Questions tagged [real-analysis]
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
2,703
questions
0
votes
0answers
36 views
Stochastic Fixpoint Approximations of Contractions
Context/Introduction
Consider a contraction $f\colon\mathbb{R}^S\to\mathbb{R}^S$ with $f(X^*)=X^*$ where function evaluation at a certain position is only possible with some stochastic error. Where Y(...
5
votes
1answer
195 views
Is there any continuous ternary function which can not be represented by composition of continuous binary functions?
Let $f : X^3 \rightarrow X$.
If $X$ is $\mathbb Z$, then there will be a couple of functions $g,h$ from $\mathbb Z^2$ to $\mathbb Z$ that satisfies $f(x,y,z) = g(h(x,y),z)$ since there is a bijection ...
3
votes
0answers
138 views
Largest weak(-like) topology with respect to which continuous functions are dense in the space of Borel functions
Let $X$ denote the space of bounded Borel functions $f\colon [0,1] \to \mathbb{R}$. Let $M$ denote the space of finite Borel measures on $[0,1]$. What is the largest family $F \subset M$ such that for ...
7
votes
1answer
211 views
Compactness of set of indicator functions
Let $\chi_A(x)$ denote an indicator function on $A\subset [0,1]$. Consider the set
$$K=\{\chi_A(x): \text{ A is Lebesgue measurable in }[0,1]\}.$$
Is this set compact in $L^\infty(0,1)$ with respect ...
-2
votes
1answer
162 views
About infinite products and Euler Gamma functions [closed]
I am interested in knowing how to calculate infinite products like (or reading any reference about it):
$$\prod_{j=1}^{\infty}\left( 1-\left( \frac{x}{a+j\pi} \right) ^2 \right)$$
Inserting it into ...
1
vote
2answers
155 views
Integral formula involving Legendre polynomial
I would like a proof of the following equation below, where $(P_n)$ denotes Legendre polynomials. I guess this formula to hold; I checked the first several values.
\begin{equation}
\int_{-1}^{1}\sqrt{...
0
votes
0answers
65 views
Does there exist $\alpha>0, \beta\in (0,1)$ such that $\dfrac{\sum_{k=1}^n a_k}{n}\le \alpha (a_1\cdots a_n)^{1/n} + \beta \max_i(a_i)$ holds?
Let $a_1\ge a_2\ge \cdots\ge a_n\ge 0$ be given non-negative numbers. My question is the following:
Is there any $\beta \in(0,1),\ \alpha>0$, such that $$\dfrac{a_1+\cdots+a_n}{n}\le \alpha (a_1\...
3
votes
1answer
204 views
On convex functions which are non constant on every segment
I have been studying for the last few weeks the paper Dirichlet problem for demi-coercive functionals by Anzellotti, Buttazzo and Dal Maso. In this work the authors introduced and studied the ...
1
vote
1answer
118 views
Trajectory leaving a set
Consider the differential equation $\dot{x}=f(x)$, where $f: \mathbb{R}^2 \to \mathbb{R}^2$ is smooth. Given a set $A \subset \mathbb{R}^2$, are there some results saying that whenever $x(0) \in A$, ...
2
votes
1answer
82 views
Uniformly Converging Metrization of Uniform Structure
This is related to trying to resolve the currently faulty second part of my answer to this question, but is by itself a purely real analysis question.
Let $X$ be a set with a uniform structure ...
4
votes
1answer
284 views
Nearly eventually almost periodic functions
Call a function $f: [0, \infty) \to \mathbb R$ nearly eventually almost periodic with period $p > 0$ if for a.e. $x \in [0, p)$, the sequence ${f(x + np)}_{n \in \mathbb N}$ converges.
Suppose $f: ...
6
votes
2answers
270 views
Maximize $L^p$ norm over sphere
For $p \in \mathbb{R}$, consider the function
$$F_p(\lambda_1, \dots, \lambda_n) = \lambda_1^p + \dots + \lambda_n^p.$$
My goal is to maximize this function under the constraints that
$$ \lambda_1^2 +...
1
vote
1answer
239 views
Prove that $x^{y^x}>y^{x^y}$ for $x>y>0.$ [closed]
Let $x>y>0$. Prove that $$x^{y^x}>y^{x^y}.$$
My attempts:
Let $1>x>y>0$.
In this case it's enough to prove that $$y^x<x^y$$ or
$$x\ln\frac{1}{y}>y\ln\frac{1}{x},$$ which is ...
2
votes
1answer
100 views
How to compute volume of parametric regions?
I guess this is something pretty standard in calculus, but I was unable to google the answer.
Assume I have unit hypercube $C_n = [0,1]^n$. I also have a function $f : \mathbb{R}^n \to \mathbb{R}^{n+...
6
votes
3answers
287 views
Computing the volume of a simplex-like object with constraints
For any $n \geq 2$, let
$$D_n [r , (a_1, b_1 ) , \ldots , (a_n, b_n) ] =
\{ (x_1 , \ldots , x_n ) \in \mathbb R^n \mid
\sum_i x_i = r \mbox{ and } b_i \geq x_i \geq a_i \, \forall i \},$$
where $r \...
4
votes
1answer
101 views
Construct a dense family in $W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$ based on distance functions
Let $\Omega$ be a sufficiently regular domain, for example $\Omega=B(0,r)$, $r>0$, and $m(x)=\textrm{dist}(x,\partial \Omega)$ be the distance to boundary function.
Suppose $\mathcal{F}$ is a ...
2
votes
1answer
214 views
An isoperimetric inequality for curve in the plane?
Let $f(x,y)=0$ be a (smooth) simple closed curve $C$ on the plane and $R$ the region bounded by $C$ (appropriately oriented). Assume the origin lies in the interior of $R$.
QUESTION. Let $r=\sqrt{x^...
4
votes
0answers
85 views
Perturbation theory compact operator
Let $K$ be a compact self-adjoint operator on a Hilbert space $H$ such that for some normalized $x \in H$ and $\lambda \in \mathbb C:$
$\Vert Kx-\lambda x \Vert \le \varepsilon.$
It is well-known ...
9
votes
2answers
347 views
Small uncountable cardinals related to $\sigma$-continuity
A function $f:X\to Y$ is defined to be
$\sigma$-continuous (resp. $\bar \sigma$-continuous) if there exists a countable (closed) cover $\mathcal C$ of $X$ such that the restriction $f{\restriction}C$ ...
2
votes
1answer
165 views
Generalised raindrop function
Given a sequence of reals $(a_n)_{n > 0}$, let $f: [0, 1] \to R$ be the generalised raindrop function defined:
$f(x) = a_q$ if $x$ is rational, with denominator $q$ in lowest form; $0$ otherwise.
...
0
votes
1answer
113 views
Reference request: Baire class 2 functions
There are many articles on Baire 1 functions, but not many on Baire 2 and above. Where can I find a nice comprehensive survey of them?
2
votes
1answer
351 views
Formal group law and Koenigs function conjecture?
Let $f(x,y)$ be a symmetric real function and a formal group law
$$G(x + y) = f(G(x),G(y)). \tag{1}$$
Consider the equation
$$ h(2x) = f(h(x),h(x)) = A(h(x)). \tag{2}$$
This equation has many ...
7
votes
1answer
512 views
Eventually almost periodic functions
Call a function $f: [0, \infty) \to \mathbb R$ eventually almost periodic with period $p > 0$ if for all $x \in [0, p)$, the sequence ${f(x + np)}_{n \in \mathbb N}$ converges.
Suppose $f: [0, \...
2
votes
1answer
60 views
Decaying of a certain ratio of binomial sums
Consider the two sequences
$$a(n)=\sum_{k=1}^n\binom{n}k\sum_{j=1}^{k/2}\binom{k}{2j}\frac{(2j)!}{j!}$$
and
$$b(n)=\sum_{k=0}^n\binom{n}k^2k!$$
QUESTION. Is this true?
$$\frac{a(n)}{b(n)}\...
5
votes
1answer
148 views
Binomial Distributions and Inequality
Suppose $c>2$ is a real number, and $x$ solves equation $f(\frac{1}{2}+x,m,2m)=(2m+1)cx$ for some $m>0$ integer number, where $f$ is probability mass function for binomial distribution, with ...
0
votes
1answer
84 views
“Find a representation [using Mellin transform] of 𝑓(𝜇,𝛽) as Gauss hypergeometric function in variable 𝜇”
This is a follow-up to the first comment (by Nemo) to the posting Compute the two-fold partial integral, where the three-fold full integral is known . (I also just asked this as a comment to that ...
1
vote
1answer
351 views
Convergence of a certain sum
Suppose $ g_i: [0, 1] \to \Bbb R$, $i\in\Bbb N$, are $C^1$ functions and that there is some $c > 0$ such that for every $0 < \epsilon < c$, the functions
$$
s(\epsilon)_i := \sum_{k=0}^i {\...
10
votes
1answer
275 views
Estimation of the Gromov–Wasserstein distance of spheres
Let $(X,d_X,\mu_X)$ and $(Y,d_Y,\mu_Y)$ be two metric measure spaces. A probability measure $\mu$ over $X\times Y$ is called a coupling if $(\pi_1)_\sharp \mu=\mu_X$ and $(\pi_2)_\sharp \mu=\mu_Y$. We ...
6
votes
1answer
166 views
Eigenvalue estimates for operator perturbations
I edited the question to a general mathematical question, since I found the answer in Carlo Beenakker's reference and think that my initial question was mathematically misleading.
What was behind ...
2
votes
1answer
161 views
Corollary of the Malgrange Preparation Theorem
(This question was previously posted on MSE and I decided to post it here too.)
Let $f:\mathbb{R}\times \mathbb{R}^n \to \mathbb{R}$ be a smooth function, such that
$$f(0,0)=0,\ \frac{\partial f}{\...
-1
votes
1answer
61 views
On probabilistic extension for Bernstein polynomials
Suppose $X_m\sim p_m(x)$ is a discrete distribution on $[0,1]$ where the value takes multipliers of $\frac{1}{m}$ (e.g., $p_m(x=\frac{k}{m})=\frac{1}{m+1})$. Suppose $p(x)=\lim\limits_{m\rightarrow\...
1
vote
1answer
119 views
Expansion of an integral
I have an integral of the form
$$ I=\int_0^{\infty}{dx}\ln \bigg(1+\exp(-\frac{f(x)}{a})\bigg) $$ where $a$ is a positive constant and $f(x)$ is a regular and positive function such that $I$ is finite ...
8
votes
4answers
269 views
Defining the value of a distribution at a point
Let $\omega\in D'(\mathbb R^n)$ be a distribution and $p\in \mathbb R^n$. If there is an open set $U\subset \mathbb R^n$ containing $p$ such that $\omega|_U$ is given by a continuous function $f\in C(...
7
votes
0answers
384 views
A discontinuous construction
Suppose we have an uncountable family of functions $f_r: [0, 1] \to R$ indexed by $r \in [0, 1]$ such that for each $r$, there exists a unique $x$ in $[0, 1]$ such that $f_{r}$ is positive on $x$ and $...
8
votes
1answer
320 views
Simultaneous Riemann Rearrangement
Here all functions are $\mathbb R \to \mathbb R$.
Fix $M$ a positive integer. For $i = 0, 1, ..., M,$ let $f_0 = Id$, and the other $f_i$ be continuous functions such that for all $0 \leq k < M$, $...
0
votes
1answer
131 views
A functional equation in real analysis
For what function $u:[0,1]\rightarrow R$ with bounded derivative, such that $\forall p\in[0,1]$,
$\lim\limits_{n\rightarrow\infty}\sum\limits_{k=0}^n\binom{n}{k}p^k(1-p)^{n-k}u(\frac{k}{n})=u(p)$
...
1
vote
0answers
73 views
Coboundary in the slow mixing systems
Given dynamical system $(X, T, \mu)$, $\mu$ is probability, $\mu \circ T =\mu$, $T$'s transfer operator $P$ is defined by following relation: $\int (P a) \cdot b d\mu= \int a \cdot (b \circ T) d\mu$ ...
0
votes
1answer
72 views
Inequality involving product-of-minus vs minus-of-product for positive integers
I'm encountering this inequality for dimensionality reduction problem. The simplified form looks as follows:
Consider positive integers $a_1$, $a_2$, $b_1$ and $b_2$ where $a_1>b_1$ and $a_2>...
12
votes
3answers
2k views
Does anyone recognize this inequality?
In some paper the authors make use of the following inequality without further explanation: Let $x\in\mathbb{R}^n$ with $x_1\le\cdots\le x_n$ and $\alpha\in[0,1]^n$ with $\sum_{i=1}^n \alpha_i=N\in\{1,...
4
votes
3answers
167 views
A Riccati type integral inequality
Let $x(t),t\in [1,\infty)$ be a nondecreasing positive function satisfying the following inequality:
$$
x'(t) \le \int_t^{+\infty} x(s)\frac{k(s)}{s^2}\,ds,
$$
for any $t \ge 1$, where $k(t),t\in [1,\...
2
votes
1answer
122 views
Must $q$ be analytic?
I have a continuous function $q:\mathbb{R}^+ \to \mathbb{R}^+$. An interesting property of this function is that
$$F(s) = \frac{e^{-q(s)}q(s+1)}{1-e^{s-q(s)}}$$
which also takes $\mathbb{R}^+ \to \...
2
votes
0answers
177 views
Shattering with sinusoids
Let $d \geq 2$ and $K$ some positive integer. Consider distinct points $\theta_1, \ldots, \theta_K\in \mathbb{T}^d$ and (not necessarily distinct) $z_1, \ldots, z_K \in \{-1,1\}$ such that $\sum\...
1
vote
1answer
64 views
Good UPPER bounds for $\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i$ where $(p_i)_i$ is a probability vector
Let $x=(z_1,\ldots,z_n)$ be real vector and $(p_1,\ldots,p_n)$ be a probability vector.
Question
$\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i \le ???$
Observation
This paper allows us to ...
3
votes
1answer
84 views
A problem with sequences with composition of $\log$s
If $(a_n)_{n \ge 1}$ is a non-negative sequence s.t., $$\sum\limits_{n = c_k}^\infty \frac{a_n}{\log^{(k)} n} < \infty, \, \forall k \ge 1 \overset{?}{\implies} \sum\limits_{n \ge 1} a_n < \...
1
vote
1answer
139 views
Why is this series summable?
Let $\delta, \epsilon \in \mathbb{R}$, $\delta >0$, $\epsilon >0$.
Let $\{ a_k\}^\infty$,$\{ b_k\}^\infty$ be sequences of positive integers such that
$\lim \sup_{k \rightarrow \infty} \frac{...
4
votes
1answer
161 views
Graphs that are not $\mathbb{R}^2$-realizable
We say that a finite, simple, undirected graph $G=(V,E)$ is $\mathbb{R}^2$-realizable if there is an injective map $\varphi:V\to \mathbb{R}^2$ such that for $v\neq w \in V$ we have $\{v,w\} \in E$ if ...
4
votes
2answers
173 views
Can one realize this as an ergodic process?
Consider the lattice $\mathbb Z^2$ and take iid random variables $Y_e$ on all edges $e$ of the graph.
We then define random variables $X_i:=\sum_{e \text{ adjacent to } i}Y_e.$
In other words: For ...
5
votes
1answer
126 views
Estimate of the difference quotients in terms of an $L^{1,\infty}$ function
Let $f \colon \mathbb R^d \to \mathbb R$ be a measurable function. Consider the following property:
(P) there exist a negligible set $N \subset \mathbb R^d$ and function $T_f \in L^p(\mathbb R^d)$ ...
11
votes
0answers
142 views
Maximizing an integral w.r.t. a measure on the unit sphere
I would like to know if the answer to the following question is known.
Let $d \ge 3$. What is the value of
$$
\theta(d) := \max_{\mu} \int_{S^{d-1}} \int_{S^{d-1}} \cdots \int_{S^{d-1}} |x_1 \...
3
votes
0answers
51 views
system of Euler like ode's
I am interested in solving some linear elliptic system like
$$ -\Delta \phi(x) + \frac{C_1 \psi(x)}{|x|^\beta} =f(x)$$
$$ -\Delta \psi(x) + \frac{C_2 \phi(x)}{|x|^\alpha} =g(x)$$ in $B_1$ (the ...
