# Questions tagged [fa.functional-analysis]

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

6,738
questions

**3**

votes

**0**answers

41 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 ...

**0**

votes

**1**answer

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 $TV(f) \le M$}\}
$?

**4**

votes

**0**answers

68 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 ...

**0**

votes

**0**answers

28 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)=...

**1**

vote

**0**answers

26 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\...

**26**

votes

**6**answers

4k views

### Nonseparable Hilbert spaces

Being nonseparable Banach space is in fact nothing special: one meets the first
examples in the standard functional analysis course, when one learns about
$\ell^p$ or $L^p[0,1]$ spaces-these spaces ...

**0**

votes

**0**answers

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
$$...

**3**

votes

**1**answer

85 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 $...

**4**

votes

**2**answers

537 views

### Weak convergence in the intersection of Lebesgue spaces or Sobolev spaces

Let $B:=B_1\cap B_2\cap...\cap B_n$, where each $B_j$ is a reflexive Lebesgue space or Sobolev space (such as $L^4$, $H^1$, etc.) on a domain in $\mathbb{R}^d$. Then $B$ is a Banach space endowed with ...

**0**

votes

**0**answers

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 ...

**3**

votes

**0**answers

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 ...

**4**

votes

**1**answer

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 ...

**3**

votes

**1**answer

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
$.
...

**22**

votes

**2**answers

5k views

### Compact embeddings of Sobolev spaces: a counterexample showing the Rellich-Kondrachov theorem is sharp

Let $U$ be an open bounded subset of $\mathbb{R}^n$ with $C^{1}$ boundary. Let $1 \leq p < n$ and $p^{\ast} = pn/(n-p)$. Then the Sobolev space $W^{1,p}(U)$ is contained $L^{p^{\ast}}(U)$ and ...

**61**

votes

**15**answers

15k views

### Is there a nice application of category theory to functional/complex/harmonic analysis?

[Title changed, and wording of question tweaked, by YC, because the original title asked a question which seems different from the one people want to answer.]
I've read looked at the examples in most ...

**2**

votes

**1**answer

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 $\...

**3**

votes

**1**answer

137 views

### Extension of positive functionals II

This is a follow-up to
Extension of positive functionals.
Assume that $X=R^n$ with the canonical order (I indicate with $K$ the positive cone, $x \in K$ iff $x_i \ge 0$ for all $i$) and let $L:M \to R$...

**0**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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 ...

**0**

votes

**0**answers

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 ...

**3**

votes

**0**answers

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. ...

**0**

votes

**0**answers

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 ...

**4**

votes

**0**answers

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=\...

**4**

votes

**1**answer

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 ...

**11**

votes

**1**answer

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 \...

**5**

votes

**1**answer

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;
...

**0**

votes

**1**answer

91 views

### Question/References on the Skorokhod M1 topology

Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for ...

**3**

votes

**1**answer

279 views

### The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$

The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$ is defined by:
$$V_k f(x)=\int_{\mathbb{R}^d}f(y)d\mu_x(y),$$
where $d\mu_x$ is a probability measure on $\mathbb{R}^d$ with support in the ...

**2**

votes

**0**answers

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; ...

**1**

vote

**0**answers

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\|...

**3**

votes

**1**answer

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) ...

**1**

vote

**2**answers

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, ...

**2**

votes

**1**answer

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)
$f_n\ge 0$
$\int_\Omega f_n=I>0$ ...

**5**

votes

**1**answer

98 views

### Critical Smoothness on Besov Spaces $B^s_{p}$: how does it evolved with $p$?

We denote by $B_{p}^s(\mathbb{T}) := B_{p,p}^s(\mathbb{T})$ the Besov space over the circle $\mathbb{T}$ with parameters $p=q \in (0, \infty]$ and smoothness $s \in \mathbb{R}$.
For $p>0$ fixed and ...

**8**

votes

**1**answer

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,
...

**0**

votes

**0**answers

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 ...

**0**

votes

**0**answers

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. ...

**2**

votes

**0**answers

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 ...

**4**

votes

**0**answers

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)\ \...

**6**

votes

**2**answers

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 ...

**2**

votes

**1**answer

85 views

### Max-norm projection of a Hermitian matrix onto the set of positive semidefinite matrices

For a given Hermitian matrix $A$ (i.e. complex matrix with $A_{ij}^{\ast}=A_{ji}$) find its max-norm projection onto the set of complex positive semi-definite matrices:
$$\Pi(A)=\mathrm{argmin}_{M\...

**1**

vote

**0**answers

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<...

**4**

votes

**1**answer

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 ...

**8**

votes

**2**answers

543 views

### If the diagonal of a positive operator is compact, is the operator itself compact?

Let $H$ be a separable Hilbert space with a fixed orthonormal basis $\{e_n\}_n$. For a bounded operator $T$ on $H$, the diagonal of $T$ is the unique operator $D_T$ on $H$ which is diagonal with ...

**9**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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{...

**4**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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 ...

**3**

votes

**1**answer

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 \...