Skip to main content

All Questions

Filter by
Sorted by
Tagged with
4 votes
1 answer
275 views

Eigenvalue of a convolution and a restriction?

Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
H A Helfgott's user avatar
  • 20.2k
3 votes
0 answers
91 views

About BMO space on smooth open bounded domain

Let $\Omega$ be any open domain in $\Bbb R^d$. Define the $\text{BMO}(\Omega)$ space as $$ \text{BMO}(\Omega)= \big\{u\in L^1_{loc}(\Omega)\,\,:\,\, |u|_{\text{BMO}(\Omega)} <\infty \big\}, $$ ...
Guy Fsone's user avatar
  • 1,101
2 votes
0 answers
78 views

Function that is (essentially) a self-convolution but not a multiple of a self-convolution

Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
H A Helfgott's user avatar
  • 20.2k
2 votes
0 answers
227 views

A deceptively simple regularity problem for functions on the plane

By various meanderings and toying with simpler problems, my current research has lead me to the following quite straightforward question, which I am wholly unable to answer: Consider a twice ...
vmist's user avatar
  • 989
0 votes
1 answer
124 views

Holomorphic functions of certain blow up at origin

Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
Ali's user avatar
  • 4,145
1 vote
0 answers
146 views

integral over the unit sphere of $\Bbb C^n$

Please, is there a way to calculate this integral $$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$ where $ z $ is a fixed point in the complex unit ball ...
zoran  Vicovic's user avatar
0 votes
0 answers
53 views

Spectral theory of compact operator for quasi-Banach spaces

Let $X$ be a Banach space and let $Y\subset X$ be a quasi-Banach space (with compact inclusion). Suppose $T:X\to X$ is a compact operator such that $1$ is not its eigenvalue and $T|_{Y}:Y\to Y$ is ...
Liding Yao's user avatar
2 votes
0 answers
194 views

Functions such that the *integral* of the Fourier transform is non-negative?

Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that $$\int_{-\infty}^x \widehat{f}(t) dt \...
H A Helfgott's user avatar
  • 20.2k
2 votes
0 answers
52 views

On distributions and kernels

Let $U\subset\mathbb{R}^{d}$ be an open set and consider $X=\mathbb{R}\times U$. Now, lets consider a smooth (regular) kernel $k_{A}\in C^{\infty}(X\times X)$ and corresponding continuous operator $A:...
G. Blaickner's user avatar
  • 1,429
0 votes
0 answers
55 views

Compactness and Leray-Schauder degree

What's the relationship between compactness of solutions in partial differential equations (PDEs) and the Leray-Schauder degree?
Davidi Cone's user avatar
2 votes
0 answers
43 views

Distributions and time-kernels

Let $U\subset\mathbb{R}^{d}$ be an open subset and set $M:=I\times U$, where $I=(a,b)\subset\mathbb{R}$ is some open subset. Lets consider a linear operator $B:C^{\infty}_{c}(M)\to C^{\infty}(M)$ that ...
G. Blaickner's user avatar
  • 1,429
1 vote
1 answer
90 views

Sobolev inequality with weight in the case $1<n\leq p$

Assume that $1<n\leq p$. Does there exist a (non-negative) measure $\mu$ (preferably with some positive density function with respect to the Lebesue measure $dx$) and $q>p$ so that for all $f\in ...
Shaq155's user avatar
  • 459
1 vote
2 answers
117 views

If $f\in C([0,\infty))$, does $\delta>0$ and $g\in C^1((0,\delta))\cap C([0,\delta])$ s.t. $g\geq f$ on $[0,\delta]$ and $g(0)=f(0)$ exist?

The question is the following: Suppose $f : [0,\infty) \rightarrow \mathbb{R}$ is a continuous function. Can I find $\delta \in (0,\infty)$ and a function $g : [0,\delta] \rightarrow \mathbb{R}$ such ...
vaoy's user avatar
  • 309
2 votes
0 answers
90 views

Representation of Dirac-delta distribution in subspace of functions

Suppose I have a subspace $V\subset L^2(\Omega)$ where $\Omega\subset \mathbb{R}^d$ is a bounded and closed set. $V$ is defined by \begin{align} V=\text{span}(\{\varphi_i(x): i=1,2,\dots,n\}) \end{...
Jjj's user avatar
  • 93
0 votes
0 answers
57 views

Double-periodic functions with (possible) poles

Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a ...
António Borges Santos's user avatar
1 vote
0 answers
45 views

Existence of optimal entropic weights for empirical modeling

Let $\mathcal{X} = [0,1]^n$ be the input space and $\mathcal{Y} = \{1, ..., n_c\}$ be a discrete output space. Let $D = \{(x_i, y_i)\}_{i=1}^N \subset \mathcal{X} \times \mathcal{Y}$ be a training ...
Damien's user avatar
  • 111
0 votes
0 answers
68 views

Family of separable Hilbert spaces over locally compact form a continuous field of Hilbert space?

Let $\{H_{x}\}_{x\in G^{0}}$ be a family of separable Hilbert spaces and $G^{0}$ be a locally compact second countable topological space. Let $\mathbb{B}_{x}$ be the orthonormal basis of $H_{x}$. If ...
K N SRIDHARAN NAMBOODIRI's user avatar
3 votes
1 answer
220 views

What we know about the function in Fefferman's Theorem

In Fefferman's many papers on Whitney's theorem he, amongst other things, constructs the existence of a smooth function $F$ which extends a function $f$ on a (say) finite set $E\subseteq \mathbb{R}^n$ ...
ABIM's user avatar
  • 5,405
2 votes
2 answers
154 views

Closure of $C([0,1]^2)$ via weak*-topology [closed]

Let $C([0,1]^2)$ denote the set of continuous functions on $[0,1]^2$. Let $L^1([0,1]^2)$ be the set of all Lebesgue integrable functions on $[0,1]^2$. The dual space of $C([0,1]^2)$, denoted by $C^*([...
tom jerry's user avatar
  • 349
1 vote
1 answer
157 views

Is finding the CDF from the Laplace transform well-posed?

In my study of Dynamic Light Scattering, I came across the following inverse problem. Let $F(s):[0,T]\rightarrow[0,T]$ be the Laplace transform of a probability distribution $f(t)$ on the real line ...
Riemann's user avatar
  • 654
2 votes
0 answers
191 views

Smoothing property of the heat kernel on the one-dimensional torus

Let $G=G(x,t)$ be the heat kernel on the one-dimensional torus $\mathbb{T}^1,$ with $x \in \mathbb{T}^1$ and $t \in (0,T].$ $G$ is given by \begin{equation} G(x,t) = (4 \pi t)^{-1/2} \sum_{k \in \...
kumquat's user avatar
  • 185
0 votes
1 answer
231 views

Questions on the compactness of $L_1([0,1]^2)$'s unit sphere

Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$ and $f(x,y)\geq 0: a.e. (x,y)\in [0,1]^2$. Recently in my study I need to study the compactness of $U$. By Riesz's theorem ...
tom jerry's user avatar
  • 349
0 votes
1 answer
101 views

Limit sequence of regular function in $L_1$‘s unit sphere

Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$. For any $f\in U$, we say it is regular if $\int_{x_0\times [0,1]}f=\int_{[0,1]\times y_0}f=1$ for a.e. every $x_0, y_0\in [...
tom jerry's user avatar
  • 349
0 votes
0 answers
121 views

Is there a good or commonly accepted short notation for the set of differentiable, but not necessarily continuously differentiable maps?

Every once in a while I find myself in need of some short notation for the set of differentiable, but not continuously differentiable maps, say, $X \to Y$. Always having to specify "...
M.G.'s user avatar
  • 7,127
-1 votes
1 answer
168 views

Space of distributions on $[0,1]^2$: weakly compact or not?

Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$. Question 1: Does $\mathcal{X}$ ...
tom jerry's user avatar
  • 349
1 vote
1 answer
330 views

Does $\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$ converge?

I am trying to study the converge of the series $$\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$$ But $e^{\sin{n}}$ is not monotone, and the Abel's test rule fails here. Can someone help me? ...
pxchg1200's user avatar
  • 287
0 votes
0 answers
100 views

Construct a bi-Lipschitz mapping that maps a cube to a ball which has the same center with the cube

A mapping $f: \mathbb{R}^n\to \mathbb{R}^n$ is said to be $K$-bi-Lipschitz, $K>1$, if \begin{equation*} \dfrac{1}{K}\leqslant \dfrac{|f(x)-f(y)|}{|x-y|}\leqslant K, \end{equation*} for any $x,y\in \...
Javier's user avatar
  • 69
7 votes
0 answers
249 views

Proving this function is convex

Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
Tom Solberg's user avatar
  • 4,049
2 votes
1 answer
128 views

Density of smooth functions in weighted Sobolev space

Let $\rho(x)=e^{-\phi(x)}$, where $\phi$ is an even polynomial with positive leading coefficient. I am interested in a proof of the fact that the space of smooth compactly supported functions $\...
Bastien's user avatar
  • 23
2 votes
0 answers
120 views

On mollifiers acting between $L^2$ and Sobolev spaces

(I'm reposting here this question from MSE as it didn't receive any answer for two weeks.) Consider a sequence of finite lattices in $\mathbb{R}^n$ defined by $$L_k= [-k,k]^n \cap 2^{-k}\cdot \mathbb{...
S.Z.'s user avatar
  • 505
3 votes
1 answer
187 views

Is this property preserved under weak$^*$ convergence?

Let $1 \le p < n$ and let $p^*$ be the Sobolev conjugate of $p$, i.e. $p^* = np/(n - p)$. Let $(\Omega_m)$ be an increasing sequence of bounded, convex and open sets such that $$ \lim_{m \to \infty}...
Cauchy's Sequence's user avatar
2 votes
1 answer
474 views

Polynomial $f(x)$ has positive coefficients and only real roots. How many polynomials formed from terms of $f(x)$ also have only real roots?

Let $$f(x)=a_n \ x^n+a_{n-1} \ x^{n-1}+\cdots+a_1 \ x+a_0$$ be a $n$-th degree polynomial with positive coefficients such that all of its roots are real. Choose any number terms from this expression ($...
Balaji Mallikarjun S's user avatar
4 votes
0 answers
140 views

Condition under a function is uniquely identifiable by the supremum values

Let $f(x),g(x)$ be two real-valued functions on $\mathbb{R}$ and $h(x,y)$ be a real-valued function on the plane. We can assume continuity (maybe piecewise differentiability also) of these functions. ...
mukhujje's user avatar
  • 271
0 votes
0 answers
84 views

Question on approximation of norms

Suppose that $E\in Int[L_{p},L_{q}]$ for some $1<p<q<\infty$ and $E$ is $w$-concave with $1<w<\infty$. It is well-known that for each $r\geq w$, we have $E=L_{r}\odot F_{r}$ for some ...
Sijie Luo's user avatar
1 vote
1 answer
130 views

Existence of solutions to a series of integral equations

I am trying to solve the following integral equation analytically: $$ \sum_{n \geq 1} \left( \int_0^te^{-n^2(t-s)} f_n(s) \, ds \right) = g(t), \quad t \in [0, T], $$ where $(f_n(t))_n$ is the unknown ...
Gustave's user avatar
  • 617
0 votes
0 answers
36 views

Derivate involving Bessel function of second type

Let. $$f := (x, y) \mapsto \text{BesselK}(1, c \cdot (a - b \cdot (x + y))) \cdot \exp(c \cdot b \cdot (y - x))$$ Is there a close formula for this $$\frac{\partial^{m+n}}{\partial y^m \partial x^n} f(...
Ryo Ken's user avatar
  • 109
2 votes
0 answers
57 views

Mappings that preserve local or global minimum

In the most general form, I'm interested in any non-trivial results of the following question. Consider metric space $X$ and $Y$, denote all real valued functions on $X$ and $Y$ as $\mathbb{R}^{X}$ ...
patchouli's user avatar
  • 275
3 votes
1 answer
176 views

Question about Lebesgue Bochner spaces

Let $T>0$ and $\Omega\subset\mathbb{R}^N$ be a bounded domain. Also $p\in (1,\infty)$ is any number. I know that $u\in L^{p}((0,T);L^p(\Omega))$ and $\nabla u\in L^{p}((0,T);L^p(\Omega))^N$. How ...
Bogdan's user avatar
  • 1,759
2 votes
1 answer
117 views

Special density on $L^2$

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain, and $u\in L^2(\Omega)$ with $0\leq u(x)\leq 1$ a.e. on $\Omega$. It is well known that $C^{\infty}_c(\Omega)$ is dense in $L^2(\Omega)$. Because $C^...
Bogdan's user avatar
  • 1,759
3 votes
0 answers
167 views

Bounding the $L^{p*}$ norm from below for functions satisfying a $p$-capacity estimate

If $1 \le p < n$, the $p$-capacity of a compact set $A \subset \mathbb{R}^n$ with respect to an open set $U$ containing it is defined as $$\text{Cap}_p(A, U) := \inf \left\{\int_U |\nabla u|^p \, ...
Cauchy's Sequence's user avatar
2 votes
0 answers
29 views

Steiner symmetrization of smooth function on non-simply connected regions

Given a smooth function $u$ defined on $\mathbb{R}^2$, restrict $u$ to a subset $\Omega \subset \mathbb{R}^2$ (possibly not simply connected) foliated by level sets of a smooth function $\psi: \Omega \...
MathLearner's user avatar
1 vote
2 answers
209 views

Approximate simple function $f$ by a sequence of continuous functions on $\mathbb{R}^d$ such that $\|f_n\|_\infty\leq \|f\|_\infty$

Let $f=\sum_{i=1}^n c_i 1_{\Delta_i}$ be a simple function on $\mathbb{R}^d$, where $c_i\in\mathbb{C}$. Then we can find sequnces of continuous functions $\{f_k^{(i)}\}$ for each $i=1,\ldots,n$ such ...
mathlover's user avatar
0 votes
0 answers
56 views

What is the maximum of $ \frac{\sin(n(x+a))}{\sin(x+a)} + \frac{\sin(n(x-a))}{\sin(x-a)}$?

I have asked this here. Due to inactivity and no satisfying answers, I am asking here. Hope that's okay. We know the global maxima of the function $\frac{\sin(nx)}{\sin(x)}$ is $n$ (thanks to this ...
RajaKrishnappa's user avatar
2 votes
1 answer
118 views

Proving that a polynomial $f(x,y)$ that is unbounded in every direction is bounded below by $1$ outside of a disc of finite radius

This is a follow up from this question. I have a polynomial function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the ...
Ryan Hendricks's user avatar
1 vote
1 answer
76 views

Proving that a function $f(x,y)$, that is unbounded in every direction, is uniformly bounded below by $1$ outside some disc of large enough radius

I have a smooth function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the curve $(ta,tb)$, then $$\lim_{t\to\infty}f(ta,...
Ryan Hendricks's user avatar
1 vote
0 answers
175 views

Solution of recurrence relation with summation

I have the following recurrence relation: $$b(n,k)=\sum _{\text{i}=0}^{2 n-1} \left(b(n-1,k-\text{i})+\frac{\text{i} (2 n-\text{i}) \binom{2 n-1}{\text{i}} \binom{(n-2)^2}{k-\text{i}}}{2 n-1} \right)$$...
Cardstdani's user avatar
0 votes
0 answers
79 views

Is the Bures metric equivalent to the Euclidean one?

Let $K=\mathbb R$ (reall numbers) or $K=\mathbb C$ (complex numbers). Define $\mathcal M_n$ to be the space of $n\times n$ matrices $A=(a_{i,j})_{1\le i,j\le n}$, with $a_{i,j}\in K$. Let $\|\cdot\|$ ...
GJC20's user avatar
  • 1,334
0 votes
0 answers
66 views

convolution of the fundamental solution with the homogeneous solution

I have a question about the convolution of the fundamental solution with the homogeneous solution. Namely if the 2 are convoluble then the homogeneous solution is necessarily zero? Let $U$ and $E$ ...
Alucard-o Ming's user avatar
2 votes
1 answer
211 views

Hölder continuity in time of heat semigroup for regular initial distribution

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} $ Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e., $$ p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \...
Akira's user avatar
  • 835
3 votes
1 answer
263 views

Hölder continuity in time of heat semigroup

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} $ We fix $\alpha \in (0, 1)$ and $c>0$. Let $\ell : \bR^d \to \bR_+$ be a probability density function such that $$ \|\ell\|...
Akira's user avatar
  • 835

1
2 3 4 5
29