDE $\partial_t u(t,x)= L(u(t,\cdot))(x)$, where $L$ is some elliptic operator. How can I compute the ...
2
votes
0answers
79 views

A question about how to use the convexity condition?

At page 5 (125), seven line after the Proof of Theorem 2.2(i), of the following article. THE HEAT EQUATION WITH A SINGULAR POTENTIAL the authors say that since $p$ is convex, we can deduce that $$ \...