Questions tagged [fa.functional-analysis]

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

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1answer
65 views

The operator of exponential derivative applied in quotients

I have an other question for a function different to the example given before in the link below: Exponential derivative operator and continuous functions We define for instance a function as: $$H(y)=\...
5
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3answers
548 views

Defects of Hamel bases for analysis in infinite dimensions

I know that Hamel bases have a couple of defects for the purposes of doing analysis in infinite dimensions: (1) Every Hamel basis of a complete normed space must be uncountable. (2) For every Hamel ...
11
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1answer
441 views

An extension of the Carlson's theorem in complex analysis

For the statement of Carlson's theorem please see, https://en.wikipedia.org/wiki/Carlson%27s_theorem. There is an extension of Carlson's theorem that says that the condition that $f$ needs to vanish ...
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1answer
83 views

About $\sigma$ strong$^*$-functionals and seminorms

I'm reading the book "Theory of operator algebras" by Takesaki. In this book, the $\sigma$-strong$^*$ topology on the space $B(H)$ (bounded operators on the Hilbert space $H$) is defined (...
3
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1answer
120 views

norm estimates for Schatten class

Let $C _p$ be the Schatten-p-classes on a separable Hilbert spaces, $p\ge 1$. Let ${\rm Tr}$ be the standard trace. Let $y\in C_p$ be a self-adjoint operator (or even a positive operator) and let $...
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1answer
72 views

Exponential derivative operator and continuous functions

I would like to know how to write down the following expression $$f(y)=\frac{1}{y^{n} e^{\frac{d}{dy}}g(y)}$$ in the form of $e^{-\frac{d}{dy}}y^{-n}(\frac{1}{g(y)})$ where $n$ is an integer and $f,g: ...
1
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0answers
62 views

Minimax type principle for a self-adjoint operator acting on a Hilbert space

Let $T\in\mathcal{B}(\mathcal{H})$ be a self-adjoint operator acting on a Hilbert space $\mathcal{H}$. Suppose $k\in\mathbb{N}$. Define $$\lambda_k(T)=\sup\limits_{\substack{\mathcal{M}\subseteq \...
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1answer
155 views

Approximation of the product $(\bar{z} - a)^{-1} \cdot (z - b)^{-1}$

I would like to construct an approximation of the product \begin{equation} f(z) = \frac{1}{\overline{z}-a} \frac{1}{z-b}, \end{equation} where $a, b \in \mathbb{C}$, and $|{a}/{z}|, |{b}/{z}| <1$. ...
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2answers
128 views

Action of Bochner integral of operator-valued functions on vectors

Consider a separable Hilbert space $\mathcal H$ and the bounded linear operators $B(\mathcal H)$. Consider a function $T: [0, \infty) \to B(\mathcal H)$, under what assumptions on $T(t)$ is it true ...
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0answers
96 views

On $L^2$ spaces which have an orthogonal basis of characters (complex exponentials)

Suppose $\Omega \subset \mathbb{R}^n$. What conditions on $\Omega$ make it so there exists a countable set $\Lambda$ such that $\{e^{2\pi i\lambda t} \}_{\lambda \in \Lambda}$ form an orthogonal basis ...
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2answers
165 views

When is the annihilator of the commutator subspace a complemented subspace?

Let $A$ be a unital Banach algebra and $C$ be its commutator subspace, i.e., $C$ is the norm-closure of the subspace spanned by the elements of the form $xy-yx$ in $A$. Notation: Let $C^{\perp}=\{f\in ...
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0answers
50 views

Lower bound for the tensor product of a semi-bounded quadratic form

When I read YVES COLIN DE VERDIÈRE's paper: Sur la multiplicité de la première valeur propre non nulle du laplacien, he gave a proposition without proof: $\mathcal{H}$ is a Hilbert space, $Q$ is a ...
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0answers
43 views

A pullback in the norm of the Sobolev space $H^{-\frac 1 2 }(\Gamma)$

Let $A$ be a $3\times 3$ real constant symmetric positive definite matrix, $\Omega\subset\mathbb{R}^3$ a bounded Lipchitz domain with boundary $\Gamma$, $\Omega'=A^{-\frac 1 2}(\Omega)$ (so we have $\...
9
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1answer
358 views

The closure of the set of injective continuous functions

Setup/Notation: Let $n,m\in \mathbb{N}$ and let $C(\mathbb{R}^n,\mathbb{R}^m)$ be the space of continuous functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ equipped with the compact-open topology. Let $...
4
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0answers
86 views

Walsh-Lebesgue type theorem in $\Bbb R^{2m}$ for $m>1$

Is someone aware of any analogue of the Walsh-Lebesgue theorem in $\mathbb{R}^{2m}$ for $m>1$ and dealing with polyharmonic polynomials? In this post, $\phi$ is said to be polyharmonic in $\mathbb{...
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1answer
78 views

If $\lambda_i$ is symmetric with $\lambda_i\{0\}=0$, why does $\int_B1-\cos\langle x,x'\rangle\:(λ_1-λ_2)({\rm d}x)=0$ imply $λ_1=λ_2$?

Let $E$ be a separable $\mathbb R$-Banach space and $\lambda_i$ be a finite symmetric measure on $\mathcal B(E)$ with $\lambda_i(\{0\})=0$ and $$\int_B1-\cos\langle x,x'\rangle\:\underbrace{(\lambda_1-...
4
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1answer
520 views

Existence of a smooth compactly supported function

Let $U$ be a bounded domain in $\mathbb R^n$. Does there exist a smooth function $f$ with compact support in $U$ such that: $$ \| f\|_{W^{k,\infty}(U)} \leq (k!)^{2-\epsilon},$$ for some $\epsilon>...
2
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0answers
135 views

Weighted Sobolev norm in terms of Spherical harmonics coefficients

Let $M = [1,\infty) \times S^2$. Consider the weighted Sobolev space $H^k_{\delta}(M)$ with the Sobolev norm: $$\lVert u \rVert_{k,\delta}^2 := \sum_{n=0}^k \int_M |D^nu \,r^{n-\delta}|^2 r^{-3} dV $$ ...
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1answer
92 views

Derivatives and exponential derivatives quotient operators on two variables

I consider for example the following function of two variables given by $$f(x,y)=\sum_{n=0}^{+\infty}\frac{\delta_{x}^{-m} \exp(\delta_{x})}{\delta_{y}^{-m}\exp(\delta_{y})}\left(\frac{x}{y}\right)^{n}...
3
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0answers
171 views

Injectivity of the covolution operator $f \mapsto (k*f(s))_{s \in S}$ via sampling at $S=\alpha \mathbb Z$

Let $S \subset \mathbb R$ be a uniform grid, say $S = \alpha \mathbb Z, \alpha >0$. Let $k$ be a square integrable function and $A$ the operator which maps $f \in L^2(\mathbb R)$ to the sequence $$ ...
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0answers
290 views

Characterization of differentiability

For a normed space $(V, \lVert\cdot\rVert_V)$ let us define: \begin{equation} \forall x, y \in V \quad [0,1] \mapsto \gamma_x^y (t) = (1-t)x + ty. \end{equation} I would like to ask whether the ...
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1answer
148 views

A consequence of the Min-Max Principle for self-adjoint operators

Let $H=(H, (\cdot, \cdot))$ be a Hilbert space. Let $T_1,T_2:D \subset H \longrightarrow H$ be a self-adjoint operators (not necessarily bounded). It's well-know that the spectrum $\sigma(T_i)$ of $...
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0answers
155 views

Link between a categorical and an algebraic characterization of (infinite-dimensional) Hilbert space

On one side, a very recent paper of Chris Heunen and Andre Kornell "Axioms for the category of Hilbert spaces" (Arxiv:2109.7418v1 latest Arxiv version) offers a characterization of the ...
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0answers
36 views

Generalization of a Gaussian measure continuity result from Hilbert to Banach space

Da Prato/Zabczyk "Second Order Partial Differential Equations in Hilbert Spaces" states the following lemma (this is a reformulation of proposition 1.3.11 in their book): Let $\mu = \mathcal ...
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0answers
71 views

Eigenvalues of Matérn covariance function

Recall that Matérn covariance function $C_\nu(d)$ is defined as $$ C_\nu(d)=\sigma^2\frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu}\frac{d}{\rho}\right)^\nu K_\nu\left(\sqrt{2\nu}\frac{d}{\rho}\right), ...
1
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1answer
56 views

Does the compactness of parameter of distribution function imply the compactness of the distribution (or probability measure) in Wasserstein space?

For a family of probability measures sharing the same form of distribution function $F(x; p)$ with different parameters (i.e., $p$'s), if the parameter falls in a compact subset of real line, can we ...
3
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0answers
42 views

Improving log-Sobolev inequalities via quadratic regularisation

Suppose that $\mu(dx) = \exp(-\psi(x)) \, \mathrm{dx}$ is a probability measure on $\mathbf{R}^d$. For suitable functions $g \geqslant 0$, define $$\text{Ent}(g) = \int \mu(dx) g(x) \log \left( \frac{...
4
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0answers
142 views

Weaker analogues of amenability for groups of piecewise projective homeomorphisms

Let $A$ be a subring of ${\bf R}$ and let $H(A)$ be the group defined/constructed in Monod's 2013 PNAS paper. Monod showed that provided $A\neq {\bf Z}$, $H(A)$ is non-amenable. (The proof breaks down ...
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0answers
78 views

How can we define $\chi_{\Omega}(A)$?

I was reading Spectrum and dynamics where Paolo Facchi discusses projection-valued measures and integrals. The discussion and constructions are all based on the fact that one can define characteristic ...
3
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0answers
132 views

Reference Aubin T.- Espaces de Sobolev sur les varietes Riemanniennes. Bull. Sc. Math. 100, (1976) 149-173

I have been searching without success for the reference: Aubin T.- Espaces de Sobolev sur les varietes Riemanniennes. Bull. Sc. Math. 100, (1976) 149-173 It is cited in many related works. In ...
1
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1answer
82 views

Limit of $u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0$ as $\epsilon \to 0$

Consider the initial-value problem associated to the PDE $u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0$. To prove that, as $\epsilon \to 0$, the weak solution ...
5
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1answer
210 views

What is the 'right' definition of zero measure subsets of Banach spaces?

Question. There are several ways of defining a notion of a 'zero measure' subset of a Banach space $X$. Which one is the 'right' or failing that, the preferred notion? [See below for a more precise ...
4
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1answer
284 views

$\|t\| = \sup_{\|z\| \le 1} \|\langle tz,z\rangle\|$ when $t=t^*$

Let $A$ be a $C^*$-algebra, $E$ be a (right) Hilbert $A$-module and $t \in \mathcal{L}_A(E)$ be an adjointable operator satisfying $t=t^*$. Is it true that $$\|t\| = \sup_{z \in E, \|z\| = 1} \|\...
11
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2answers
853 views

A variation of the Ryll-Nardzewski fixed point theorem

Is there a fixed-point theorem that implies the following result? Let $F$ be a nonempty convex set of functions on a discrete group with values in $[0,1]$. Suppose $F$ is invariant with respect to ...
2
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0answers
154 views

Sobolev (Triebel-Lizorkin) norm estimate for $F \circ u - F \circ v$

Let $F \in C^1(\mathbb R^d;\mathbb R)$ be such that $F(0) = 0$ and $$|F'(\tau v + (1 - \tau)w)| \leq \mu(\tau)(G(v) + G(w))$$ for some $\mu \in L^1([0,1])$ and some non-negative $G \in C^0(\mathbb R^d;...
3
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0answers
193 views

Geometric characterisation of polynomials between normed spaces

Let $(X, \| \cdot \|_X)$ $(Y, \| \cdot \|_Y)$ be normed space. A function $f \colon X \to Y$ shall be called an $n$-th degree (single variable) polynomial ($n \in \mathbb{N}\cup \{ 0\})$ if there ...
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1answer
95 views

How to prove the reverse Hölder inequality for Laplace equations?

Let $ u\in H^1(2B) $ be a weak solution of $ \Delta u=0 $ in $ 2B $, where $ B=B(0,1) $ is a ball with center $ 0 $ and radius $ 1 $. Then there exists some $ p>2 $ such that \begin{eqnarray} \left(...
2
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1answer
138 views

Determine the sign (positive or negative) of an integral with the fractional Laplacian

Let $u,v:\mathbb R \to \mathbb R$ and $\phi: \mathbb R \to \mathbb R_+$ be smooth bounded functions. Assume also $\phi' \ge 0$. Assume that $u(0) - v(0) = 0$ and that $0$ is a strict global minimum of ...
1
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1answer
89 views

Averaging and fractional Laplacian

Let $u,\phi:\mathbb R \to \mathbb R$ be smooth functions and $\Omega_\epsilon$ be a bounded domain in $\mathbb R$ with diameter $\epsilon>0$ (consider for exaple the ball $B_{\epsilon/2}(0)$). Is ...
2
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1answer
116 views

Does the space of Lipschitz functions have the Radon-Nikodym property?

Context. Space of Lipschitz functions. Denote by $Lip_0(D)$ the space of all Lipschitz functions on a metric space $D$ vanishing at some base point $e \in D$. The norm in $Lip_0$ is defined as ...
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0answers
127 views

A characterization of the Dunford-Pettis property

A Banach space $X$ is said to have the Dunford-Pettis property if for any Banach space $Y$ every weakly compact operator $T:X\rightarrow Y$ is completely continuous. Recall that $T$ is completely ...
1
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1answer
161 views

Eigenvalues of operator

In the question here the author asks for the eigenvalues of an operator $$A = \begin{pmatrix} x & -\partial_x \\ \partial_x & -x \end{pmatrix}.$$ Here I would like to ask if one can extend ...
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0answers
49 views

Computation of the trace of a convolution operator

I try to understand a particular step on page two from the research paper "A multidimensional analog of a limit theorem of G. Szegö". https://iopscience.iop.org/article/10.1070/...
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0answers
90 views

Is there a generalization of the Agmon-Douglis-Nirenberg regularity theorem for elliptic equations to domains with corners?

The Agmon-Douglis-Nirenberg theorem(s) state(s) that whenever $f\in W^{m,p}(\Omega)$ where $\Omega$ is a bounded open set of class $C^{m+2}$, then there is a unique solution $u\in W^{m+2,p}(\Omega)$ ...
4
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0answers
80 views

For what $C^*$ algebras $A$ do different types of projection equivalence agree?

For example, For what $C^*$ algebras $A$ is unitary equivalence the same as mvn equivalence for projections. For what $C^*$ algebras $A$ is unitary equivalence the same as homotopy equivalence for ...
1
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1answer
124 views

Non-unital algebras in geometric algebra, smooth envelopes

In Nestruev's (2000) Smooth Manifolds and Observables, the authors define an $\mathbb{R}$-algebra as a commutative, associative algebra with unit (p. 21). A natural generalization of this definition ...
1
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1answer
150 views

The mean ergodic theorem for weakly mixing extension

I asked this question in https://math.stackexchange.com/q/4236870/528430, but did not get any help. I got stuck with the following while going through the proof of Lemma 3.21 from the book 'Ergodic ...
4
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0answers
205 views

Is there a notion of „flatness” in point-set topology?

In algebraic geometry, flat morphisms are usually associated with the intuition that they have „continuously varying fibers”. Is there a notion in topology formalizing the same intuition? Consider for ...
2
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0answers
217 views

A question on convergence rates of Fourier series and strict convergence

Consider BV functions on a torus. The Fourier partial sum using the first $n$ coefficients will converge to the function at every point of continuity, as $n\to\infty$. The convergence rate is $O(1/n)$....
2
votes
1answer
168 views

Small ball Gaussian probabilities with moving center

I would like to prove (if possible, otherwise find a counterexample for) the following lemma: Let $(X,\|\cdot \|_X)$ be a separable Banach space. Additionally, we have a centred Gaussian measure $\mu$ ...