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Can't figure out "standard application" of the Garsia-Rodemich-Rumsey Lemma

I'm currently reading the paper http://arxiv.org/abs/0908.2473 and can't figure out what they call a "standard application" of the Garsia-Rodemich-Rumsey lemma (see p.8). Summed up, they have a ...
r_faszanatas's user avatar
3 votes
1 answer
6k views

Classical Derivative, Weak Derivative and Integration by Parts

Hello, While studying Sobolev spaces, the following question came to my mind. Any help in this direction is appreciated. QUESTION Let $U\subseteq\mathbb{R}^n$ be open. Does there exist a function $...
Tatin's user avatar
  • 895
0 votes
1 answer
104 views

Operator identity for convergent series

Let $T_i$ and $S_i$ be a sequence of bounded operators such that $$ \sum_{k,i,j=0}^{\infty} S_j^* T_i^* T_i S_k$$ converges unconditionally in operator norm on some Hilbert space. The limit is then ...
Jason O Neil's user avatar
2 votes
1 answer
104 views

Limits of a quasiperiodic function with two pseudoperiods

Let $\beta$ be a real number such that $\beta^2\notin\mathbb{Q}$. For any smooth function $f$ on $\mathbb{R}$ that decreases sufficiently at infinity, for example a Gaussian function, let us define $$ ...
Sylvain Ribault's user avatar
0 votes
2 answers
720 views

Is there a probability density function satisfying the following conditions?

I find myself in need of the solution of this problem in finding a probability density function. I had asked this question in Math Stack Exchange but I did not get an answer so I am posting it here. ...
Nilotpal Kanti Sinha's user avatar
2 votes
0 answers
136 views

To find a positive function with compact spectrum

Let $e_1=(0,1)^T$, $$ S=\left\{x\in \mathbb{R}^2\Big| \frac{|\langle x, e_1\rangle|}{|x|}>\delta>0\right\}, $$ is a cone in $\mathbb{R}^2$. I want to find a non-trivial smooth function ...
John Zhao's user avatar
1 vote
1 answer
238 views

Does the bounded extension of the Fourier multiplier operator agrees with its original explicit definition?

We consider the Fourier multiplier operator $T_0$ defined by the explicit expression $$(T_0f)(x)=\int_{\mathbb{R}^n}{e^{ix\cdot \xi}m(\xi)\hat{f}(\xi)d\xi}, \ f\in S(\mathbb{R}^n),$$ where $S(\mathbb{...
Mr.right's user avatar
  • 171
3 votes
1 answer
146 views

Radial Kernel with Bounded Support and Norm of Gradient Bounded by a Dimension-free Constant

I was wondering if it is possible to construct a compactly supported radial kernel function in $\mathbb{R}^d$ such that the norm of the gradient is bounded by some dimension-free constant. That is, ...
Steve's user avatar
  • 1,127
4 votes
0 answers
716 views

Can one integrate around a branch-cut?

How meaningful is it to try to integrate around the branch-cut of a function? For example lets say I have the function $\log(z^2+a^2)$ for $a>0$ and I choose my branch-cuts to be starting at $\pm ...
user6818's user avatar
  • 1,893
1 vote
1 answer
124 views

On a weaker condition of summability for Fourier series

The Wiener algebra $W:=W(\mathbb{T}^n)$ on the torus is defined as the algebra of all continuous fonctions $f$ on $\mathbb{T}^n$ such that $(\widehat f(k))_{k\in \mathbb{Z}^n} \in \ell^1(\mathbb{Z}^n)$...
Phil-W's user avatar
  • 1,035
2 votes
0 answers
78 views

Generalization of supersymmetry to dimension 3

in two dimensions there is a simple trick to study the spectrum of operators of the form $$\textbf{A}:=\left( \begin{matrix}0 && A^* \\ A && 0 \end{matrix}\right)$$ The trick is to ...
Zehner's user avatar
  • 167
4 votes
0 answers
500 views

Properties of the solution of the heat equation

Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention. Note 2: This is my research problem, but the original problem ...
JumpJump's user avatar
  • 679
3 votes
1 answer
2k views

A question about a formal power series manipulation

I want to find a function $f(x,y)$ which can satisfy the following equation, $\prod _{n=1} ^{\infty} \frac{1+x^n}{(1-x^{n/2}y^{n/2})(1-x^{n/2}y^{-n/2})} = exp [ \sum _{n=1} ^\infty \frac{f(x^n,y^n)}{...
Anirbit's user avatar
  • 3,541
0 votes
1 answer
348 views

Request for references about computing or estimating Rademacher complexity

Is Rademacher complexity defined for any space of functions? Or are there restrictions on the function space over which this can be defined? For example is the Rademacher complexity defined or has ...
Student's user avatar
  • 617
2 votes
0 answers
186 views

Is this simple oscillatory integral operator uniformly bounded on $L^2$?

Let $\phi(t,s)$ be a real-valued function smooth away from the diagonal, and equal to 0 on the diagonal. Assume that $0\le \phi(t,s)\le |t-s|$ for $t,s\in \mathbb{R}$. Let $$T_\lambda f(t)=\int \frac{\...
Mr.right's user avatar
  • 171
2 votes
2 answers
1k views

Approximation of smooth compactly supported functions on $\mathbb{R}^2$ using sums of products of one variable functions

Let $f \in C^{\infty}(\mathbb{R}^2)$ be smooth and compactly supported. Can we approximate $f(x,y)$ by sums of the form $\sum_{i=1}^m g_i(x) h_i (y)$ where $g_i, h_i \in C^{\infty}(\mathbb{R})$ are ...
ebg's user avatar
  • 33
0 votes
1 answer
217 views

Reproducing Kernel Hilbert Spaces with positive kernels

In my research I'm dealing with the following question. Let $E$ set, $K:E \times E \to \mathbb R$ a positive type function, and $\mathcal H := \mathcal H(1+K)$ (in the sense of the Moore theorem). ...
3Matrolod's user avatar
5 votes
1 answer
187 views

Getting out a system of linear ODEs by knowing the Magnus expansion

Assume we are given for a transition between two time points $t_0 = 0$ and $t_1$ a matrix relationship, eventually describing the solution of a system of linear with non-constant coefficients, $$Y(t_1)...
tobias's user avatar
  • 749
1 vote
2 answers
923 views

Spectrum of Mathieu equation

I have the differential equation $-f''(x)-q \cos(x) f(x) = \lambda f(x)$ and I want to find all the eigenvalues of this equation analytically on $[0,2\pi]$ that satisfy the boundary condition $f(0) = ...
user avatar
1 vote
0 answers
45 views

Shifting Sobolev norms in a hyperbolic estimate

Suppose $\Omega$ is a bounded domain and $\omega \subset \Omega$. Suppose we have the following estimate: $$ \|u\|_{H^1((0,T) \times\Omega)} \leq C (\|u\|_{H^1((0,T) \times \omega)} + \|\Box u\|_{L^2((...
Ali's user avatar
  • 4,143
1 vote
0 answers
124 views

Inequality about the Fourier transform: $\Vert u \Vert_{L^k} \le \Vert \mathcal{F}(u) \Vert_{L^m}$ (where $1 \le m \le 2$ and $m,k$ Holder conjugates)

How can I prove the following inequality about the Fourier transform? $$\Vert u \Vert_{L^k(\mathbb{R}^N)} \le \Vert \mathcal{F}(u) \Vert_{L^m(\mathbb{R}^N)}$$ for $1 \le m \le 2$ and $m,k$ Holder ...
user avatar
1 vote
0 answers
50 views

Comparison of (square) of a function and its Fourier transform in an integral

I am completely stuck on a comparison between $f(t)^2$ and $\hat{f}(t)^2$ in an integral. Considering $f(t)$ of rapid decrease at infinity such that near zero: $f(t) \sim_0 t^{-\frac{1}{2}- \alpha}+o(...
Bertrand's user avatar
  • 1,199
0 votes
1 answer
731 views

Reproducing Kernel of a RKHS of continuous functions may not be continuous in two variables together

Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$. That is $\mathcal{K}$ is RKHS, ...
partha's user avatar
  • 3
4 votes
1 answer
224 views

Urysohn type cut off function

I am looking for a cutoff function. The Urysohn's Lemma says Let $X$ be a $T_{4}$ space and $A,B \subset X$ be two closed and disjoint subsets of $X$. Then there exists a continuous function $f:X \...
sharpe's user avatar
  • 721
1 vote
1 answer
162 views

Does there exist a class of real-valued upper semicontinuos functions on $X$ such that $\mathcal{F}$ is countable?

Ian Morris quoted the following: For any upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a nonempty topological space $X$ there exists a nonempty set $\mathcal{F}\...
Idonknow's user avatar
  • 623
1 vote
1 answer
390 views

Square Integrable Harmonic Functions in an Infinite Strip

Suppose $S= \left\{x \in \mathbb{R}^3 : a <x_1< b \right\} $ is an infinite strip the three dimensional Euclidean Space. Is it true that the only $L^2$ harmonic function in this strip is the ...
Ali's user avatar
  • 4,143
3 votes
0 answers
177 views

Interesting stipulation about completely monotone functions

This question relates to a question I asked here. I thought of a well thought out generalization which appears to follow in the situations I've encountered it. I tried to generalize the answer ...
user avatar
8 votes
1 answer
597 views

complete metric space

Hallo, I have the following question: Let $(X,d)$ be a complete metric space. Is then $(X,\operatorname{dist})$ also complete? Here by $\operatorname{dist}$ I mean the metric induced by $d$ by: $\...
denis's user avatar
  • 83
2 votes
2 answers
197 views

About preserving real-rootedness of multivariable polynomials

Say $f_i(z_1,z_2,..,z_m)$ are polynomials real rooted in the $z$s for a bunch of polynomials indexed by $i$. When can one say that $\sum_{i} p_i f_i(z_1,z_2,..,z_m)$ is also real rooted? If ...
guest's user avatar
  • 31
2 votes
0 answers
125 views

Constant periodic Sobolev embedding

Dear mathoverflowers, I would like to have a reference regarding the optimal constant in the Sobolev embedding $$ \|u\|_{L^q}\leq C_{s,q}\|u\|_{\dot{H}^s}, $$ ($H^s$ denotes the standard L^2 ...
guacho's user avatar
  • 843
-1 votes
1 answer
346 views

An infinite set in a compact space

Let $X$ be a topological space. Is there any characterization for the property that says "for every infinit subset $A$ of $X$ there exists $a\in A$ such that if $f$ be an arbitrary real continuous ...
robert caro's user avatar
2 votes
0 answers
139 views

Existence of solution of a variational inequality

Let $K\subseteq \mathbb{R} ^n$ be closed and convex, and let $F:K \to \mathbb R^n $ be a continuous function. If for every $x,y \in K$ we have $$(x-y)^T(F(x)-F(y))\ge \alpha ||x-y||^2 \, ;\quad \...
a.a's user avatar
  • 21
3 votes
0 answers
306 views

Metric analogues of bounded variation

A function $f:[a,b]\to\mathbb{R}$ is said to be of bounded variation if $$ \sup_I \sum_{i=1}^n |f(x_i)-f(x_{i-1})| \le V $$ for some finite $V>0$, where the supremum is over all finite partitions $...
Aryeh Kontorovich's user avatar
1 vote
0 answers
194 views

Cotlar-Stein's Lemma and the Dirichlet kernel

It is well-known that Cotlar-Stein's Lemma can be used to prove the $L^2$ boundedness of the Hilbert transform. See e.g. $L^2$ boundedness of the Hilbert transform via Cotlar-Stein Lemma. Then using ...
Mr.right's user avatar
  • 171
7 votes
1 answer
306 views

An indicator of a planar subset as an element of a tensor product

Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that $$ f^2=f $$ (that ...
limanac's user avatar
  • 452
1 vote
1 answer
114 views

Reference request: regularity of functionals on the space of probability measures

Let $\mathcal M=\mathcal M(\mathbb R^d)$ be the space of finite measures on $\mathbb R^d$, and $\mathcal P=\mathcal P(\mathbb R^d)\subset\mathcal M$ be the space of probability measures. Let $F:\...
CodeGolf's user avatar
  • 1,835
5 votes
1 answer
229 views

Does this infinite sum arising from separation of variables converge?

This problem came up in a PDE where I used separation of variables to formally get a solution. Now I need to know whether that formal solution is sensible. Let $a_k >0$ be an increasing sequence ...
RealMax's user avatar
  • 53
1 vote
0 answers
105 views

Generalize characterization of upper semicontinous functions

Let $X$ be a metric space and denote $f:X \rightarrow \mathbb{R}.$ It is easy to show that the following two statements are equivalent: $(1)$ For any real number $c$, we have $f^{-1}(-\infty,c)$ is ...
Idonknow's user avatar
  • 623
2 votes
4 answers
3k views

Splitting a space into positive and negative parts

Let $V$ be a vector space over $\mathbb R$. A symmetric bilinear pairing on $V$ is a linear map $a: V\otimes V \to \mathbb R$. Because $\mathbb R$ is characteristic not-two, I will freely confuse ...
Theo Johnson-Freyd's user avatar
1 vote
0 answers
182 views

The real method of interpolation and operator ideals

Let $\overline{A} \mbox{ and } \overline{B}$ be n+1-tuples of Banach spaces and $T:\overline{A}\rightarrow \overline{B}$ be an interpolation operator; let $J(\overline{A})$ and (the corresponding, ...
Alexi Quevedo S.'s user avatar
2 votes
0 answers
79 views

One-dimensional integral equation uniquely solvable?

I recently met a question similar to this one and I would like to post it here, because I basically found nothing: We define the (possibly unbounded) integral operator $T:D(T) \subset C_0(\mathbb{R}) ...
BaoLing's user avatar
  • 329
2 votes
0 answers
115 views

Does this Sobolev-space like construction have a name?

Take $\Omega \subset \mathbb{R}^n$ arbitrary then define as $X$ the closure of $C^1(\Omega) \cap W^{1,1}(\Omega)$ w.r.t. the norm $f \mapsto \left\lVert f \right\rVert_{\infty} + \left\lVert \nabla f \...
Kinzlin's user avatar
  • 305
2 votes
2 answers
2k views

Does the Fourier series of an $L^1$ function converge to the function *weakly* in $L^1$?

Let $f$ be a periodic $L^1$ function, and $S_n[f]$ the $n$-th partial sum of its Fourier series. I am aware that $S_n[f]$ might not converge toward $f$ in $L^1$ (i.e., in norm). However, does it at ...
Gro-Tsen's user avatar
  • 32.5k
2 votes
0 answers
58 views

Absolute continuity of DOS measure for Schrödinger operators

Kotani theory gives roughly that for ergodic operators there is a certain equivalence between absolutely continuous spectrum and an absolutely continuous density of states measure. I would like to ...
DDriggs's user avatar
  • 21
1 vote
2 answers
226 views

Smooth but non-analytic kernel functions

Does there exist a (stationary) covariance kernel function which is $C^\infty$-smooth but not real analytic? If so, could you please provide an example?
Tom LaGatta's user avatar
  • 8,512
2 votes
0 answers
183 views

Fourier series and regular distribution

Assume you have a distribution $K$ on $\mathbb{T}$, the torus, such that $\sum_{n=-\infty}^{\infty} |K(e_n)|^2$ is finite, where $e_n := e^{in\cdot}$ are the Fourier basis. Does this imply that the ...
plain's user avatar
  • 95
2 votes
0 answers
86 views

when is the average of a function with Gaussian inputs bounded away from zero

Define a function $\phi(x):\mathbb{R}\rightarrow\mathbb{R}$. Consider the expected value function defined as follows \begin{align*} \mu(\beta)=E[g\phi (\beta g)]\quad with \quad g\sim\mathcal{N}(0,1)\...
Anahita's user avatar
  • 363
1 vote
0 answers
331 views

Verifying a claim regarding $H^1$ weak convergence and $L^2$ strong convergence on a surface

I'm reading a paper whose first section discussed $H^1$ maps defined on star-shaped sets, but I got stuck in verifying a claim for quite a while. I'm now thinking the claim is wrong, but it's hard to ...
student's user avatar
  • 1,350
3 votes
1 answer
210 views

Using $H^2$ to find a cyclic vector in $\ell^2$

Let us consider $\ell^p(\mathbb{Z})$. We know that the vector $e_1=(\dots,0,0,1,0,0,\dots)$ is a cyclic vector in sense that given the right shift operator $S:(\dots,x_0,x_1,x_2,\dots)\mapsto (\dots,...
Michael's user avatar
  • 31
0 votes
1 answer
705 views

Continuity of a Functional

A certain functional $T$ is defined as: $$T(F)=\int_{(0,1)}F^{-1}(s)M(ds)$$ where $M$ is a probability measure with support $[\alpha,1-\alpha]$,for $\alpha>0$. The result that above functional is ...
Dinesh's user avatar
  • 111

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