# Questions tagged [fa.functional-analysis]

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

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

8,502
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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} =: \...

3
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1
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177
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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 ...

0
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0
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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 ...

2
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0
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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 ...

1
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1
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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 ...

0
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1
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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
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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
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1
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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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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
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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
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1
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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
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206
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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
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1
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185
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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 $\Omega$...

0
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1
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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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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
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1
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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 ...

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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 ...

0
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0
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53
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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 ...

3
votes

1
answer

161
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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{...

8
votes

1
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439
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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 ...

2
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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
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(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
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0
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56
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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 ...

0
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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 \...

2
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2
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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 ...

0
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0
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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
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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 ...

0
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1
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200
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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
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0
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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 ...

1
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1
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197
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Does there exist a simple smooth closed curve $\gamma:S^1\to \mathbb C$ such that
$$ \int_{0}^{2\pi} e^{\gamma(e^{it})} \, |\gamma'(e^{it} )|\,dt =0?$$

6
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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 ...

0
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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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134
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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-...

0
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68
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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\...

2
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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 ...

3
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3
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284
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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 (...

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0
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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{...

1
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1
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49
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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
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4
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484
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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 ...

4
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0
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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 ...

3
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1
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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 ...

5
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2
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285
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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
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1
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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
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0
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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}...

1
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1
answer

196
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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 ...

0
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1
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58
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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}
...

1
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1
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168
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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
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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 $...

1
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0
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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^{...

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0
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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}
\...