Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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2 votes
1 answer
148 views

Construction of the Lipschitz function with a given Lipschitz constant, given two values and with small Lipschitz norm

Let the function $f\colon [a,b] \to\mathbb{C}$ be Lipschitz and let $|f(a)| \geq c,$ $|f(b)| = c$ and $\varepsilon > 0.$ It is easy to see that if $\|f\|_{\infty}< \frac{\varepsilon}{2} =: \...
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3 votes
1 answer
177 views

General version of Weyl's lemma

The classical Weyl's lemma say, suppose $u \in L^1_{loc}(\Omega­)$ satisfies $$\int_{\Omega}u \Delta \phi dx=0\ \ \forall \phi\in C_c^{\infty}(\Omega),$$ then $u$ is harmonic in $\Omega.$ What I want ...
  • 379
0 votes
0 answers
48 views

Construct RKHS $H$ on $\mathbb R^d$ s.t for every $C \gt 0$, there exists $f_0 \in H$ with $\|f_0\| = 1$ and $\|\nabla f_0\|_{L^2(\gamma_d)} \gt C$

Fix a positive integer $d$, and let $\gamma_d$ be the standard Gaussian measure on $\mathbb R^d$. Question. Construct (or prove non-existence of) a reproducing kernel Hilbert space (RKHS) $H \subseteq ...
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2 votes
0 answers
82 views

Noether's theorem in the critical heat equation

I initially posted this question on Stackexchange Mathematics (see here) but as I got no answer, maybe someone here will be able to help me. I am watching a serie of lectures on "Blow up solution ...
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1 vote
1 answer
102 views

Construction of the Lipschitz function with a given Lipschitz constant and given two values

Let the function $f\colon [a,b] \to\mathbb{C}$ be Lipschitz and let $|f(a)| \geq c$ and $|f(b)| = c$. Is there a Lipschitz function $g$ such that $|g| \geq c,$ $g(a)=f(a),$ $ g(b)=f(b)$ and Lipschitz ...
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0 votes
1 answer
132 views

Let $X^n$ be a collection of smooth functions so that their $\alpha$-Holder norms are uniformly bounded

Let $X^n$ be a collection of smooth functions so that their $\alpha$-Holder norms for $\alpha \in (1/3,1/2)$ are uniformly bounded - that is $\sup_n \|X^n\|_\alpha<\infty$. Define the standard ...
2 votes
0 answers
46 views

The initial sigma-algebra on the dual of a Banach lattice

Let $E$ be an AL space (i.e. a Banach lattice whose norm is additive on the positive cone $E_+$) that satisfies Mazur's condition (every sequentially weak$^*$-continuous functional on $E'$ is weak$^*$-...
3 votes
1 answer
109 views

General form of bounded linear functionals on Banach spaces

It is a famous result due to Riesz that every bounded linear functional $f$ on a Hilbert space $\mathcal{H}$ is of the form $f(x)=\langle x,z \rangle$ for a unique $z\in H$. On p.188 of Introductory ...
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1 vote
0 answers
72 views

Find a finite partition of unity in the centralizer of a von Neumann algebra

Let $M$ be a von Neumann algebra and $\varphi$ is a faithful normal state on $M$. Suppose that $M_{\varphi}$ is a type II$_1$ factor. Suppose we have the following conclusion: if $p$, $q$ are any two ...
7 votes
0 answers
142 views

The spectrum of the Banach algebra of certain arithmetic functions under Dirichlet convolution

I was thinking about using the tools of functional analysis to study some subring of arithmetic functions under Dirichlet convolution. If I let $D_s$ be the ring of arithmetic functions with finite ...
2 votes
1 answer
53 views

Is the measure density condition a necessary condition for bounding the Sobolev norm $W^{n,p}(\Omega)$ by the extremal terms?

Let $\Omega \subseteq \mathbb{R}^M$ be a measurable subset of positive measure. R. A. Adams and J. Fournier in their article have proven that if $\Omega$ satisfies the so-called weak cone property, ...
3 votes
1 answer
206 views

A pexiderization of the sine addition law on semigroups

Can we solve the follwing functional equation $$f(xy)=g(x)h(y)+g(y)h(x)$$ on semigroups for unknown complex valued functions $f,g,h$ ?
1 vote
1 answer
185 views

Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$?

Let $\Omega$ be a metric space, $C_b(\Omega)$ the space of all real-valued bounded continuous functions on $\Omega$, and $\mathcal{M}(\Omega)$ the space of all finite signed Borel measures on $\Omega$...
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0 votes
1 answer
94 views

Complex Borel measures: relation between the total variation norm of a measure and those of its real and imaginary parts

Let $X$ be a metric space and $\mathcal B$ its Borel $\sigma$-algebra. For $B \in \mathcal B$ we denote by $\Pi(B)$ the collection of all finite measurable partitions of $B$, i.e., $$ \Pi(B)=\left\{\...
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-2 votes
0 answers
226 views

Meromorphic functions conjecture

For a meromorphic function $f$ and a complex number $z$ in the domain of $f$, we define $d(f,z) := \inf\{ \lvert z-a\rvert\colon a \neq z, f(a) = f(z)\}$. If there is no $a \neq z$ in the domain of $f$...
2 votes
1 answer
146 views

Approximating a function by a convolution of given function?

Let $g:\mathbb{R}\to \mathbb{R}$ be a given differentiable function of exponential decay on both sides. Now let us be given a function $f:\mathbb{R}\to \mathbb{R}$, also of exponential decay, if you ...
4 votes
2 answers
169 views

Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation

I'm reading a proof of below theorem from this paper. Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...
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0 votes
0 answers
53 views

Let $E$ be Banach, $\mu_n\to\mu$ weakly on a locally compact $X$, and $f \in C_b(X, E)$. Does $\int f\mathrm d\mu_n\to\int f\mathrm d\mu$ in norm?

Let $X$ be a metric space, $(E, |\cdot|)$ a Banach space $\mathcal M(X)$ the space of all finite signed Borel measures on $X$, $\mathcal C_b(X)$ be the space of real-valued bounded continuous ...
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3 votes
1 answer
161 views

On the domain of the Neumann Laplacian

Let $U$ be a bounded domain of $\mathbb{R}^d$, and write $m$ for the Lebesgue measure on $U$. For $k=1,2$, we denote by $H^k(U)$ the set of all locally $m$-integrable functions $u\colon U \to \mathbb{...
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8 votes
1 answer
439 views

Bounding the discrete $l^p$ norm by the continuous $L^p$ norm for trigonometric polynomials

Let $ X_N = \text{span} \{\cos(2\pi lx): l=0, \cdots, N-1 \} $ with $ x \in [0, 1] $ and $ Y_N = \{v =(v_0, \cdots, v_{N-1}): v_j \in \mathbb{C}\} = \mathbb{C}^N $. Then $ X_N $ is the space of ...
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2 votes
0 answers
91 views

Eigenfunction of $h\mapsto H(h')|_{[-1,1]}$?

Let $H$ be the Hilbert transform. Is there a continuous, even function $h:\mathbb{R}\to \mathbb{R}$ with support on $[-1,1]$ such that, for some $\lambda\in \mathbb{R}$, $$H(h')(t) = \lambda h(t)$$ ...
2 votes
0 answers
237 views

A periodic integral inequality

(This problem comes in connection with a geometric problem exposed here.) Let $\gamma(x,y)$ be a (real) function on the unit disk such that \begin{align} \frac{\partial^2\gamma}{\partial x \partial y}=...
0 votes
0 answers
56 views

A little question about the derivation of the generalized alternating projection (GAP) algorithm

This question is actually just a question about Euclidean projection. I've been studying some articles on the generalized alternate projection (GAP) algorithm recently, but have a little question ...
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0 votes
0 answers
50 views

Equivalence of discrete and continuous $ L^p $ norms of trigonometric polynomials

Let $ \Omega = [-\pi, \pi] $ and $ X_N = \text{span}\{e^{i k x}:k=-N, \cdots, N-1\} $ where $ x \in \Omega $ be the space of trigonometric polynomials. On $ X_N $, we can define usual $ L^p(1 \leq p \...
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2 votes
2 answers
200 views

Minimal conditions on random vector $X \in R^n$ to ensure that $\lim_{t\to 0^+}\sup_{\|w\|_p = 1}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t)=0$

Let $X$ be a random variable on $\mathbb R^n$ and let $S_p^n := \{w \in \mathbb R^n \mid \|w\|_p = 1\}$ be the unit-sphere w.r.t to the $\ell_p$-norm in $\mathbb R^n$. We will be particularly ...
  • 6,094
0 votes
0 answers
45 views

Weyl's law for hyperbolic operators

Let $\Omega \subset \mathbb{R}^n$ be a smooth, bounded domain and consider the operator $T: L^2([0,1]\times\Omega) \to L^2([0,1]\times\Omega)$ so that $Tf = v$ if $$ \begin{cases} \frac{d}{dt}v - \...
2 votes
1 answer
112 views

On a core for Neumann Laplacian on $C(\overline{D})$

Let $D \subset \mathbb{R}^d$ be a bounded $C^1$ domain. We consider a reflected Brownian motion $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in \overline{D}})$ on $\overline{D}$. Let $\{p_t\}_{t>0}$ denote ...
  • 499
0 votes
1 answer
200 views

Deduce that a function is zero on interval $[0,M]$

I have been thinking about this for the last few days but I was not able to produce a definitive answer. Take an integrable function $g$ that maps in $\mathbb{R}$ and with domain contained in $[0,M]$ (...
0 votes
0 answers
40 views

What are general Horizon functions?

I was reading a paper from arxiv. There I found this definition of horizon functions Definition 2.2. Let $d \in \mathbb{N}_{\geq 2}$. Given any function $b:[0,1]^{d-1} \rightarrow[0,1]$, define the ...
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1 vote
1 answer
197 views

On a property of complex exponentials

Does there exist a simple smooth closed curve $\gamma:S^1\to \mathbb C$ such that $$ \int_{0}^{2\pi} e^{\gamma(e^{it})} \, |\gamma'(e^{it} )|\,dt =0?$$
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6 votes
2 answers
153 views

Does the (distributional) support of the Fourier transform of an $L^p$-function with $p<\infty$ have positive measure?

Suppose that $f \in L^p(\mathbb R^n)$ such that $1\leq p < \infty$. Let $\hat f$ be the Fourier transform of $f$. Clearly, if $p=1$ or $p=2$ then the support of $\hat f$ has positive Lebesgue ...
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0 votes
0 answers
21 views

Sufficient conditions on the density of a random vector $X$ to ensure that the density of $X^\top w$ is Holder-continuous for all $w \in \mathbb R^d$

Let $X$ be a random vector $\mathbb R^d$, with density $f$. For any any unit-vector $w \in \mathbb R^d$, let $\rho_w$ be the density of the random vector $X^\top w$. Question 1. Under what minimal ...
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0 votes
0 answers
134 views

Finding a unitary operator on L^{2}(\mathbb{R}^{2},dxdy)

I have a one parameter (r) family of self-adjoint representations of the universal enveloping algebra of some nilpotent Lie group on $L^{2}(\mathbb{R}^{2},dxdy)$ as follows: $$\hat{X}^{r}=\hat{x}-i(r-...
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0 votes
0 answers
68 views

A good estimation for the norm of a matrix

For a given natural number $n$, let us consider $E_n=\{0\leq k\leq n-1 : k\equiv_41 \}$. Suppose that $E_n$ including $k_1<k_2\cdots < k_m$. Consider the following matrix $A$: $$A=\left(\cos\...
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2 votes
0 answers
34 views

Selecting some linearly independent columns of a particular matrix

Let us consider the matrix $C=A_1+A_2$ where : $A_1=(a_{k,l})_{k,l=0}^{n-1}$ is the $n$ by $n$ matrix given by $a_{k,l}=\frac{2}{\sqrt{n}}(\cos\frac{2kl\pi}{n})$ $A_2$ is the the $n$ by $n$ block ...
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3 votes
3 answers
284 views

Do these properties characterize Hilbert spaces?

Suppose $X$ is a Banach space with the following property: For any $x\in X$ there exists a two dimensional subspace $E$ isometric with $l_2^2$ such that $x\in E$. Does this property characterize a (...
  • 1,103
1 vote
0 answers
100 views

On some integral equation

Let $M$ be the set of continuous and increasing functions $h: [0,1)\to\mathbb R_+$ s.t. $h(0)=0$ and $h(1-)=+\infty$. Consider the equation as follows: $$ 1-t= \int\limits_0^\infty p(x)\Phi\left(\frac{...
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1 vote
1 answer
49 views

Is the Lorentz space $L_{W,1}(0,1)$ isomorphic to $L_1(0,1)$?

Let $W$ be a positive non-increasing continuous function on $(0,1]$ so that $\lim_{t \rightarrow 0} W(t)=\infty$, $W(1)=1$ and $\int_0^1 W(t) dt =1$. For $1 \leq p <\infty$, the Lorentz function ...
8 votes
4 answers
484 views

Uniform density of Lipschitz maps is space of continuous function — for general metric spaces

Let $X$ and $Y$ be metric space, $X$ be compact, $C(X,Y)$ denote the set of continuous functions from $X$ to $Y$ with uniform convergence on compacts topology, and $\operatorname{Lip}(X,Y)$ denote the ...
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4 votes
0 answers
69 views

Banach spaces $X$ with $L_2(X)$ isomorphic to $\ell_2(X)$

I've recently come across this interesting thread Banach spaces $X$ with $\ell_2(X)$ not isomorphic to $L_2([0,1],X)$ I'm interested in the opposite question. Are conditions known that ensure a Banach ...
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3 votes
1 answer
81 views

Support projection vs closed support projection of a normal state in enveloping von Neumann algebra

I preface this by saying that I am fairly new to the enveloping von Neumann algebra scene, so there may be some gaps in my understanding. Given a $C^*$-algebra $A$ and a state $\phi$ on $A$, one may ...
  • 93
5 votes
2 answers
285 views

Why is $\frac{1}{|x|^{n-2}}u(\frac{x}{|x|^2})$ harmonic if $u$ is harmonic?

I found myself trying to prove the following, but I had to compute everything explicitly. It is well known that if $u:\mathbb{R}^n\to\mathbb{R}$ is an harmonic function on $\mathbb{R}^n$, then the so-...
3 votes
1 answer
264 views

Endpoint Calderon-Zygmund inequality of nonlocal fractional laplacian

For $s\in(0,1],$ consider the following non-local fractional laplacian: $$(-\Delta)^sv= f ~~\text{on } \mathbb{R}^n.$$ Then how to use "the standard elliptic estimate" to obtain: for $p\in[...
3 votes
0 answers
35 views

Sufficient conditions for the weight function to have compact embedding of a weighted Sobolev space

Let $\rho$ be a smooth density function on $\mathbb{R}^N$, that is, $\rho(x)\ge 0$ for all $x\in\mathbb{R}^N$ and $\int_{\mathbb{R}^N}\rho(x) dx=1$. Let $L^p_\rho(\mathbb{R}^N)=\{f: \int_{\mathbb{R}^N}...
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1 vote
1 answer
196 views

Are separable, continuous, monotonic and scale invariant real-valued functions everywhere differentiable?

Consider a function $f:\mathbb{R}_+^2\rightarrow\mathbb{R}$ of two non-negative real variables (or more generally of several real variables) that is increasing in each argument, continuous, additively ...
  • 379
0 votes
1 answer
58 views

A solution satisfying an integral inequality is bounded [closed]

Let $y$ be a positive function and $c>0$. If $y$ satisfies the following integral inequality \begin{equation} y(t)+\int_{0}^{t}y(s)ds\leq c_{1}\int_{0}^{t} y^{\frac{3}{2}}(s)ds+c_{2} \end{equation} ...
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1 vote
1 answer
168 views

Finding $W^{1,\infty}$ solutions to an integral equation by fixed point

Let $f \in L^\infty((0,1)\times(0,1))$ and consider the integral equation \begin{align*} u(x, y)= \int_0^y \int_0^{x-y} u(x-y+\tau, s+\tau) f(s+\tau, \tau) d s d \tau -\int_0^y f(x-y+s, s) d s. \end{...
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1 vote
1 answer
75 views

Tight upper-bounds for the Gaussian width of intersection of intersection of hyper-ellipsoid and unit-ball

Let $\Lambda$ be a positive-definite matrix of size $n$ and let $R \ge 0$, which may depend on $n$. Consider the set $S := \{x \in \mathbb R^n \mid \|x\|_2 \le R,\,\|x\|_{\Lambda^{-1}} \le 1\}$ where $...
  • 6,094
1 vote
0 answers
51 views

Properties of spectrally defined Sobolev spaces

Let $\overline{M}$ be a compact manifold with smooth boundary, and consider the Laplace operator $\Delta: H_0^1(M) \to H^{-1}(M)$ and its inverse $\Delta^{-1}: H^{-1}(M) \to H_0^{1}(M)$. Then $\Delta^{...
1 vote
0 answers
83 views

How to prove a concentration isoperimetric inequality for a non-Lipschitz function

Definition $1$. A probability measure $\mu$ on $\mathbb{R}^{d}$ satisfies c-isoperimetry if for any bounded L-Lipschitz $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$, and any $t \geq 0$, \begin{align} \...
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