# Questions tagged [real-analysis]

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

2,498 questions

**1**

vote

**0**answers

39 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: $\int (P a) \cdot b d\mu= \int a \cdot (b \circ T) d\mu$ ...

**12**

votes

**3**answers

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

**2**answers

103 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

**1**answer

112 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

**0**answers

145 views

+50

### 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\...

**0**

votes

**0**answers

45 views

### switching the order of composition of two functions [on hold]

Let $f_a(x)=x+\exp(-ax)/a$ be a function, defined on $R_+$ for $a$ positive.
Fix $a$ and $b$ positive, can we show that the sign of $f_a\circ f_b - f_b\circ f_a$ does not change on $R_+$, with $\circ$ ...

**1**

vote

**0**answers

29 views

### Convergence estimates for approximation with Gaussians / radial basis functions

tl;dr: Are there known convergence estimates for approximating a function with a radial basis family?
Details: Let $\mathcal{G}$ be a family of radial basis functions, e.g. $\mathcal{G}=\{\exp(-\...

**1**

vote

**1**answer

50 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 ...

**-1**

votes

**0**answers

45 views

### Theorem of Lebesgue point [closed]

Let $h \in {L^1}\left( {0,T} \right)$, then for almost $t \in \left( {0,T} \right)$ prove that
$$n\int\limits_t^{t + \frac{1}{n}} {\left( {1 - n\left( {s - t} \right)} \right)h\left( s \right)ds} \to ...

**3**

votes

**1**answer

78 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

**1**answer

122 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

**1**answer

146 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

**2**answers

153 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

**1**answer

84 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 $T_f \in L^p(\mathbb R^d)$ ...

**9**

votes

**0**answers

110 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

**0**answers

48 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 ...

**1**

vote

**0**answers

53 views

### Regularity of superposition operator generated by function between Banach spaces

Let $E$, $F$ be Banach spaces, $D$ be open in $E$, and $K=[0,1]$. Given $\varphi\colon K\times D\to F$ I call
$$
\varphi^\sharp\colon D^K\to F^K,\quad u\mapsto \varphi(\cdot,u(\cdot))
$$
the ...

**0**

votes

**1**answer

49 views

### Empty interior lack of minima

Suppose that $U \subseteq \mathbb{R}^d$, and satsifies
$U$ is dense in $\mathbb{R}^d$,
U has empty interior,
Then is it possible that
$$
\inf_{x \in U} f(x) >\inf_{x \in \mathbb{R}^d} f(x),
$$
...

**5**

votes

**1**answer

277 views

### Logarithmic and polynomial functions with two roots

This is a question that I came across a few days ago，Although it is not particularly like a research problem, the following problem is that I study the zero distribution of a class of elementary ...

**0**

votes

**1**answer

70 views

### Existence of a certain type of function

Trying to find functions with the given property:
Given $M>0, K$ compact in $\mathbf{R^n}$,$f:U\rightarrow\mathbf{R}$ a $C^2$ function, where $U$ open in $\mathbf{R^n}$ and $K\subset U$such that $...

**2**

votes

**2**answers

193 views

### Ergodic theorem and products

If $f_n (\omega) = \sum_{i=1}^n f_1 (T^i \omega)$ and $T$ is an ergodic action with respect to the measure $\mu$ then it is know as Birkhoff's theorem that
$$ \lim_{n \rightarrow \infty} \frac{f_n}{...

**-1**

votes

**0**answers

55 views

### interchange between integral and limsup

Let $\{u_n\}_{n\in\mathbb N}$ be a sequence in $L^p$ (with $p>1$) such that $\|u_n\|$ converges to $+\infty$ and $u_n/\|u_n\|$ weakly converges to zero.
Under what conditions
$$
\int \limsup_n f(...

**0**

votes

**0**answers

32 views

### Feller semigroups and fractional operators

Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...

**4**

votes

**1**answer

148 views

### Proving two inequalities involving the gamma and digamma functions

I'm having trouble proving the following inequality:
$$\forall p>1 \quad \forall m\geq 0 \quad \dfrac{m^2\Gamma(\dfrac{2m}{p})\Gamma(\dfrac{2m}{q})}{\Gamma(\dfrac{2m+2}{p})\Gamma(\dfrac{2m+2}{q})}\...

**4**

votes

**1**answer

228 views

### Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $?

This question is an old question from mathstackexchange.
Let $f_- (n) = \Pi_{i=0}^n ( \sin(i) - \frac{5}{4}) $
And let
$ f_+(m) = \Pi_{i=0}^m ( \sin(i) + \frac{5}{4} ) $
It appears that
$$\sup f_-...

**2**

votes

**0**answers

91 views

### Can a local extremum of a function be an asymptotically stable equilibrium of corresponding gradient dynamics?

Let's first describe the setup: we consider a(say smooth enough) function $f: \mathbb{R}^d \to \mathbb{R}$ and write it as $(x,y) \to f(x,y)$, where $x \in \mathbb{R}^{d_x}$, $y \in \mathbb{R}^{d_y}$ ...

**6**

votes

**1**answer

357 views

### Continuous functions of three variables as superpositions of two variable functions

Could we always locally represent a continuous function $F(x,y,z)$ in the form of $g\left(f(x,y),z\right)$ for suitable continuous functions $f$, $g$ of two variables? I am aware of Vladimir Arnold's ...

**27**

votes

**2**answers

728 views

+50

### Rademacher theorem

If $f:\mathbb{R}^n\to\mathbb{R}^m$ is of class $C^1$ and $\operatorname{rank} Df(x_o)=k$, then clearly $\operatorname{rank} Df\geq k$ in a neighborhood of $x_o$. It is not particularly difficult to ...

**1**

vote

**0**answers

30 views

### How is the dominated convergence theorem applied in the proof of Lyapunov’s criterion?

Let $$\Gamma(f,g):=\frac12f'g'\;\;\;\text{for }f,g\in C^1(\mathbb R),$$ $\mu$ be a probability measure on $(\mathbb R,\mathcal B(\mathbb R))$ with a continuously differentiable and positive density $\...

**5**

votes

**0**answers

93 views

### Is there any geometrical/homological intuition behind symmetrized gradient?

The gradient/differential/exterior differential/divergence/curl are all strictly related first order differential operators. As far as I understood, they are the base of (co)homological theories in ...

**0**

votes

**0**answers

83 views

### A certain ratio condition for polynomials with real coefficients

Let $p:\mathbb{C} \longrightarrow \mathbb{C}$ be a polynomial with real coefficients and suppose that $p$ satisfies
\begin{equation}
\frac{p(y)}{y} \le \frac{p(x)}{x} \tag{*} \label{ratcond}
\end{...

**44**

votes

**4**answers

7k views

### Why isn't integral defined as the area under the graph of function?

In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also ...

**1**

vote

**0**answers

83 views

### Baker map-like problem

Let $S =[0, 1]^2$ denote the unit square in $\mathbb R^{2}$. For any set $A$ in $S$ let $A^{c}$ denote its compliment in $S$, and $cl$ $A$ it’s closure in $S$. Given a measurable map $g: W \to V$ ...

**0**

votes

**1**answer

101 views

### Short question on functions of bounded variation

For a function $f: \mathbb R \to \mathbb R$ of locally bounded variation, when is $$\liminf_{e \to 0} V(f)[x, x+e]/e $$finite everywhere? Here $V(f)[a, b]$ denotes the total variation of the function ...

**6**

votes

**2**answers

191 views

### How can we show that if $f$ is convex, then $\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0$?

Let $d\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1.$$ How can we show that $$\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0?$$ $f$ ...

**0**

votes

**0**answers

26 views

### Cross entropy loss is not twice differentiable?

I was reading a recent theory paper in machine learning by Kenji Kawaguchi and Leslie Pack Kaelbling
https://arxiv.org/pdf/1901.00279.pdf
and the authors seem to suggest in section 2.2 that cross-...

**0**

votes

**0**answers

81 views

### Harnack Inequality for uniformly elliptic PDE via constructing a singularity

I am trying to prove a Harnack inequality for a nonnegative subsolution $u \in H^1(B_2)$ to the PDE $\text{div}(A Du) \ge 0$, where $A = A(x)$ is uniformly elliptic. The proof outline I am following ...

**2**

votes

**1**answer

189 views

### Density in fractional Sobolev space

Suppose $s∈(0,1)$, $D$ is an open set in $\mathbb{R}^d$. Define
$$
H^s=(1−\Delta)^{-s/2}L^2\left(\mathbb{R}^d\right),
$$
$$
H^s_D=\left\{f\in H^s:f=0 \mbox{ a.e. on } D^c\right\}.
$$
Q: Is $C^\...

**3**

votes

**1**answer

123 views

### Uniformly differentiable functions

Note: Here all functions are $\mathbb R \to \mathbb R$. $Id$ denotes the identity function.
Let $g_i$ be a family of functions indexed by some (potentially uncountable) index set $I$. Given a ...

**7**

votes

**1**answer

203 views

### A strictly decreasing function between uncountable subsets of the reals

By a standard technique of inductive killing everything relevant (in this case decreasing homeomorphisms between uncountable $G_\delta$-subsets of the real line) it is possible to prove the following ...

**0**

votes

**0**answers

94 views

### Quantitative Lusin’s theorem

We consider only the set $M$ of a.e. essentially locally bounded measurable functions $[0, 1] \to \mathbb R$. Here $m(S)$ denotes the Lebesgue measure of $S$.
Let $f$ be measurable. For every $e$ in $...

**6**

votes

**1**answer

310 views

### Looking for infinite series resembling an exponential

I'm looking for some $f(x)$ that has the following property:
$\sum_{x=1}^\infty f(kx) = r^k$
for some real $0 < r < 1$, and at least for strictly positive integer $k$.
Does such an $f(x)$ ...

**0**

votes

**0**answers

103 views

### How to prove the uniform boundary？

Assume that $a_1<1$, $a_3<1,$ $a_1+a_2+a_5>1$, $a_3+a_4+a_5>1,$ $a_1+a_2+a_3+a_4+a_5>2,$ and $b_1, b_2>0$. For $x,y>0,$ define a fucntion
$$H(x,y)=\frac{x^{\frac{1}{2}}\int_0^{\...

**0**

votes

**1**answer

99 views

### Integrable function [closed]

Suppose that $a, b, c_1$ and $c_2$ are real constant.
Is there the necessary and sufficient conditions of $a ,b, c_1,c_2 $ for the following integration is integrable? i.e.
$$\int_1^{\infty}\int_1^{\...

**12**

votes

**2**answers

227 views

### Show that $f(t)=\sum_{i=1}^n a_i e^{-(x_i-t)^2}-c$ has at most $2n$ zeros

Let
\begin{align}
f(t)=\sum_{i=1}^n a_i e^{-(x_i-t)^2}-c
\end{align}
where $x_1<x_2<...< x_n$ and $a_i>0$. For some positive constant $c$.
Can we show that $f(t)$ has at most $2n$ ...

**1**

vote

**0**answers

41 views

### Continuous injection of metric ball into Euclidean ball

This is a follow-up to this post.
Suppose that $(X,d_X)$ is a compact metric space with (finite) metric-capacity, defined by
$$
\kappa_X(\epsilon)\triangleq\sup\left\{
k : \exists x_0,\dots,x_k \...

**2**

votes

**1**answer

176 views

### Necessary and Sufficient conditions for integrable function [closed]

Suppose that $a, b$ and $c$ are constant.
Is there the necessary and sufficient conditions of $a ,b, c$ for the following integration is integrable? i.e.
$$\int_0^\infty \int_0^\infty \int_0^\infty ...

**0**

votes

**1**answer

185 views

### Norm of a tuple of operators

Let $F$ be a complex Hilbert space and $\mathcal{B}(F)$ be the algebra of all bounded linear operators on $F$.
For ${\bf A} = (A_1,...,A_d) \in \mathcal{B}(F)^d$, the norm of ${\bf A}$ is given by
$...

**1**

vote

**1**answer

81 views

### Continuous inclusion of metric spaces of smaller capacity

If $(X,d_X)$ is a compact metric space, and $(Y,d)$ is another metric space. Moreover, suppose that the metric capacity of $(Y,d)$ is at-least that of $(X,d_X)$, that is
$$
\kappa_X(\epsilon)\leq \...

**0**

votes

**0**answers

145 views

### How do we introduce a signed finite measure on the space of curves confined into the box $[0,1]^{n}$?

Given $\Omega_{n} = \{\alpha:[0,1]\rightarrow[0,1]^{n}\,|\,\alpha\,\,\text{is smooth}\}$, consider the equivalence relation:
\begin{align*}
& \alpha_{1} \sim \alpha_{2} \Leftrightarrow \int_{0}^{1}...