Questions tagged [fa.functional-analysis]

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

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10 views

Examples/applications of parabolic PDEs that are not posed on domains or manifolds

Are there any examples of parabolic PDEs $$u' - Au = f$$ posed in a Gelfand triple setting $V \subset H \subset V^*$ with $V$ and $H$ chosen NOT as spaces of functions (or distributions) over a domain ...
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46 views

Poisson summation formula for infinite dimensional spaces

Let $M$ be an orientable, compact smooth manifold with and $H^{-1}(M)$ be the dual space of $$H^{1}(M)=\{f:\int |f|^2+(\nabla f)^2<\infty\}$$ I know it is well known that (see Julien Dubedat, page ...
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1answer
45 views

Kolmogorov entropy of a subset of $L^1$

How can we estimate the Kolmogorov $\epsilon$-entropy $$H_\epsilon (A,L^1(\mathbb R))$$ where $ A = \{f:\mathbb R \to [0,K] \text{ s.t. $f \in L^1$ and has total variation $TV(f) \le M$}\} $?
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29 views

Lower bound on iterated matrix application

Let $n \in \mathbb Z^2$ such that the non self-adjoint weighted Laplacian is $$(\Delta u)(n)=u(n_1+1,n_2)-u(n_1-1,n_2) + i( u(n_1,n_2+1)- u(n_1,n_2-1))$$ the adjoint operator is then $$(\Delta^* u)(n)=...
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28 views

Relation between minimizer of regularized risk & risk in statistical learning theory

In supervised machine learning, we typically take a Risk Minimization (RM) point of view when formulating a problem. So, what we typically solve for is the following: $$ R^L(h) = \underset{h\in\...
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54 views

Condition for sum of linear surjections to have right-inverse

Question: Let $L_1,\dots,L_K:V\rightarrow W$ be continuous linear maps between Hilbert spaces (of at-least dimension $1$) such that $L_K(V)=W$. Let $\beta_1,\dots,\beta_K\in (0,\infty)$. and define $$...
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69 views
+50

Condition on kernel convolution operator

I am studying O'Neil's convolution inequality. Let $\Phi_1$ and $\Phi_2$ be $N$-functions, with $$ \Phi_i(2t)\approx \Phi_i(t), \quad i=1,2 $$ with $t\gg 1$ and let $k \in M_+(\mathbf R^n)$ is the ...
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33 views

biorthogonal sequence property inheritance

Suppose the sequence $(x_n)_{n\ge1}$ in a Banach space $X$ is minimal and complete. In this case we know there is a unique biorthogonal sequence $(x^*_n)_{n\ge1}$ in the dual $X^*$, associated with ...
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1answer
87 views

Stone-Weierstrass theorem for modules of non-self-adjoint subalgebras

In "Weierstrass-Stone, the Theorem" by Joao Prolla, there is a Stone-Weierstrass theorem for modules, stated as the following: Let $\mathcal{A}$ be a subalegebra of $C(X, \mathbb{R})$ and $...
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61 views

“Weakly” nuclear operators (terminology)

Recently, I'd come across the following kind of operators and I wonder if they have been considered before and given a name. Let's say that a linear map $T:V\to W$ between locally convex topological ...
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1answer
101 views

A question about series involving a Sobolev functions

Let $\Omega\subset\mathbb{R}^n$ open, bounded and smooth. Let $\lambda_j$ and $e_j$, $j\in\mathbb{N}$, be the eigenvalue and the corresponding eigenfunctions of the Laplacian operator $-\Delta$ in $\...
4
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1answer
88 views

Canonical multiplication representation of self-adjoint operator in quantum chemistry and coding theory research

In my applied math research group, we are studying and going over functional analysis results from papers and theses from our institution to generalize their results and apply them in our discrete ...
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23 views

Topological constraints on a compact convex set admitting a strictly convex and subdifferentiable real function

It is a theorem of Hervé that A compact convex set $K$ admits a strictly convex and continuous real function only if $K$ is metrizable. (The converse is also true.) I'm wondering if any results of ...
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35 views

Functions on dense subgroups of $\mathbb{R}^n$

Let $G$ be a finitely generated dense subgroup of $\mathbb{R}^n$, and $f$ be a character on $G$. In the situation I'm looking at $f$ is either $1$ or $-1$ at any point. Function $f$ can be extended to ...
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54 views

Green operator of elliptic differential operator and radius of convergence

Let $E \to X$ be a hermitian vector bundle over a compact Kähler manifold and let $L$ be a self-adjoint elliptic linear differential operator on $E$. Suppose that $E \to X$ and $L$ are real-analytic. ...
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38 views

Regarding supremum over a set

Let $\Delta$ be a Compact Hausdorff space in $\mathbb{C^n}$. Let $A$ be a closed sub algebra of $C(\Delta)$(space of all complex valued continuous functions on $\Delta$) which contains the constant ...
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1answer
228 views
+50

Is this operator continuous?

Let $I=[0,1]$ and $E$ a Banach space. We note by $X:=\mathcal {C}(I,E), $ the space of all continuous functions from $I$ to $E$, with $\left \| x \right \|_X=\sup_{t\in I }\left \| x(t) \right \|_E $. ...
4
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1answer
113 views

Trying to recover a proof of the spectral mapping theorem from old thesis/paper with continuous functional calculus

In my research group in functional analysis and operator theory (where we do physics and computer science as well), we saw in an old Russian combination paper/PhD thesis in our library a nice claim ...
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69 views
+50

A bilinear estimates involving critical Sobolev norms

Given $q>1$, consider the critical Sobolev space $W^{n/q,q}(\mathbb{R}^n)$, which fails to embed in $L^{\infty}(\mathbb{R}^n)$. I'm wondering if we can recover some critical estimate by considering ...
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104 views

Learning from eigenvalues of Hilbert-Schmidt integral operator

Do eigenvalues of the Hilbert-Schmidt integral operator determine the underlying measure up to translation, reflection and rotation? Details: Suppose we have a measure $\mu$ on a Euclidean space $X=\...
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1answer
311 views

A density question for the Hilbert transform

Let $\mathscr Hf$ denote the Hilbert transform of a function $f$ defined on the real-line $\mathbb R$. Are the set of functions $$ \{(f+\mathscr Hf)_{|_{(0,1)}}\,:\, f \in C^{\infty}(\mathbb R)\quad \...
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51 views

A strong duality for convex functional optimization that admits Lipschitz continuity constraints?

Problem Statement I am looking for formal proof---hopefully textbook material---of two items: an analogue to Slater's condition [1] that obtains strong duality for optimization of convex functionals; ...
3
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1answer
81 views

A function $V : \mathbb{R}^2 \to \mathbb{R}$ is a (logarithmic) potential

I'm looking for references given some sort of inverse problem in logarithmic potential theory. That is, given a function $V : \mathbb{R}^2 \to \mathbb{R}$, what is a sufficient (and perhaps necessary) ...
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69 views

Uniformly local Sobolev spaces and interpolation

Let $d\in\mathbb{N}^+$, $s\geq 0$, and consider the uniformly local Sobolev space $$H^s_{u,loc}(\mathbb{R}^d):=\{f\in H^s_{loc}(\mathbb{R}^d)\,s.t.\,\|f\|_{H^s_{u,loc}}:=\sup_{x\in \mathbb{R}^d} \|f\|...
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54 views

Isolated points of the spectra of self-adjoint operators on Hilbert spaces

Let $T$ be a (everywhere defined) self-adjoint operator on a complex Hilbert space $\mathcal{H}$. I am interested in results that give (non-trivial, possibly mild) sufficient conditions on $T$ to ...
2
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1answer
89 views

Positive subharmonic functions with constant integral blowing up at boundary

Say, we're given smooth functions $f_n$, $n=1,2,3,...$ defined on a smooth bounded domain $\Omega\subset\mathbb{R}^d$ satisfying $\Delta f_n\ge 0$ (subharmonic) $f_n\ge 0$ $\int_\Omega f_n=I>0$ ...
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40 views

Does anyone know if it's possible to construct Moduli space of J holomorphic curves using Holder spaces?

let Y be a contact (3) manifold and X be its symplectization. let's say the Reeb dynamics is at least Morse Bott. let $u: \Sigma \rightarrow X$ be a $J$ holomorphic curve. I know the usual model for a ...
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81 views

Can we show equivalence of two distributions based on their statistics?

Let $p,q$ be two distributions on $\mathbb{R}^d$. Let $f:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}$. Under what conditions does $\mathbb{E}_{x\sim p}f(x,z)=\mathbb{E}_{x\sim q}f(x,z)\ \...
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2answers
204 views

Is the restriction map $C^1\ni f\mapsto\left.f\right|_K$ a continuous map?

Let $E$ be a $\mathbb R$-Banach space, $\Theta\subseteq C^{0,\:1}(E,E)$ be a $\mathbb R$-Banach space and $\iota$ be a continuous embedding of $\Theta$ into $C^1(E,E)$. I would like to show that, ...
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44 views

How can we deal with singular integral equation that is a sum of the Cauchy kernel and Hilbert kernel

I am a theoretical physicist dealing with Matrix Models. In this context, one often encounters singular integral equations whose solution gives the eigenvalue density in the so-called large $N$ limit. ...
8
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1answer
179 views

Sobolev inequalities on manifolds: dependence of the constants on the Riemannian metric

Let $g$ be a smooth Riemannian metric on the 2-torus $T^2$. $g$ induces the Sobolev space $W^{2,2}_g(T^2)$ via the norm $$ \|f\|_{W^{2,2}_g}^2 = \int_M |f|^2 + g(\nabla^2 f,\nabla^2 f)\, \text{vol}_g, ...
6
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2answers
115 views

holomorphy in infinite dimensions (holomorphic families of operators)

Let $X$ be a Banach space (over $\mathbb C$), and let $\mathcal L(X)$ be its algebra of bounded linear operators. Let $U\subset \mathbb C^N$ be an open subset, and $f:U\to \mathcal L(X)$ a function ...
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0answers
90 views

Weak upper semi-continuity with orthogonal condition

Let $B \subset \mathbb{R}^d$ be the ball of radius one, and consider the map defined on $L^2(B,\mathbb{R})$ \begin{align*} f(\phi) = \underset{\substack{ \varphi \in H^1(B,\mathbb{R}) \\ \left<...
9
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1answer
242 views

Scottish Book Problem 172

The problem is formulated using old terminology and I want to understand what it actually says. The problem reads: "A space $E$ of type (B) has the property (a) if the weak closure of an ...
3
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50 views

Strong maximum principle for a PDE with coefficient in $L^1$

Let $U$ be a bounded open subset of $\mathbb{R}^n$ with smooth boundary, $n \geq 3$. Set $N = \frac{2n}{n-2}$. I am interested in the following equation: $$ -\Delta \phi + R \phi + \phi^{N-1} = 0 $$ ...
4
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1answer
68 views

If $F$ is a countably normed, nuclear Fréchet space, can I then find a fundamental system which exhibits both of these properties at once?

Let $F$ a Fréchet space. This means that $F$ is a complete Hausdorff topological space whose topology can be generated by an increasing family of seminorms $\{ p_{n} \}_{n \in \mathbb{N}}$. Let's ...
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0answers
17 views

Well-posedness of hyperbolic system with constant coefficients in finite domains

I'm studying the PDE $$ \frac{\partial u}{\partial t} + A_x\frac{\partial u}{\partial x} + A_y\frac{\partial u}{\partial y} + A_z\frac{\partial u}{\partial z} = 0 $$ with $A_x, A_y, A_z$ being ...
4
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1answer
54 views

Inverting convolutions over finite intervals

There are well-known techniques for inverting convolutions over the whole or half real line with Fourier and Laplace transformations, but on the face of it they can't be applied to an integral ...
5
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1answer
171 views

Weak convergence in a product space

Given a function $f: Y\longrightarrow Y$ ($Y$ is a Banach space). Assume that $f$ satisfies: If $y_n \rightharpoonup y $, then $f(y_n)\rightharpoonup f(y) \text{ in } Y$; $f$ is weakly compact; ...
3
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1answer
73 views

Measurability of superposition operator with non-separable Banach space

Let $f\colon I \times X \to \mathbb{R}$ be a map where $I \subset \mathbb{R}$ is an interval, $X$ is a Banach space (possibly non-separable) and we have $$t \mapsto f(t,x) \text{ is measurable}$$ $$x \...
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0answers
22 views

Find a fractional integral operator for the Weyl-Marchaud fractional derivative

It is known that the left-side Weyl-Marchaud fractional derivative $$ D^\mu f(x) = \frac{\mu}{\Gamma(1-\mu)}\int_0^\infty \frac{f(x)-f(x-t)}{t^{\mu+1}}dt $$ for continuous and integrable $f(x)$ on $\...
4
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1answer
77 views

Family of Pettis integrals functions “uniformly approximated” by sums

In this book (proof of $4.1.3.$ Lemma. exactly), one can find this passage, that I tried to rephrase here: Let $f:I\times E\rightarrow E$ a Pettis integrable function, where $I:=[0,T]\subset \mathbb{...
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0answers
75 views

Representation of radial functions

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be radial. Then can $f$ necessarily be represented as $$ A(\|x\|)x = f(x), $$ for some function $z\mapsto A(z)$ from $\mathbb{R}$ into $SO(n)$? Here, by ...
3
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1answer
137 views

Commutation between integrating and taking the minimal eigenvalue

Let $S = (f_{ij})_{ij}$ be a $n \times n$ real symmetric matrix, with functions $f_{ij} \in L^1(\mathbb{R}^d,\mathbb{R})$ in it. We define $\left(\int u S \right)_{ij} = \int u S_{ij}$ as the ...
6
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0answers
235 views

Hamiltonian dynamics on cotangent bundle

I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that $N$ is a differential manifold and $S_0 ,S_1\subseteq N$ two ...
8
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2answers
204 views

When does $\left\Vert f(\mathbf{N}) - f(\mathbf{M})\right\Vert_{\mathrm{op}} \leq k\left\Vert \mathbf{N} - \mathbf{M}\right\Vert_{\mathrm{op}}$ hold?

Define the Frobenius norm of a matrix as $\left\Vert A \right\Vert_{\mathrm{F}}=\sqrt{\sum_{i,j} A_{ij}^2}$ and the operator norm as $\left\Vert A \right\Vert_{\mathrm{op}}=\sup_{x \not = 0} \frac{\...
1
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1answer
128 views

Find this reference or an alternative where I can find this result

I need this reference, but I couldn't find it online as a PDF. Any help please? J. Sun, X, Zhang, The fixed point theorem of convex-power condensing operator and applications to abstract semilinear ...
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0answers
28 views

Eigenvalues associated to differential opertator over natural spline spaces

Let $S_n^k$ denote de linear space of natural splines of degree $k$ and uniform grid $a < t_1 < \ldots < t_n < b$ over the bounded interval $[a,b] \subseteq \mathbb R$. For any $m \in \...
5
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1answer
462 views

Analytic functions where all derivatives vanish at infinity and which are bounded

Let $C_0(\mathbb{R})$ denote the analytic functions $f : \mathbb{R} \rightarrow \mathbb{R}$. I wonder whether there a functions $f \in C_0(\mathbb{R})$ with $f \neq 0$, such that there is a constant $...
3
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0answers
61 views

Non-linear weak*-continuous left inverses

Let $T\colon X\to Y$ be a continuous linear surjection between Frechet spaces. Then it is possible to use Michael's selection theorem to show that there exists a continuous (non-linear) function $g\...

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