All Questions
5,672 questions
2
votes
0
answers
56
views
Inequality for a weighted bilinear form in Fourier variables
Let $\phi:\Bbb R^d\setminus\{0\}\to [0,\infty)$ be a continuous and symmetric, i.e., $\phi(-\xi)=\phi(\xi)$. Let $F:\Bbb R\to[0,\infty)$ be increasing and $L-$Lipschitz with $F(0)=0$.
Consider the ...
- 1,101
0
votes
0
answers
107
views
$\log$-classes of irrationals
Let $\mathbb{N}$ denote the set of non-negative integers. For $A\subseteq \mathbb{N}$ we let the (upper) density of $A$ be defined by $d^+(A) = \lim\sup_{n\to\infty} \frac {|A\cap \{0,\ldots, n\}|}{n+...
- 48.9k
8
votes
3
answers
701
views
Regularity of Newtonian potential along smooth boundary
Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with $C^\infty$ boundary, $n\ge 3$. Define
$$V(z)=\int_\Omega \frac{1}{|z-y|^{n-2}}dy$$
Is it true that $V(z) \in C^{\infty}(\partial \Omega)$?
...
- 1,350
4
votes
0
answers
114
views
Find at least one square-boxed subcontinuum
Recall that a plane continuum is a closed, bounded,
connected subset of the plane.
It is non-degenerate if it contains at least two points.
(We may sometimes just say "continuum" even if
we ...
- 1,375
4
votes
1
answer
202
views
Finding a real-analytic diffeomorphism
Let $U_1\subset \mathbb R^3$ be a simply connected bounded open set with a smooth boundary and let $U_2$ be a neighborhood of $U_1$. Does there exist a real-analytic diffeomorphism $\psi: U_2 \to W_2$ ...
- 4,143
5
votes
2
answers
564
views
Stone-Weierstrass without the "subalgebra" condition
Suppose I consider $C_0(\mathbb{N})$ consisting of function on the natural numbers vanishing at $\infty$. For an irrational $1<\alpha<2$, let $p_{m\alpha}(\cdot)$ be the function $p_{m\alpha}(n)=...
- 161
2
votes
0
answers
66
views
K functional of $L^{1,\infty}$ and $L^\infty$ real interpolation
I want to know where can I find how to compute, given a function $f$ and $t>0$
$$K(t,f;L^{1,\infty};L^\infty)$$
where $L^{1,\infty}=\sup_{t>0}t f^*(t)$
and $K$ is the Petree K-functional ...
- 21
3
votes
0
answers
176
views
A variant of the Laplace principle
$\newcommand{\R}{\mathbb R}\newcommand{\eps}{\varepsilon}$In $\R^d$ I am given a sequence of smooth functions $f_\eps(x)$ that converges uniformly to some $f(x)$, which is assumed to be a good rate ...
- 5,380
2
votes
0
answers
159
views
Are there hereditarily square-boxed plane continua?
A plane continuum is a bounded, closed and connected subset of the plane.
A bounding box $B$ for a plane continuum $C$ is
a rectangle $B=[a,b]\times[c,d]$ (including sides and interior)
such that $C$ ...
- 1,375
0
votes
1
answer
61
views
What can we say about the order of convergence of a critical point of Gaussian mixture density to its limit when the parameter $h$ goes to $0?$
Density of Gaussian mixture with $n$ components is given by:
$$f(x):=C \sum_{i=1}^{n}e^{-\frac{1}{2}||\frac{x-x_i}{h}||^2}, x_i \in \mathbb{R}^d, h > 0$$
where $C$ is a normalization constant ...
- 1,467
8
votes
2
answers
655
views
An extension problem
Let $\Omega$ be an open subset of $\mathbb R^n$ for $n \geq 2$, and $p \in \Omega$.
Let $k$ be a positive integer. Suppose that $f: \Omega \setminus \{p\} \to \mathbb R$ is in $C^k$, and $\lim_{x \to ...
- 6,215
1
vote
1
answer
139
views
Which kind of convergence can we get from Laplace transform convergence?
This question is a related question see this post Vague convergence VS Laplace transform convergence. But now we assume that
\begin{equation}
\int_0^\infty e^{-sx}\mu_n(dx)\to \int_0^\infty e^{-sx}...
6
votes
1
answer
408
views
On an asymptotic integral decay
Let $f\in C^{\infty}([-1,0])$ be real-valued and suppose that
$$ \left| \int_{-1}^{0} f(t)\,e^{\lambda t} \,{\rm d} t \right| \leq e^{-\sqrt{\lambda}},$$
for all $\lambda > 0$. Does it follow that $...
- 4,143
2
votes
1
answer
309
views
Reverse estimate on the Riesz potential $I_\alpha : L^{n/\alpha}\to \mathrm{BMO}$
$\newcommand\BMO{\mathrm{BMO}}$Consider the Riesz potential on $\mathbb{R}^n$ given by
$$
I_\alpha f(x) = c_{n,\alpha} \int_{\mathbb{R}^n} \frac{f(y)}{\lvert x-y\rvert^{n-\alpha}} dy.
$$
It is known ...
- 363
1
vote
1
answer
161
views
Is there a two-dimensional unimodal function with fractal level sets
Is there an open simply connected $U\subset\mathbb{R}^2$ and a continuous non-constant function $f: U\to \mathbb{R}$,
such that for all $c\in \mathbb{R}$ both sets
$$ f_{<c}~=~ f^{-1}\left( (-\...
- 1,676
2
votes
0
answers
147
views
"Almost rational" irrational
This is a follow-up to an older question.
Let $r\in \mathbb{R}\setminus\mathbb{Q}$, let $\mathbb{N}$ denote the set of non-negative integers and let $\mathbb{N}_+=\mathbb{N}\setminus\{0\}$. For $n\in\...
- 48.9k
5
votes
1
answer
340
views
How to give a counterexample of this estimate related to Paley-Littlewood theorem?
I am studying Paley-Littlewood theorem in Harmonic analysis, and I met an exercise. I would like to construct a function $f$ as a counterexample to show that the inequality
\begin{equation}
\| f \|^...
- 187
2
votes
1
answer
152
views
The asymptotic behavior of $F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}$, $b>a>0$
Let $b>a>0$. Given $ \lambda>0$, what is the asymptotic behavior of
$$F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}$$
as $\lambda\to 0^{+}$ and as $\lambda \...
- 852
1
vote
1
answer
223
views
Analytic expression for $\int_0^\infty \mathrm{d}p\frac{e^{-p \sin (\phi )} \sin (p \cos (\phi ))}{p \left(e^{c p}+1\right)}$
I am looking for ways to do this integration analytically
\begin{equation}
\int_0^\infty \mathrm{d}p\frac{e^{-p \sin (\phi )} \sin (p \cos (\phi ))}{p \left(e^{c p}+1\right)}
\end{equation}
For ...
- 199
9
votes
0
answers
165
views
Changing coordinate to smoothen a function
Let $U\subset \mathbb{R}^2$ be an open neighborhood of the origin $0$, and let $f:U\to \mathbb{R}$ be a continuous function which is smooth on $U\setminus\left\{0\right\}$. Let's say that $f$ is ...
- 2,178
1
vote
0
answers
99
views
Proving more stronger fomula for discrepancy of a sequence [closed]
I am reading famous book about uniform distribution of sequences by Kuipers and Niederreiter and have questions about solving below exercise from that book. Before going to main exercise I will write ...
- 111
1
vote
0
answers
165
views
Question about stationary phase with Hessian close to $0$
Let $\phi$ be a smooth real function in one variable and say $w$ is a smooth function with compact support say $[- 1, 1]$. Let me define
$$
I_{\lambda} = \int_{\mathbb{R}} w(t) e^{i \lambda \phi(t)} ...
- 3,625
1
vote
1
answer
301
views
Vague convergence VS Laplace transform convergence?
If we assume that $\int_0^\infty e^{-sx}\mu_n(dx)\to \int_0^\infty e^{-sx}\mu(dx), \forall s\geq0$, it is possible to show that $\mu_n\to\mu$ vaguely. Where $\mu_n$ is a measure. Please check here for ...
1
vote
0
answers
115
views
Value of $\pi$ and algorithm for Bernoulli numbers
Chowla and Hartung provide an "algorithm" for computing Bernoulli numbers in this paper.
In particular, if the Bernoulli numbers are defined by
$$\frac{x}{e^x-1}=1-\frac{x}{2}+\sum_{n=1}^\...
- 41
2
votes
1
answer
106
views
Is $f^{-a}$ locally integrable if $f\geq 0$ has a unique stationary point ( a minimum) at which the Hessian is positive definite, $0<a<d/2$
Let $0<a<d/2$, let $B$ be the unit ball in $\mathbb{R}^{d}$ centered at the origin, and let $f:B \to [0,\infty[$ be a a smooth function such that
(1) $f(x)\geq f(0).$
(2) $\nabla f(x)\neq 0,\...
- 852
3
votes
1
answer
147
views
What exactly is the topology on $O_M$ that makes the convolution map $S \times S' \to O_M$ hypocontinuous?
Let $O_M(\mathbb{R}^n):= \mathcal{S}'(\mathbb{R}^n) \cap C^\infty(\mathbb{R}^n)$ be the space of slowly increasing smooth functions on $\mathbb{R}^n$.
Following p.294 proposition 9.10 of the "...
- 3,477
0
votes
1
answer
157
views
Does rapid convergence of the Cesaro sums imply convergence of the original sequence?
Question: Let $a_n$ be a sequence of real numbers. Is it true that for every $\varepsilon > 0$, if
$$\left \lvert \frac{1}{N} \left ( \sum_{n=0}^{N-1} a_n \right )\right \rvert < \frac{1}{N^{1+\...
- 6,215
12
votes
1
answer
437
views
Slick proofs using the Henstock–Kurzweil integral?
I enjoyed Iosif Pinelis's slick answer to another MO problem using the Henstock–Kurzweil integral. Are there other examples of problems whose statement does not explicitly involve the Henstock–...
5
votes
1
answer
167
views
What structure is preserved by pseudo-homeomorphisms of pseudo-Euclidean spaces?
Let us recall that for integer numbers $t,s\ge 0$ the pseudo-Euclidean space $\mathbb R^{t,s}$ is the vector space $\mathbb R^{t+s}$ endowed with the quadratic form $q_{t,s}:\mathbb R^{t+s}\to\mathbb ...
- 41.9k
4
votes
1
answer
209
views
Why is there a $\mathcal{H}^d$-null set in the definition of d-rectifiable set?
Given a set $A \subset \mathbb{R}^n$, this is called d-rectifiable if it can be covered by a countable union of images of lipshitz functions from $\mathbb{R}^d $ to $ \mathbb{R}^n $ and a $\mathcal{H}^...
- 697
2
votes
2
answers
382
views
Asymptotics of an integral requested
Given an integer $n\geq2$, consider the following integral
$$I_n:=\int_0^1nx^{n-1}\sqrt{\left\vert \frac{\log(1-x)}{\log n}\right\vert} \, dx.$$
QUESTION. Is this true? It appears to be so.
$$\lim_{n\...
- 43.2k
6
votes
2
answers
503
views
Computing a limit on the unit sphere: Riemann Lebesgue?
Let $u\in L^1(\mathbb{S}^{d-1})$. I want to show that
\begin{align*}
\lim_{|\xi|\to \infty}
\int_{\mathbb{S}^{d-1}}(1-\cos(\xi\cdot w))u(w)d \sigma_{d-1}(w)
= \int_{\mathbb{S}^{d-1}}u(w)d \sigma_{d-1}(...
- 1,101
0
votes
0
answers
108
views
linear span of linear in C_0(R)
We consider the set $C_0(\mathbb R)$ of real continous functions $f:\mathbb R\rightarrow \mathbb R$ with $\lim_{|x|\rightarrow \infty}f(x)=0$ endowed with the supremum norm.
Is there $f\in C_0(\mathbb ...
- 1
0
votes
1
answer
246
views
Riemann-Liouville integral of $f$ is zero implies $f =0$ a.e
The Riemann-Liouville integral is defined by
$$
I^\alpha f(x)=\frac{1}{\Gamma(\alpha)} \int_a^x f(t)(x-t)^{\alpha-1} d t
$$
where $\Gamma$ is the gamma function and $a$ is an arbitrary but fixed base ...
- 141
2
votes
0
answers
75
views
On Dirichlet eigenfunctions of a domain
Given any bounded domain $\Omega\subset \mathbb R^n$, $n\geq 2$, with a Lipschitz boundary, let $\{(\lambda_k,\phi_k)\}_{k\in \mathbb N}$ be the Dirichlet eigenvalues and eigenfunctions of $-\Delta$ ...
- 4,143
15
votes
1
answer
2k
views
Real polynomials that go to infinity in all directions: how fast do they grow?
Let $f(x_1, \cdots, x_n) \in \mathbb{R}[x_1, \cdots, x_n]$ be a polynomial. Define property $\mathbf{P}$ to be the property that there exists a compact set $K \subset \mathbb{R}^n$ and a positive ...
- 26.9k
0
votes
0
answers
317
views
What is the "best" good kernel?
A family of functions $k_n(x):[-\pi,\pi]\to \mathbb R$ for $n\in \mathbb N$ is said to be a good kernel if all the following are satisfied:
$\frac{1}{2\pi }\int_{-\pi}^\pi k_n(x) \, \mathrm d x=1$,
$...
- 3,062
3
votes
2
answers
382
views
Singular support: equivalent definition
Let $U\subset\mathbb{R}^{d}$ be an open set. The singular support of a distribution $u\in\mathcal{D}^{\prime}(U)$ is defined to be the compliment of the set of points, which have a neighbourhood in ...
- 1,171
0
votes
0
answers
120
views
The sufficient condition for a function equals to zero when it vanishes in an open set
For the class od analytic functions in one variable, there is an identity theorem: "If two holomorphic functions $f$ and $g$ on a domain $D$ agree on a set $S$ which has an accumulation point $c$ ...
- 101
2
votes
0
answers
164
views
What is known about "anti polynomials"?
I recently encountered a problem whose solution required solving $f(x):=\sum\alpha_i r_i^x\, =\, c;\ \alpha_i,r_i,x\in\mathbb{R},i\in I\subset\mathbb{N}, $ for $x$.
While the Newton method solves the ...
- 13.2k
0
votes
0
answers
175
views
Does l2 projection of sequences preserve l1 norm convergence?
Let $\ell^2$ denote the set of square summable sequences with inner product $\langle x,y\rangle=\sum_{i=1}^{n}x(i)y(i)$ and $\ell^2$ norm $\|x\|_2=\sqrt{\langle x,x\rangle}$. Let $\|x\|_1=\sum_{i=1}^{\...
- 45
4
votes
1
answer
172
views
Existence of a domain with simple Dirichlet eigenvalues
Let $g$ be a smooth Riemannian metric on $\mathbb R^3$ that coincides with the Euclidean metric outside a compact set $K$. Does there exist some domain $\Omega$ with smooth boundary such that $K \...
- 4,143
10
votes
1
answer
936
views
Derivative without extrema is monotone
This is a cross-post from Math.SE.
The question was asked there 3 months ago but didn't receive much attention aside from one comment asking for clarification. I feel like it might be non-trivial and ...
- 103
5
votes
2
answers
223
views
Continuous functions on $[0,1]^\omega$ and a product lower bound
I have a concrete question about continuous functions on $X = [0,1]^\omega$ (with the product topology).
The map $f:X\to [0, 1]$ given by $(x_i)\mapsto \prod x_i$ is well-defined and Borel but not ...
- 53
3
votes
0
answers
151
views
Is there a space of smooth functions dense in the domain of Coulomb-like potentials in dimension two?
Let $V : \mathbb{R}^2 \to \mathbb{R}$ be compactly supported, bounded away from the origin, and obey
$$ |V(x)| \lesssim r^{-\delta_0}, \qquad 0 < |x| \le 1, \qquad r : =|x|,$$
for some $0 < \...
- 481
8
votes
1
answer
273
views
Smooth approximation of Hölder functions "from below"
We assume that we have a $\alpha$-Hölder continuous function $f$ on an interval $[0,1].$
I am wondering if there exists an explicit construction of a sequence $f_{n} \in C_c^{\infty}(\mathbb R)$ such ...
8
votes
3
answers
545
views
Approximation of pseudogeometric progression
Let $f_n(x)=1+x+x^{\sqrt{2}}+x^{\sqrt{3}}+x^{\sqrt{4}}+\cdots+x^{\sqrt{n}}$ be a sequence of functions on the interval $[0, 1]$. Is there a good closed form approximation for such a function ( ...
- 728
4
votes
2
answers
245
views
On the monotonicity of the ratio of two logarithmic expressions
According to this comment and this comment, a positive answer to this recent question (about Bernoulli numbers) would be sufficient to prove the following:
$r:=f/g$ is increasing on $(0,\pi/2)$ from $...
- 128k
5
votes
2
answers
708
views
Approximation of Hölder continuous functions "from below"
We assume that we have a $\alpha$-Hölder continuous function $f$ on an interval $[0,1]$ with $f(0)=0$.
I am wondering if there exists an explicit construction of a sequence $f_{n} \in C_c^{\infty}(\...
1
vote
2
answers
123
views
Whether the integral $t^2(\iint_{\mathbb{R}^2}|\xi|e^{\frac{-a\xi^4t}{\xi^2+b}} \,d\xi_1 \,d\xi_2)$ is bounded?
Is the integral
$$
t^2\left(\iint_{\mathbb{R}^2}|\xi|e^{\frac{-a\xi^4t}{\xi^2+b}}\,d\xi_1\,d\xi_2\right)$$ bounded when $t\rightarrow\infty$? Here
$\xi=(\xi_1,\xi_2)\in\mathbb{R}^2$,
$|\xi|=\sqrt{\...
- 13