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Questions tagged [real-analysis]

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

1
vote
0answers
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
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
2answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
2answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
2answers
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
0answers
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
0answers
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
0answers
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
4answers
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
0answers
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
1answer
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
2answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
2answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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}...