Questions tagged [real-analysis]
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
5,902 questions
4
votes
1
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Order-preserving injection $\iota:[0,1]\to X$ for large $X\subseteq [0,1]$ [duplicate]
Inspired by this older question: If $X\subseteq [0,1]$ such that $|X|=2^{\aleph_0}$, is there necessarily an order-preserving injection $\iota:[0,1]\to X$?
- 54.4k
2
votes
1
answer
241
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Non-topological argument for the non-existence of an order-embedding $\iota: [0,1]\to ([0,1]\setminus \mathbb{Q})$
Starting point. I was toying around with the following question: If $A, B\subseteq [0,1]$ with $A\cup B = [0,1]$, does $[0,1]$ order-embed into at least one of $A, B$? Quickly I focused on $A = [0,1]\...
- 54.4k
0
votes
1
answer
117
views
Cesàro convergence rates for products
If I have given two sequences $(a_n)_n$ and $(b_n)_n$ in $\mathbb{R}$, rates $\alpha, \beta > 0$ and constants $C_1, C_2 > 0$ such that
\begin{equation}
\frac{1}{n} \sum_{k = 0}^{n - 1} a_k \le ...
- 505
2
votes
1
answer
88
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Poincaré inequality on domains bounded in one direction via compactness
Let $\Omega \subset \mathbb{R}^n$ be an open set which is bounded in one direction, i.e. there exist a unit vector $e \in \mathbb{R}^n$ and constants $a<b$ such that
$$
a < x\cdot e < b \...
- 1,185
1
vote
0
answers
105
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Steklov spectrum on the unit ball and its relation to the spherical Laplacian on $\mathbb{S}^3$
Let $B^4 \subset \mathbb R^4$ be the Euclidean unit ball with boundary $S^3$.
Consider the Steklov eigenvalue problem
$$
\begin{cases}
\Delta u = 0 & \text{in } \mathbb{B}^4,\\
\partial_\nu u = \...
- 569
-2
votes
1
answer
333
views
How small can $|\frac{a}{b-c}| + |\frac{b}{a-c}| + |\frac{c}{a-b}|$ get for distinct positive $a,b,c$? [closed]
Let $\newcommand{\Rplus}{\mathbb{R}_+}\Rplus$ denote the set of positive reals. What is the value of
$$\inf\Big\{\Big|\frac{a}{b-c}\Big| + \Big|\frac{b}{a-c}\Big| + \Big|\frac{c}{a-b}\Big|:
a,b,c \in \...
- 54.4k
5
votes
1
answer
294
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Are the powers of a set with the Baire Property in a Polish group a set with the Baire Property?
Let $G$ be a Polish group and let $A\subseteq G$ be a subset with the Baire Property. Does it follow that for any $n\in \mathbb{N}$, the power $A^{n}$ also has the Baire Property?
Of course, if $A$ is ...
5
votes
1
answer
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Can positive decreasing functions $f$ with $f' \geq -f$? be such that $\int_{0}^{x} f(t) \mathrm{d}t$ diverges arbitrarily slow for $x \to \infty$?
Given an increasing function $G \colon [0, \infty) \to [0, \infty)$ with $G(x) \rightarrow \infty$ for $x \to \infty$, is there a (strictly) decreasing (differentiable) function $f \colon [0, \infty) \...
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2
votes
1
answer
161
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Is there a increasing, convex, superlinear $f$ with $c_1 f(x)y \leq f(xy)\leq c_2 f(x)f(y)$ such that $\mathbb{E}[f(X)] < \infty$?
The following version of a de la Vallée Poussin - criterion would be very helpful to me if it would be true. Can you say something about the truth value or give a reference?
Given a positive random ...
- 818
5
votes
1
answer
458
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Continuous vs discrete
Suppose that $z_1,\dotsc,z_n$ are complex numbers situated on the unit circle
$C:=\{z\colon |z|=1\}$. Let $|\cdot|$ denote the uniform, continuous,
probabilistic measure on $C$, and let $\mu$ be the ...
- 23.5k
5
votes
1
answer
231
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Hausdorff dimension of graphs of singular functions
Let $f: \mathbb R^n \to \mathbb R^m$ be continuous, and differentiable almost everywhere with $Df = 0$ almost everywhere.
Question: What is the maximal Hausdorff dimension of the graph of $f$?
- 9,980
4
votes
1
answer
310
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Testing equal count between pairs of sets
For each fixed positive integer $N\in\mathbb{N}$, let's define two sets
\begin{align}
A_N:=&\{(a,b)\in\mathbb{N}^2: N=a(2b-1)+(2a-1)(b-1)\}, \\
B_N:=&\{(c,d)\in\mathbb{N}^2: N=c(2d-1)+(d-1)(d-...
- 43.9k
6
votes
2
answers
377
views
Can a Lipschitz function have derivative 0 on a dense set of small dimension?
Let $f: \mathbb R^n \to \mathbb R$ be Lipschitz continuous. Denote by $Z(f)$ the set on which $f$ is differentiable with derivative $0$.
Suppose $f$ is such that $Z(f)$ is topologically dense.
...
- 9,980
4
votes
1
answer
270
views
Convergence from some recursive hierarchy inequality (II)
Let $f^N_n: [0,T]\to [0,2]$ be continuous functions such that for all $n=1,\ldots, N-1$
$$f_n^N(t)
\le Cn\int_0^t \big(f_{n+1}^N(u)+f_n^N(u)\big)du
+ \frac{Cn^2}{\sqrt{N-n}}t,\quad \forall t\in [0,T]...
- 1,468
1
vote
0
answers
94
views
solve explicitly an integral equation
Let $\lambda$ be the Lebesgue measure. Let $f \in L_1([0,1])$, I would like to construct a $g$ function in $L_1(\mathbb{R}^+)$ such that
$$
\mathbf{1}_{[0,1]}\lambda(dx)\text{-a.e., }\quad f(x)
= \...
- 696
1
vote
1
answer
69
views
Convergence from some recursive hierarchy inequality
Let $f^N_n: [0,T]\to [0,2]$ be continuous functions such that for all $n=1,\ldots, N-1$
$$f_n^N(t)
\le C_{n}\int_0^t \big(f_{n+1}^N(u)+f_n^N(u)\big)du
+ \frac{C_{n}}{\sqrt{N-n}}t,\quad \forall t\in [...
- 1,468
2
votes
0
answers
95
views
Inverting the conditional expectation for some coupling
Let $\nu$ be a probability measure equivalent to $\mathbf{1}_{\mathbb{R}_+}(y) \, \lambda(dy)$. Let $\pi$ be a probability measure on $\mathbb{R}^2$ of second marginal $\nu$, such that $\nu(dy)$-a.e.,
...
- 696
2
votes
0
answers
78
views
Control of an ODE to attain target
Consider the following ODE system:
$$
\begin{aligned}
dR_1(t) &= -\lambda_1 R_1(t)\,dt + \lambda_1 \left( \beta_0 - \beta_1 R_1(t) + \beta_2 R_2(t) \right) C\,dt, \\
dR_2(t) &= -\lambda_2 R_2(...
- 696
0
votes
0
answers
137
views
Integrability of a solution to a boundary value problem
Let $a>0$ and $\lambda$ be real constants and $\mu$ a parameter. Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain with smooth boundary. Assume $\phi\in C^{\infty}_c(\Omega)$ and $\alpha\in C^1(...
- 43.9k
9
votes
1
answer
363
views
Hausdorff dimension of the stretch set of a Lipschitz map
Let $f: \mathbb R^n \to \mathbb R^m$ be a non constant Lipschitz map, for $m < n$. We denote by
$$\text{Lip}(f):= \sup_{x, y \in \mathbb R^n} \frac{|f(x) - f(y)|}{|x - y|}$$
the best Lipschitz ...
- 9,980
0
votes
0
answers
78
views
Characteristic function of a domain to have higher order variation
For any $k\in\mathbb{N}$, the space $BV^k(\mathbb{R}^n)$ is defined as the set of all functions $f\in L^1(\mathbb{R}^n)$ such that there exists a Radon masure $D^\alpha f$ and for any $\phi\in C^\...
- 519
2
votes
1
answer
128
views
How to prove the convergence of the maximum point random variable of random concave function sequence？
I am really wondering how to prove this lemma from the book 'Counting processes and survival analysis'. No need for the first and second point, just the third point, why does the maximum random ...
- 35
6
votes
1
answer
439
views
Staying away from points/minimizing potential energy
Let $x_1,\dotsc,x_n$ be points on $\mathbb{R}/\mathbb{Z}$. Write $\|x-y\|$ for the distance between two points $x,y$ in $\mathbb{R}/\mathbb{Z}$. Let $V$ be one of the following functions $V_j:\mathbb{...
- 22k
2
votes
1
answer
118
views
When is a mapping that is both a measure isomorphism mod 0 and an order isomorphism unique mod 0?
Suppose we have two measure spaces, $(\Omega, \mathscr{F}, \mu)$ and $(\Psi, \mathscr{G}, \nu)$, with $\mu(\Omega) = \nu(\Psi) < \infty$, and we consider the set of measure isomorphisms mod 0 ...
- 319
0
votes
0
answers
103
views
Monotonicity of the convex sum of two binary entropy functions
Let $T$ be some random variable on $[0,1]$, and define
\begin{equation}
\alpha(t) \triangleq \mathbb{E}[T \vert T\le t],\\
\beta(t) \triangleq \mathbb{E}[T \vert T>t], ~t\in[0,1].
\end{equation}
...
- 570
11
votes
2
answers
1k
views
Which sets are "persistently measurable"
Let $\mathcal{E}$ be the class of Lebesgue-measurable subsets of $\mathbb{R}$. The notion of $(\mathcal{E},\mathcal{E})$-measurable function is a bit pathological since lots of continuous functions ...
- 19.6k
5
votes
1
answer
127
views
The definition of Q-curvature on $(S^3,g_0)$ and general definition $n\geq 1$
I am reading the paper "On Conformal Deformations of Metrics on $\mathbb{S}^n$" by Juncheng Wei and Xingwang Xu, and I am trying to understand how equation \eqref{1} is derived. The authors ...
- 569
5
votes
1
answer
295
views
Spectral theorem for multivalued operators
For a compact bounded self-adjoint operator on a Hilbert space, one has the usual spectral theorem: there exists an orthonormal basis of eigenvectors with real eigenvalues converging to $0$, and every ...
- 506
1
vote
1
answer
221
views
Positive roots of real exponents function
Let $$f(x) = a+b(x+3)^r+c(x+3)^r(x+2)^r+d(x+3)^r(x+2)^r(x+1)^r,$$ where $r>1$ and $a,b,c,d$ are real numbers. I need to prove that $f(x)$ can't have more than $3$ positive zeros.
Attempts -
By ...
- 258
0
votes
0
answers
63
views
Concentration for Markov chain with spectral gap
Sub-Gaussian concentration for reversible Markov chains with spectral gap
Setup.
Let $(X_i)_{i\ge1}$ be a stationary, $\pi$-reversible Markov chain on a measurable space with spectral gap $\gamma>0$...
- 1
1
vote
0
answers
64
views
Asymptotic behavior of integrals of fast-oscillating functions via empirical measure convergence
Setting
Let $f\ge 0$ be a measurable function on $[0,\infty)$ and define $g(s,t)=ste^{-s-st}$. For $\lambda>0$ set
$$
I(f,\lambda)=\int_0^\infty g(s,f(\lambda s))ds.
$$
Let $\mu_T$ be the ...
- 19
1
vote
1
answer
112
views
Limit of a sequence defined via return frequencies to a measurable set
Let $(X, \mathcal{B}, \mu)$ be an arbitrary measure space and $T: X \to X$ a measure-preserving transformation. Fix a measurable set $A \in \mathcal{B}$ with $0 < \mu(A) < \infty$, and fix $t \...
- 135
0
votes
1
answer
278
views
Pointwise estimate for a multilinear operator
Let $0<\alpha<1$, and let $$f=\frac{\chi_{[0,1]}}{x^{a}},\quad 0<a<1,$$
$$g=\chi_{[0,1]},$$
$$h(x)=\frac{\chi_{[0,1]}}{|x-1|^{b}},\qquad 0<b<1.$$
I have a multlinear operator on $L^{...
- 868
0
votes
0
answers
54
views
Stability of (stochastic process) separability under composition
The separability that is used in the context of stochastic processes is typically already defined specifically for stochastic processes. I will define this for deterministic functions instead such ...
- 343
2
votes
0
answers
149
views
Possible flaw in Karp and Margulis's proof of convexity of null quadrature domains?
I was reading L. Karp and A. Margulis's proof of the convexity of the complement of a null quadrature domain. This paper is cited by many others, for example, S. Eberle, A. Figalli and G. Weiss.
...
- 189
0
votes
0
answers
140
views
Weak L2 norm in proof of Carleson's theorem
I am reading the paper of Michael Lacey called "Carleson's theorem: proof, complements, variations" 1, on Carleson's theorem in Fourier analysis. Under Lemma 2.18 on page 10 it says: "...
- 237
0
votes
0
answers
126
views
Necessary condition for a partial sum in the limit
For $s\in\mathbb{C}$, let $S(s)$ be a series that checks for the Hurwitz Theorem,
If $S(s)=0$, then by Hurwitz theorem, there exists a sequence $s_n$ so that $s_n \xrightarrow[n\to \infty]{} s$ ...
- 17
2
votes
1
answer
163
views
For which $x, c$ does $\sum_{n=0}^{\infty}\binom{x}{n}\sum_{k=0}^{n} \binom{n}{k}(-1)^{n-k} |k - c|^{\alpha}$ converge?
Consider the Newton series
$$ \sum_{n=0}^{\infty}\binom{x}{n}\sum_{k=0}^{n} \binom{n}{k}(-1)^{n-k}|k - c|^{\alpha} $$
for $x, c\in\mathbb{R}$ and $\alpha\in (0, 1)$. For given values of $c$ and $\...
- 171
4
votes
1
answer
758
views
Quantitative differentiation via Littlewood-Paley theory
In an article by Cheeger, he presents an elementary method, which he refers to as quantitative differentiation, for obtaining a quantitative version of Rademacher's theorem. While these arguments use ...
- 343
5
votes
1
answer
150
views
Do the derivatives of the entropy along the heat flow have alternating signs?
Let $X$ be a random vector in $\mathbb{R}^d$ and $X_t = X + \sqrt {t} Z$, $t >0$, where $Z$ is an independent standard normal vector. Denote by $f_t$, $t>0$, the density of $X_t$ with respect to
...
- 59
0
votes
0
answers
76
views
How to use tube union volume estimates to infer Kakeya conjecture
Wang Hong and Zahl's work "
Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions“ says
Why this is OK ? (We know Kakeya maximum function estimates can ...
- 873
0
votes
0
answers
125
views
Properties of a function defined as an eigenfunction expansion and a generalized Dirichlet series
I am trying to find advice or references concerning the following function defined on $[0,\infty)\times[0,1]$ by the following combined eigenfunction expansion/generalized Dirichlet series:
\begin{...
- 41
3
votes
1
answer
181
views
What are the conditions for a function to be expressed as a sum of multiplicatively separable functions?
Let $f$ be a smooth real-valued function defined on a product domain $ U\times V $ of $\mathbb{R}^{n}\times \mathbb{R}^{m}$. I am interested in the conditions under which f can be written as a finite ...
- 103
5
votes
1
answer
210
views
Remez-type inequality
This question is about a statement on a Remez-type inequality from the paper Polynomial Inequalities on Measurable Sets and Their Applications by M. I. Ganzburg (Constr. Approx. (2001) 17: 275–306).
...
- 53
0
votes
0
answers
53
views
Pointwise supremum representation of bounded functions on a strengthened topology
Let $(X, \tau)$ be a topological space and let $\varphi \colon X \to \mathbb{R}$ be a function. We define $\tau_\varphi$ as the smallest topology containing $\tau$ such that $\varphi$ is continuous. A ...
- 341
1
vote
0
answers
89
views
weak containment of representation
The following is the definition of weak containment:
Let $\pi$ and $\rho$ be two unitary representations of the group $G$. Then, we say $\pi$ is weakly contained in $\rho$ denoted as $\pi \prec \rho$ ...
0
votes
1
answer
80
views
Global Integrability from local integrable function and scaling degree
I am asking myself the following question:
$f \in C^{\infty}(0,1]$ and
$\text{sd(f)}=\inf_{s \in \mathbb{R}} \{ \lim_{\lambda\to 0} \lambda^s f(\lambda x)=0 | \forall x \in (0,1]\}< 1$ (Scaling ...
- 1
2
votes
1
answer
183
views
Non linear Gronwall inequality
Let $K_1,K_2 >0$ be two non negative constant. And I consider $\phi : [0,T] \rightarrow R$ a non negative continuous function such that
\begin{align*}
\forall t \le T, \phi(t) \le K_1 \int_{0}^{t} \...
- 143
5
votes
2
answers
295
views
Any compact subset of smooth functions on the line has a "maximal rate of growth"
Let $K\subset \mathcal{C}^\infty(\mathbb{R})$ be a compact subset in the usual Fréchet topology. Is it known that there exists an everywhere positive function $\varphi\in \mathcal{C}^\infty(\mathbb{R})...
- 1,536
2
votes
0
answers
551
views
Integral of product of $\sin(x/n)/(x/n)$
I recently watched a video of 3b1b's about Borwein integral. I got interested when the integral product goes to infinity
What is the integral of
$$\int_0^{\infty} \prod_{n=1}^{\infty}{\sin(x/n)\over x/...
- 29