All Questions
1,487 questions with no upvoted or accepted answers
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
$$P^\varepsilon_k(X) = \tfrac{T_k\left(\tfrac{2(X+\varepsilon)}{\rho+\varepsilon}-1 \right)}{\left|T_k\left(\tfrac{2(1+...
- 21
2
votes
0
answers
126
views
How does the area affect the integral?
Let $\Omega\subset\mathbb{R}^n$ a open bounded set. For any $r>0$ consider the integral:
$$J_\Omega(r)=\int_{\Omega}\frac{|x^s|dx}{r^c+\sum_{i=1}^m|x^{p_i}|r^{d_i}},$$
where $s,p_i\in\mathbb{N}^n$ ...
- 561
2
votes
0
answers
58
views
Tail asymptotics of Durfee square identity
This post is related to the problem Asymptotics of a combinatorial series
According to the Durfee square identity:
$$\sum_{k \ge 0} \frac{q^{k^2}}{(q;q)_k^2} (q;q)_{\infty} = 1,$$
where $(q;q)_k$ is ...
- 51
2
votes
0
answers
211
views
Theory of mollifiers on the boundary of a $C^2$ domain
Let $D\subseteq\mathbb{R}^d$ be a nice but not smooth domain, somewhere between Lipschitz and $C^2$. I am looking for a reference on the theory of mollifiers and regularization for functions on $\...
- 171
2
votes
0
answers
182
views
Inequalities between $M_k=\sup \limits_{\mathbb{R}}|f^{(k)}(x)|$
1. Let $f\in C^{(n)}\left(]-1,1[\right)$ and $\sup \limits_{-1<x<1} |f(x)|\leq 1$. Let $m_k(I)=\inf \limits_{x\in I} |f^{(k)}(x)|$, where $I$ is an interval (by interval here I mean closed, open ...
- 330
2
votes
0
answers
136
views
Banach limit with added properties
Let $c=\{a:\mathbb{N}\rightarrow \mathbb{C}: \exists \alpha\in \mathbb{C} \textrm{ so that }
\lim\limits_{n\rightarrow \infty} a(n)=\alpha=:L_c(a)\}\subset \ell^\infty$, where $\ell^\infty$ is the ...
- 21
2
votes
0
answers
163
views
Bochner's formula for fractional Laplacian
Is there an analogue of the classical Bochner formula
$\frac{1}{2} \Delta |\nabla u|^2 = |\nabla^2 u|^2$ for harmonic functions that holds for $s$-harmonic functions?
- 161
2
votes
0
answers
553
views
$\ell_\infty$-norm covering number of RKHS ball $\{f\in\mathcal{H}: \|f\|_\mathcal{H} \leq R\}$
For any $\epsilon \in (0,1)$, let $N_\infty(\epsilon, \mathcal{H}, R)$ denote the $\epsilon$-covering number of the RKHS norm ball $\{f\in\mathcal{H}: \|f\|_\mathcal{H} \leq R\}$ with respect to the $\...
- 121
2
votes
0
answers
70
views
Monotone rearrangements of function, constrained optimization in $\mathcal{L}^p$
By $\mathcal{S}$ let us denote the set of such step functions $f:[0,1]\to [0,1]$ that additionally satisfy:
$$\forall_{ x>\frac{1}{2}} \ \ \lambda\Big(f=x\Big) \ = \ x\cdot \Big[\lambda\Big(f=x\Big)...
user153000
2
votes
0
answers
72
views
Product of Besov and Lorentz functions
Let us fix $n\in\mathbb{N}^+$ and $p,q\in [1,\infty)$. Given $r_1,r_2,r_3\in[1,\infty)$, I would like to understand whether we have the bound
$$
\|fg\|_{L^{q,r_3}(\mathbb{R}^n)}\lesssim \|f\|_{B^{n/p}...
- 282
2
votes
0
answers
251
views
A sequence of compact operators, weak convergence versus strong convergence
Let $(X,\mu)$ be a finite measure space.
We consider two sequences of bounded linear operators $\{T_n\}_{n=1}^\infty$ and $\{S_n\}_{n=1}^\infty$ on $L^2(X,\mu)$. We denote by $\mathcal{L}$ the space ...
- 721
2
votes
0
answers
132
views
When does a nonnegative $C^1$ function on $[a,b]$ have finitely many zeros?
It is well known that if $f:\mathbb R\to \mathbb R$ is real analytic then we have that either $f\equiv 0$ or $f$ has finitely many zeros on the interval $[a,b]$.
It is also known that you can have a ...
- 133
2
votes
0
answers
104
views
Weak convergence rates for integral operators
Suppose $q=\sum_{i=1}^m\pi_i\delta_{x_i}$ is a discrete measure on $\mathbb{R}^n$ and let $q\ast \varphi_\epsilon$ denote the convolution of $q$ with some mollifier $\varphi_\epsilon$, so that $q\ast\...
- 75
2
votes
0
answers
102
views
Wave equation with infinite time
Let $\Omega \subset \mathbb R^n$ be a compact domain with smooth boundary. Let $f \in H^1_\delta((0,\infty)\times \partial \Omega)$, where
$$ \|f\|_{H^1_\delta}^2= \|f\|^2_{L^2_\delta}+\|Df\|^2_{L^2_\...
- 4,153
2
votes
0
answers
52
views
A certain expectation of a function of independent gammas
Suppose that $Y_1...,Y_n$ are independant gamma random variables: $Y_i \sim \Gamma(\alpha_i,\beta_i)$, with density $f_i(t) = \frac{t^{\alpha_i-1} e^{-\frac{x}{\beta}}}{\Gamma(\alpha) \beta^{\alpha}}$....
- 686
2
votes
0
answers
109
views
Finding $\Omega$ such that the 1-form $\Omega^2 \omega$ is $L^2$ orthogonal to conformal killing vector fields on $S^2$
Consider the space $\mathcal{A}$ of functions $\Omega$ such that $\Omega^2 \gamma_0$ is isometric to the round sphere, where $\gamma_0$ is the round sphere. (so $\Omega^2 \gamma_0$ is of constant ...
- 969
2
votes
0
answers
52
views
A question for regularity of solutions to wave equation
let $T>0$ and suppose $\Omega$ is a bounded domain with smooth boundary. Let $g \in L^\infty(0,T;H^1(\partial \Omega))$ and consider the wave equation
\begin{equation}\label{pf0}
\begin{aligned}
\...
- 4,153
2
votes
0
answers
84
views
A problem of uniqueness
Let $f\in\mathcal{S}(\mathbb{R}^n)$, $a\in(-1,1)$, how i can prove that the following problem:
$$\text{div}(t^a\nabla u)=0,\quad\text{in }\mathbb{R}^n\times(0,\infty),$$
$$ u(x,0)=f(x),\quad\forall x\...
- 339
2
votes
0
answers
76
views
wave equation with L^2 boundary data via spectral decomposition
It is classical that if $\Omega \subset \mathbb R^n$ is a bounded domain with smooth boundary, then the equation
\begin{equation}\label{pf2}
\begin{aligned}
\begin{cases}
\partial^2_{t}u- \Delta u=0\,\...
- 4,153
2
votes
0
answers
86
views
Eigenvalues of the operator $A = -v'' + B(x) v$
How can I prove that for the eigenvalues of the operator $$A := -v'' + B(x) v$$ on $(0,L)$ with zero Dirichlet boundary condition it holds that
$$
\left| \lambda_n - \frac{\pi^2n^2}{L^2}\right| \le ||...
- 217
2
votes
0
answers
163
views
Hilbert transform on weighted Sobolev spaces
Let $\mathscr H\,f$ denote the Hilbert transform of a function $f \in L^2(\mathbb R)$. We know that $\mathscr H$ is an isometry on $L^2(\mathbb R)$, but I want to know to what is the mapping ...
- 4,153
2
votes
0
answers
141
views
For a 1-parameter family of metrics, how do we compute the derivative of the intrinsic geometrical objects like curvature, Hessian, etc
Consider a family of metrics and functions $(g(t), u(t))$ on $M:= \mathbb{R}^3 \setminus B_1$ satisfying
$$ g(0) = g_0, \quad g'(0) = \tilde g, \quad u(0) = u_0, \quad u'(0) = \tilde u$$
where $g_0$, $...
- 969
2
votes
0
answers
164
views
What are (the different aspects of) harmonic analysis good for?
Let $G$ be a locally compact group. To the best of my understanding, harmonic analysis has three legs that all work perfectly in the case that $G$ is in addition compact and abelian, but have ...
- 2,071
2
votes
0
answers
115
views
Showing that for measurable $\Omega \subseteq \mathbb{R}^n$, $L^1(\Omega; C_0(\mathbb{R}^n))$ is separable
Here we're integrating "Banach-valued" functions $u: \Omega \rightarrow C_0(\mathbb{R}^n))$ , and by $u \in L^1(\Omega; C_0(\mathbb{R}^n))$ I mean that
$$\int_{x \in \Omega} \| u(x) \|_{\...
- 21
2
votes
0
answers
42
views
Generalized Hardy operator and Lorentz gamma spaces
I would like to find an inequality which would 'place' the generalized Hardy operator $\int_0^th(y)dy\int_y^tk^*(s)ds$ in between two Lorentz gamma spaces.
Any literature or ideas would be greatly ...
- 109
2
votes
0
answers
128
views
Set of discontinuities for Thomae's function in $\mathbb{R}^2$
For this question, label:
Part A: Is $f(x,y)$ integrable? Question $3$-$7$ from Spivak's Calculus on Manifolds
Part B: Prove $f:[0,1]\times[0,1]→\mathbb{R}$ is integrable.
For part A and B question, ...
- 21
2
votes
0
answers
99
views
Lower bound on iterated matrix application
Let $n \in \mathbb Z^2$ such that the non self-adjoint weighted Laplacian is
$$(\Delta u)(n)=u(n_1+1,n_2)-u(n_1-1,n_2) + i( u(n_1,n_2+1)- u(n_1,n_2-1))$$
the adjoint operator is then
$$(\Delta^* u)(n)=...
- 192
2
votes
0
answers
74
views
Distorted elementary functions
Let $f(x)$ be an elementary function defined on $X\subseteq\mathbb{R}$ and $\xi(x), \eta(y)$ strictly monotone for $x\in X,\, y\in f(x)$.
Questions:
is there an established name for functions of the ...
- 13.2k
2
votes
0
answers
37
views
Binary law on pairs of finite unions of segments
Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does ...
- 1,209
2
votes
0
answers
88
views
$1$-parameter analytic functions are almost everywhere Morse
Let $I = [t_{0}, t_{1}]$ be a closed interval with $t_{0} < t_{1}$ and let $M$ be a compact real analytic $n$-dimensional manifold without boundary. Furthermore, let $f:I \times M \rightarrow \...
- 21
2
votes
0
answers
84
views
How to define Lebesgue Integrability of functions assuming values in an arbitrary topological vector space over an arbitrary topological field?
I have already asked this question in this MSE thread, but some people suggested me to ask to the MO community also.
Preliminaries
An algebra of sets in a set $X$ is an $\mathcal{X}\subseteq\mathcal{P}...
- 129
2
votes
0
answers
89
views
Prove integral inequality for divergence-free vector fields
Let $u$ be a divergence-free vector field $u:\mathbb R^n \to \mathbb R^ n$. Does the following inequality hold?
$$\Big( \int_{\mathbb R^n} |u|^2 dx\Big)^2 \le C\Big(\int_{\mathbb R^n} |u|^2|x|^2 dx \...
- 839
2
votes
0
answers
126
views
Mixed partial derivatives of planar functions converging to delta distribution
Given a sequence $(f_k)_{k\in\mathbb{N}}\subset C^2(\mathbb{R}^2)$ of strictly positive functions $f_k\equiv f_k(x,y)$ with $\|f_k(x,\cdot)\|_{L^1}=1$ for all $x\in\mathbb{R}$ and such that for each $...
- 51
2
votes
0
answers
120
views
Hilbert transform on a Besov space
Consider the usual Hilbert transform of periodic functions
$$H(f) = \frac{1}{2\pi}P.V.\int_{-\pi}^{\pi}\cot(\frac{x-y}{2})f(y)dy.$$
We know $H$ does not map $L^\infty$ continuously to $L^\infty$. Now ...
- 903
2
votes
0
answers
65
views
Request for resources or techniques for bounding the infinity norm of an infinite product convolved with a simple function
I'm attempting to bound an expression of the form.
$$
\lVert(\prod_{i=1}^{\infty} \phi_i) * s \rVert_{\infty}
$$
Where $\phi_i$ are bounded periodic step functions which can be replaced by smoothed ...
- 41
2
votes
0
answers
156
views
On sup over boundaries of Sobolev functions
In Gilbarg-Trudinger's section on the maximum principle for weak solutions, the sup of a boundary of a Sobolev function defined as follows:
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. For $u, ...
- 353
2
votes
0
answers
158
views
Estimate involving Besov norm
When reading some old notes of my advisor on interpolation spaces, I bumped into a problem I can't quite wrap my head around. Here are the details.
For $p\in(0,\infty)$ a $p$-variation semi-norm of a ...
- 421
2
votes
0
answers
100
views
What is the weak limit of $f_n \ \mathrm{sign}(f_n - 1)$ if $f_n \to f$ weakly in $L^p([0,1])$?
Let $f_n: [0,1] \to \mathbb R$ be a uniformly bounded sequence in $L^p$. Then there exists a subsequence such that $f_{n_k} \to f$ weakly in $L^p([0,1])$. What is the weak limit of the sequence of ...
- 217
2
votes
0
answers
42
views
Analysis of coefficients for quickly vanishing analytic vector field
Let $u = (u_1, u_2, u_3): \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a divergence-free analytic vector field for $n =3$ or $n =4$, i.e., $u_i : \mathbb{R}^n \rightarrow \mathbb{R}$ are analytic ...
- 749
2
votes
0
answers
85
views
Are a map with constant singular values and its inverse always conjugate through isometries?
Let $U \subseteq \mathbb R^2$ be open, connected and bounded, and let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1 \sigma_2=1$.
Suppose that $f:U \to U$ is a diffeomorphism whose singular values (of $...
- 6,741
2
votes
0
answers
160
views
Approximation in fractional Sobolev space
Assume $\Omega\subset \Bbb R^d$ is Lipschitz open set. Let $p\geq 1$ and $0<s\leq 1/p$.
How to prove that $C_c^\infty(\Omega)$ is dense in $W^{s,p}(\Omega)$?
Recall that,
$$|u|^p_{W^{s,p}(\Omega)}= ...
- 1,101
2
votes
0
answers
196
views
Have you seen this PDE before?
Consider the second-order nonlinear PDE
$$(\partial_x\partial_y\varphi)\cdot\varphi = \partial_x\varphi\,\partial_y\varphi.$$
This PDE is solved by all ('separable') functions $\varphi\in C^2(\Omega)$ ...
- 463
2
votes
0
answers
186
views
Bounding the condition number of a matrix associated with an even symmetric positive definite function
Define a set $A = \{x_i/x_i\in\mathbb{R}^m, i = 1,2,3..n\}$. Let $f:\mathbb{R}^m\to(0,\infty)$ be an even symmetric positive definite function.
Let $D = [d_{i,j}]$ be an $n\times n$ matrix such that $...
- 698
2
votes
0
answers
190
views
What is the smallest dimension that allows finding $n$ points at distances $|x_i-x_j|^{\delta/2}$, where $0<\delta<1$, and $x \in \mathbb{R}^n$?
Let $x_1,\cdots,x_n \in \mathbb{R}$, are there $\xi_1,\cdots,\xi_n \in \mathbb{R}^s$, such that
$|x_i-x_j|^{\delta}=||\xi_i-\xi_j||^2$, $0<\delta<1$, what is the smallest $s$ to guarantee the ...
- 178
2
votes
0
answers
162
views
Conditions for absolute continuity in the Bochner-Schwartz theorem
Suppose that $f$ is a positive-definite Schwartz distribution, that is,
$$\langle\phi,f*\phi\rangle\geq0\qquad\text{for every }\phi\in C_0^\infty(\mathbb R^n).$$
By the Bochner-Schwartz theorem, there ...
- 891
2
votes
0
answers
176
views
Approximate identities: a converse question
Let $K(x)$ be a positive-valued function such that
$$\int_{-\infty}^{\infty} K(x) \ dx = 1$$
and let
$$K_{\lambda}(x) = \lambda K(\lambda x), \ \ \ (\lambda > 0);$$
that is to say, the family of ...
- 71
2
votes
0
answers
107
views
Proof that Littlewood-Paley vertical square function is NOT bounded on L^infinity
The classical heat semigroup on $\mathbb{R}$ is given by
$$
W_t f(x)=\frac{1}{t}\int_{\mathbb{R}}e^{-\pi (\frac{x-y}{t})^2}f(y)dy, \qquad t>0.
$$
Then the Littlewood-Paley vertical square ...
- 421
2
votes
0
answers
56
views
A question about Holder exponents of a function at different points in its domain
Suppose that $f(x)$ is continuous on $[0,1]$. We make an agreement that if there exists an interval $[a,b]\subseteq[0,1]$ including point $y$ such that $f(x)$ satisfies $\alpha$-Holder condition on $[...
- 419
2
votes
0
answers
65
views
Measure of the convex hull of a ball and a point
I need to prove the following statement:
Let $B_s(z)$ be a ball centered at $z$ of radius $s$ s.t. $0\not\in B_s(z)$. Moreover let $K_s(z)$ the convex hull of $\{0\}\cup B_s(z)$.
Then
$$ \...
- 73
2
votes
0
answers
77
views
Second derivative estimates for a subsolution of linear elliptic equation
Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$.
Note. Saying that $u$ is semiconvex is ...
