All Questions
1,485 questions with no upvoted or accepted answers
2
votes
0
answers
117
views
Bounding integral expression with BV norm of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\...
user139844
2
votes
0
answers
71
views
measure corresponding to certain orthogonal polynomials
Given a scalar $\lambda\in (0,1)$, consider a sequence of monic polynomials $\{p_n(x)\}_{n\geq 0}$ over real variables satisfying the following recurrence relations:
$xp_n(x)=p_{n+1}(x)+\lambda p_{n}(...
- 165
2
votes
0
answers
155
views
Second differential of total variation
I am trying to give meaning to the notion of second differential of total variation.
For sufficiently regular $u:\Omega \subset \mathbb{R}^2 \to \mathbb{R}$ let the total variation be given by
$$TV(u)=...
- 319
2
votes
0
answers
87
views
Maximal function to high power
Consider the following maximal function : in dimension $n$ consider $B(0,1)\subset \mathbb{R}^n$ the unit ball, if $f\in L^1(B(0,1))$, $\alpha\geq 0$ :
$$
M_\alpha f : x \mapsto \sup \left\{ \frac{1}{...
- 363
2
votes
0
answers
161
views
The Laplace transform and the Lagrange compositional inversion formula
I'm looking for references which derive the Lagrange inversion formula, given below (in bold), for the Taylor series coefficients of the compositional inverse of a function $f$ analytic at the origin ...
- 10.5k
2
votes
0
answers
130
views
Smoothness of Radon transform
Let $f:\mathbb R^n \to \mathbb R$ be density function (i.e nonnegative function which integrates to $1$), and consider its Radon transform $R[f]$ defined by
$$
R[f](w,b) := \int_{\mathbb R^n}\delta(x^\...
- 6,853
2
votes
0
answers
321
views
Distribution of $\frac{(\sin(n))^2}{2^n}$ in dyadic intervals?
Good morning all,
I was wondering what kind of methods could help in order to tackle the following problem :
Define the set $A = \left\{ \frac{(\sin(n))^2}{2^n}\right\}$ for $n$ integer. So A is a ...
- 125
2
votes
0
answers
148
views
Finding an asymptotically flat manifold with ${\rm Ric}_{r\phi} = \frac{\sin\theta}{r^2}$
Let $(r,\theta,\phi)$ be the spherical coordinates on $\mathbb{R}^3$ where $\theta \in (0,\pi)$ and $\phi\in (0,2\pi)$.
Does there exist an asymptotically flat metric $g$ on $\mathbb{R}^3\setminus B_1$...
- 969
2
votes
0
answers
66
views
Existence of saddle points under a $C^0$-perturbation of a continuous function
Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function and has a strict maximum point $a$ and strict minimum point $b$. Define $g(x,y)=f(x)+f(y)$ and $h_\varepsilon(x,y)$ be a family of continuous ...
- 379
2
votes
0
answers
152
views
Riesz transform of constant function
My one-line question would be, what is the Riesz transform of the constant function, identically equal to 1 on $\mathbb{R}^2$?
But more fundamentally, my question stems from some confusion about the ...
- 287
2
votes
0
answers
150
views
Closeness of a rational approximation
What is
$$p_*:=\inf\big\{p\in\mathbb R\colon\,\inf_{n\in\mathbb N}n^p\,\inf_{k\in\mathbb N}
|2\sqrt{3n}-9\pi/4-k\pi|>0\big\},$$
where $\mathbb N:=\{1,2,\dots\}$?
In other words, I would like to ...
- 128k
2
votes
0
answers
138
views
Problems arising from a paper on the radial symmetry of the global solution of semilinear PDE $\Delta u+f(u)=0$ in $\Bbb{R}^{n}$
I am reading the paper [1] by Congming Li.
I want to talk about the typical case that the author gives as follows ([1], §1, pp. 590-):
In this section, we study positive solutions of the following ...
- 809
2
votes
1
answer
137
views
Convergence of the average weight of an infinite path through a weighted directed graph
Consider a directed graph $G = (V, E, w)$, where $V$ is the set of vertices, $E \subseteq V \times V$ is the set of directed edges (with self-loops allowed), and $w : E \to \mathbb{R}_+$ is a weight ...
- 21
2
votes
0
answers
73
views
Separately continuous functions of the first Baire class
Problem. Let $X,Y$ be (completely regular) topological spaces such that every separately continuous functions $f:X^2\to\mathbb R$ and $g:Y^2\to \mathbb R$ are of the first Baire class. Is every ...
- 4,535
2
votes
0
answers
166
views
Green's function for elliptic PDE with potential
$\newcommand{\div}{\operatorname{div}}$Suppose I have an elliptic operator $\mathcal{L} u = -\div (A \nabla u) $ on some open set $\Omega \subseteq \mathbb{R}^d$ where here $A$ is uniformly elliptic ...
2
votes
0
answers
369
views
Components of the complement of a compact set
Suppose $K$ is a compact subset of $\mathbb{R}^m$ ($m>1$), and $0<r<R$ are fixed numbers. Let $A$ be the set of points having a distance $<R$ and $>r$ from $K$. My questions are
If $K$ ...
- 411
2
votes
0
answers
85
views
Proving an eigenvalue bound without resorting to Weyl's law
Suppose $(M,g)$ is a smooth compact Riemannian manifold of dimension $n\geq 2$ with smooth boundary and denote by $\{\phi_k,\lambda_k\}_{k\in \mathbb N}$ its Dirihclet spectral decomposition for the ...
- 4,115
2
votes
0
answers
77
views
Weyl's law and eigenfunction bounds for weighted Laplace-Beltrami operator
I would appreciate any answers or even references for the following problem.
Let $(M,g)$ be a complete smooth Riemannian manifold with an asymptotically Euclidean metric (let's even say that the ...
- 4,115
2
votes
0
answers
98
views
Has this "optimal constrained transport" notion of convergence of measures been named and/or studied?
Let $(X,d)$ be a compact metric space, and let $\{\mu_n\}_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures on $X$.
Fix $L \geq 1$. I will say that $\mu_n$ converges in ...
- 708
2
votes
0
answers
108
views
Absolute lower bound on derivative of generalized trigonometric polynomial at zeroes
By a generalized trigonometric polynomial, I shall mean a function $f:\mathbb R^+ \rightarrow \mathbb R$ given by an expression of the form
$$f(x) := \sum_{j=1}^k a_j \cos(\alpha_j x) + b_j \sin(\...
- 739
2
votes
0
answers
87
views
Is there a finite set of polynomials generating all rational numbers by iteration?
In this previous post I asked for the smallest set of continuous real functions that could generate $\mathbb{Q}$ by iteration starting from 0. Surprisingly one continuous function suffices.
The ...
- 4,862
2
votes
0
answers
80
views
A complicated equation of integro-differential type
Consider the following equation of $\beta$ : $\beta(0)=2$ and
$$\dot\beta(t)\beta(t)^2 = \frac{1}{2}\int_0^t\frac{\dot\beta(s)(t-s)}{A_{\beta}(s,t)^{3/2}}f\left(\frac{t-s}{\sqrt{A_{\beta}(s,t)}}\right)...
- 1,334
2
votes
0
answers
68
views
Core for Neumann Laplacians
Let $d$ be a positive integer. We write $\mathbb{H}^d$ for the closed $d$-dimensional upper-half space: $\mathbb{H}^d=\{(x_1,\ldots,x_d) \in \mathbb{R}^d,\,x_d \ge 0\}$. We consider the Neumann ...
- 721
2
votes
0
answers
55
views
An integral average condition and its relationship with BMO, VMO, and Sobolev spaces
Let $V: \mathbb R^n \to \mathbb R^n$ be a vector field which satisfies
$$
\lim_{l \to \infty} \sup_{x \in \mathbb R^n} \left|\frac{1}{l^n} \int_{[0,l]^n}V(x+y) dy \right| = 0
$$
What is the ...
- 839
2
votes
0
answers
131
views
Taylor series with less than differentiability
I have a function $f^0\colon (0;\infty) \to \mathbb R$ with the property that the following limit exists and is finite
$$
F^1 := \lim_{x\to \infty} x \cdot f^0(x)
$$
Then I consider $f^1(x) := x \cdot ...
- 335
2
votes
0
answers
123
views
Strange convexity condition
I have the following condition of the function $f : \mathbb R \to \mathbb R$: $$\exists\, h:\mathbb R_+ \to \mathbb R_+ \text{ bijective s.t., } \forall s>0, \frac{\partial^2}{\partial^2 s}\left(\...
- 686
2
votes
0
answers
104
views
Coercivity of an integral operator in control theory
Let us consider the integral operator $T:\mathbb{R}^{n\times d}\to [0,\infty)$ such that for all $K\in \mathbb{R}^{n\times d}$,
$$
T(K)=\int_0^1 \operatorname{tr}(KK^\top \Sigma_t) \,d t,
$$
where $\...
- 503
2
votes
0
answers
221
views
Turán–Nazarov's lemma for algebraic polynomials?
Nazarov proved a version of Turán's lemma in Complete Version of Turan’s Lemma for Trigonometric Polynomials on the Unit Circumference, which is now known by the name Nazarov–Turán's lemma. A special ...
- 399
2
votes
0
answers
57
views
Is the lattice of bounded Henstock Kurzweil integrable functions countably complete?
The set of HK integrable functions with an integrable upper bound $f$ forms a lattice, and satisfies the MCT and DCT. Does this mean that the lattice is countably complete?
Indexing any countable set, ...
- 1,947
2
votes
0
answers
229
views
Weighted Sobolev norm in terms of Spherical harmonics coefficients
Let $M = [1,\infty) \times S^2$.
Consider the weighted Sobolev space $H^k_{\delta}(M)$ with the Sobolev norm:
$$\lVert u \rVert_{k,\delta}^2 := \sum_{n=0}^k \int_M |D^nu \,r^{n-\delta}|^2 r^{-3} dV $$
...
- 969
2
votes
0
answers
115
views
Least positive value of a random polynomial
Fix a positive even integer $d$ and consider the polynomial $f(x)=c_d x^d+\ldots+c_1x+c_0$, where the $c_i$ are independent random variables that follow the uniform distribution in the interval $[-1,1]...
- 3,062
2
votes
0
answers
298
views
A question on convergence rates of Fourier series and strict convergence
Consider BV functions on a torus. The Fourier partial sum using the first $n$ coefficients will converge to the function at every point of continuity, as $n\to\infty$. The convergence rate is $O(1/n)$....
- 698
2
votes
0
answers
164
views
(Generalized) Uncentered Maximal Function $\tilde Mf$ in Stein's Harmonic Analysis
It is well known that on $\Bbb R^n$, equipped with the usual Lebesgue measure, the standard Hardy-Littlewood maximal function $Mf(x)$ (with respect to averaging on cubes or balls centered at $x$) is ...
- 1,245
2
votes
0
answers
58
views
Uniqueness for a certain semilinear equation
Suppose that $(M,g)$ is a smooth compact Riemannian manifold with smooth boundary $\partial M$. Let $a \in C^{\infty}(M)$, let $k \in \mathbb Z$ and consider the equation
$$
\begin{aligned}
\begin{...
- 4,115
2
votes
0
answers
108
views
Are a.e. derivatives of continuous $VBG_*$ functions Denjoy–Perron integrable?
I would like to ask a question pertaining to the Denjoy–Perron (Henstock–Kurzweil) theory of integration. It is simple enough that I have entertained the idea that perhaps an answer is known, but I ...
2
votes
0
answers
67
views
A polar open set in a topological subspace?
Suppose $U$ is a bounded open set in $\mathbb{R}^m$ with ($m\geq2$). Is it possible to have a non-empty set $E$ in the boundary $ \partial U$ of $U$ that is open in $ \partial U$ and is polar?
A set $...
- 411
2
votes
0
answers
45
views
Additivity of squared Schatten $p$-norm with respect to spatial partition
Consider a Hilbert-Schmidt operator $A$ on $L^2(\mathbb R^d)$ with integral kernel $A(x,y)$. Let $\Omega\subset \mathbb R^d$ and $1_{\Omega}(x)$ denote its characteristic function as well as the ...
- 91
2
votes
0
answers
448
views
Conjecture: The sequence {$π(2n+1)!$} is equidistributed in the interval (0,1)
Let $n\in\mathbb{N}$.
From the book "Uniform Distribution of Sequences" (available here) by L. Kuipers and H. Niederreiter, (from pg. 8) I found that for any irrational $\theta$, the ...
2
votes
0
answers
297
views
Examples of harmonic functions
I am looking for non-trivial examples (in the sense to be described below) of harmonic functions, which can be represented as cubes of smooth functions ($C^1$ would be also OK if this is important).
...
- 246
2
votes
0
answers
153
views
unique continuation in a disk
Let $D$ be the unit disc in $\mathbb R^2$ centered at the origin. Let $w \in C^{\infty}_c(D)$ satisfy
$$ (1-r^2)^2\Delta w +w =0.$$
Prove that $w \equiv 0$.
- 4,115
2
votes
0
answers
65
views
On a question relating integral equation:
I don't know if the following question qualifies as research level. If it isn't, sorry.
Set the following terminology:
$ \alpha_1 =\alpha_1(t,x)=t(\tan^{-1}(x)+c)$
$\alpha_2=\alpha_2(s,x)=s(\tan^{-1}(...
- 41
2
votes
0
answers
254
views
Dense property of intersection of Sobolev space
I'm using Muscalu and Schlag's textbook (online notes) to study Littlewood-Paley theory in harmonic analysis, where I encounter the following claim:
Pick an arbitrary real number $s$, we have that the ...
- 197
2
votes
0
answers
54
views
Sequence tending to modulus function
I am looking for a sequence of smooth functions $f_n: \mathbb R^2 \rightarrow \mathbb R$such that the
1.) the nodal set of $f_n$, i.e. $N_n:=f_n^{-1}(0)$ converges to the graph of the absolute value ...
2
votes
0
answers
470
views
Norm of a Taylor approximation of a multivariate function
I have a function $f:\mathbb{R}^n\to\mathbb{R}^m$. My goal is to bound the first order Taylor approximation of $f$. Given $x,x'\in\mathbb{R}^n$ I have that
\begin{equation}
f(x)-f(x')\approx (x-x')^...
- 21
2
votes
0
answers
65
views
Reference request for type of specific integral equation in two variable:
Consider the following integral equation:
$$\int_0^\infty K(t,y)\phi(t,x)dt=0$$
Here, $K(t,y)$ is a trigonometric kernel and
$\phi(t,x)$ is monotonic wrt $x$ ( for fixed $t$).
I want to find the ...
- 41
2
votes
0
answers
144
views
Does this geometric PDE have a solution?
Let $s(\theta), b(\theta)$ be two smooth non-constant real-valued functions on $\mathbb{S}^1$, and assume that $s$ never vanishes.
Does there exist a map $h:(0,1) \times \mathbb{S}^1 \to \mathbb{S}^1$,...
- 6,741
2
votes
0
answers
132
views
Green's identity with a different norm
Let $\Omega \subset \mathbb{R}^n$ be a domain with a smooth boundary $\Gamma$. Suppose that $f, g \colon \mathbb{R}^n \to \mathbb{R}$ are of class $C^\infty( \overline{\Omega})$. Then Green's first ...
- 820
2
votes
0
answers
44
views
Derivatives of $G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt$ when $h$ is positive-homogeneous
Let $h:\mathbb R \to \mathbb R$ be a continuous which is positive-homogeneous of order $p \ge 1$, and define $G_h:[-1,1] \to \mathbb R$ by
$$
G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt.
$...
- 6,853
2
votes
0
answers
71
views
Strict Riesz's rearrangement inequality when function is not nonnegative
The strict Riesz rearrangement inequality (Lieb- and Loss's book Analysis, Section 3, Theorem 3.9 ,page 93) says that if the functions $f,g,h,$ are all nonnegative and $g$ is strictly symmetric ...
- 379
2
votes
0
answers
93
views
Variation of the sum of absolute values of coefficients for shifted Chebyshev polynomials
Setting
Let $\rho \in ]0,1[$, $\varepsilon\in[0,\rho]$, $k \in \mathbb{N}^*$ and
$$P^\varepsilon_k(X) = \tfrac{T_k\left(\tfrac{2(X+\varepsilon)}{\rho+\varepsilon}-1 \right)}{\left|T_k\left(\tfrac{2(1+...
- 21