Questions tagged [fa.functional-analysis]

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

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1 answer
140 views

Check an equation on the Heisenberg group $H_1$

The Heisenberg group $H_1$ is the set $\mathbb C\times \mathbb R$ endowed with the group law $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right); \quad \forall z,w \in \mathbb C\,...
0 votes
0 answers
23 views

Maximizing the integral of a transformation that depends on a neighborhood of values of the original function

I'm not an expert in analysis whatsoever, so I might be posing a well-established question, or even an unanswerable one. We are working with non-negative real functions over a sufficiently nice region ...
2 votes
1 answer
162 views

Functions with asymmetrically decreasing Fourier transform?

$\def\ii{{\rm i}}\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\bbNo{\mathbb N_0}\def\Fou{\mathscr F}$Specifically, I would like to have a compactly supported continuous function $f=u+\ii\,v:\bbR\to\bbC$ ...
8 votes
1 answer
370 views

The cardinality of projections of subsets of the Hilbert cube by inner products

I have three related questions. Question 1: Is there a subset $X$ of the Hilbert cube $[0,1]^{\Bbb N}$ of cardinality continuum, such that for each sequence $a\in [0,1]^{\Bbb N}$ with $\sum a_n$ ...
0 votes
1 answer
109 views

Is the space $C_0^{k}(\Omega)$ a Montel space?

I asked this question in the MathStackExchange, but I think I'm not get any answer. I'm trying to find a reference for the following result: Theorem: Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ ...
4 votes
2 answers
169 views

Simple proof that exactness implies strong mixing

Let $f$ be a continuous map defined on a compact metric space $X$. Suppose that $f$ preserves the Borel probability measure $\mu$ and that, for every positive-measure set $A\subseteq X$, we have $$\...
0 votes
1 answer
37 views

Norm of a $2$-tuple of operators

Let $E$ be a complex Hilbert space and $K_1,K_2$ are bounded linear operators on $E$. Let $\omega(K_1)$ and $\omega(K_2)$ be the numerical radius of $K_1$ and $K_2$ respectively. That is \begin{align*}...
2 votes
1 answer
185 views

Combination of simple tensors - II

This is a follow-up question to Combination of simple tensors. I am interested in devising an alternative norm (I mean, other than the usual $\pi$ or $\epsilon$ norms) in the tensor product of two ...
1 vote
0 answers
63 views

Uniform approximation over compacts using weighted function spaces

I'm interested in approximations over the so-called weighted function spaces. Let $(X,\tau_X)$ be some completely regular Hausdorff topological space. Additionally, consider some map $\psi: X \to (0,\...
5 votes
1 answer
571 views

Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution

Examples of infinite dimensional involutions Edit 2/25/23, as suggested by YCOR below: (Start) The first return on a Google search on involution--from late Latin 'a rolling up'--gives the Oxford ...
2 votes
1 answer
127 views

Example of a conditionally convergent series $\sum_{n=1}^\infty b_n$ such that $n^2(b_n-b_{n+1})$ is bounded

Let $(b_n)_{n \in \mathbb{N}}$ be a real sequence such that $(nb_n)$ is bounded. I know that if the series $\sum_{n=1}^\infty b_n$ is conditionally convergent, then $(n^2b_n)_n$ is not bounded. But, ...
3 votes
1 answer
116 views

Does the union of fractional Sobolev spaces fills $L^p$?

Let $\Omega\subset \Bbb R^d$ be any open set. Recall that for $s\in (0,1)$, the fractional Sobolev space $W^{s,p}(\Omega)$ is the collection of function in $L^p(\Omega$ such that \begin{align*} \iint_{...
0 votes
0 answers
83 views

Analogue of $\ell^2(X)$ over an arbitrary Banach ring

Let $X$ be a set. Over the Banach fields $F=\mathbb{R}$ or $F=\mathbb{C}$ we can define the Banach space$$\ell^2(X)=\{\xi\colon X\to F\mid \sum_{x\in X}|\xi(x)|^2<\infty\}$$which satisfies a list ...
0 votes
1 answer
87 views

How to understand the unique continuation result

Let $E$ be the closure of $C_c^{\infty}\left(\mathbb{R}^N\right)$ ($N \geqq 3)$ under the norm $$ \|u\|_E=\left(\int_{\mathbb{R}^N}|\nabla u|^2\right)^{1 / 2}. $$ Suppose $K(x) \in C^1\left(\mathbf{R}^...
9 votes
1 answer
2k views

Chain rule for distributional derivative

Let $V \subset H \subset V^*$ be a Gelfand triple (eg. $H^1 \subset L^2 \subset H^{-1}$). Let $u \in L^2(0,T;V)$ have a distributional derivative $u' \in L^2(0,T;V^*)$. So $\int_0^T u(t)\varphi'(t) = ...
1 vote
0 answers
100 views

Mulitplicity one property for $\mathcal{D}'$ and $L^2$ over a homogeneous space

Let $G$ and $G_0$ be Lie groups, and suppose that a homogeneous space $X=G/G_0$ have a $G$-invariant measure. It is known (E.G.F. Thomas showed) that there is an admissible parametrization $\{\mathcal{...
5 votes
3 answers
1k views

Constant rank theorem for Banach spaces

Is there a similar statement to the constant rank theorem for finite dimensional real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dimensional ...
7 votes
2 answers
280 views

Integral means vs infinite convex combinations

Let $(X,\mathcal A, \mu)$ be a probability space, $\mathbb E$ a Banach space, and $f:X\to\mathbb E$ a Bochner integrable function. Does there exist a sequence $(x_k)_{k\ge 1} $ in $X$, and a ...
8 votes
0 answers
149 views
+100

L^2 polynomial approximation in higher dimensions

Let $\mu$ be a measure on $\mathbb{R}^n$ I'm looking for known upper-bounds on $$\| f-P_m \|_{L^2(\mu)} $$ where $P_n$ is the orthogonal projection of $f$ onto the polynomials of degree less than $m$. ...
7 votes
2 answers
955 views

Method of characteristics for higher order PDEs in more than two variables

I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
1 vote
0 answers
26 views

Can a lift satisfy Chen's relation, geometric condition but not be a rough path?

Let $(X,\mathbb X):[0,1]^2\to \mathbb R^d\oplus\mathbb R^{d\times d}$ satisfy the following four properties: \begin{align} &X_{s,t}=X_{0,t}-X_{0,s}\\ &\sup_{t\neq s}\frac{|X_{s,t}|}{|t-s|^\...
2 votes
1 answer
79 views

On compactly supported functions with prescribed sparse coordinates

Let $\{\phi_n\}_{n=1}^{\infty}$ be an orthonormal basis for $L^2((0,1))$ consisting of Dirichlet eigenfunctions for the operator $-\partial^2_x + q(x)$ where $q \in C^{\infty}_c((0,1))$ is fixed. ...
2 votes
0 answers
29 views

Is there any example of linear operator which is bounded on all Besov spaces but not on Triebel-Lizorkin spaces

Is there any linear operator $T:S'(\mathbb R^n)\to S'(\mathbb R^n)$ such that $T:B_{pq}^s(\mathbb R^n)\to B_{pq}^s(\mathbb R^n)$ for all $0<p,q\le\infty$ and $s\in\mathbb R$, but there exist a $F_{...
4 votes
1 answer
146 views

Bound in terms of harmonic oscillator

I wonder if the following is true: Let $\alpha >0$ be a positive real number, do we have $$\Vert H^{\alpha} \psi''\Vert \le \Vert H^{\alpha+1} \psi\Vert,$$ where $H = -\frac{d^2}{dx^2} + x^2$ is ...
1 vote
0 answers
38 views

If a weighted Laplacian's eigenfunction is zero in an open set, when is it identically zero?

Let $m, s \in ([0, 1]^d \rightarrow \mathbb{R}_{\geq 0}$). Define a weighted Laplacian $\Delta_{m, s}f$ evaluated at $x \in [0, 1]^d$ to be: $m(x) \cdot \text{div} ( s(x) \nabla f(x))$. What ...
5 votes
1 answer
274 views

Does the Poincaré inequality hold on annular domains?

Does the following Poincaré inequality hold $$\int_{B_{r_2}\setminus B_{r_1}} |f-\bar{f}|^2 dx \leq C(r_2-r_1)^2 \int_{B_{r_2}\setminus B_{r_1}} |\nabla f|^2 dx,$$ where $B_r$ denotes a ball of radius ...
3 votes
1 answer
98 views

Can gradient zero implies that a function is constant with Hörmander vector fields

Let $X=(X_1,\cdots,X_m)$ be a system of Hörmander vector fields defined on $\mathbb{R}^n$. The Sobolev space $W_{X}^{1,p}(\Omega)$ is defined by $$W_{X}^{1,p}(\Omega):=\{u\in L^p(\Omega)|X_iu\in L^p(\...
1 vote
1 answer
45 views

Zeros of Gram-Schmidt derived polynomials in weighted integral space

This is a problem out of Chapter 3 of Luenberger's Optimization by Vector Space Methods that I have been having trouble with. Any guidance would be appreciated. Let $w(t)$ be a positive (weight) ...
9 votes
2 answers
988 views

Do Hausdorff locally convex inductive limits always exist?

The following is from Schaefer, "Topological Vector Spaces", 1999, p. 56/57: Let $(E_\alpha)_{\alpha \in A}$ be a family of locally convex spaces with $\alpha$ in a directed poset $A$ and $h_{\beta \...
1 vote
1 answer
2k views

Monge–Ampère operator

I'm studying the article of Bedford–Taylor "Fine topology, Šilov boundary…" but I don't understand the proof of the following proposition. Let $u$, $v$ be plurisubharmonic functions defined ...
0 votes
0 answers
20 views

Directions of differentiability of log-concave measures with infinite-dimensional support

I recently came across the very nice review "Differentiable Measures and the Malliavin Calculus" by Bogachev (1997) which begins by discussing measures $\mu$ on locally convex spaces $X$ ...
7 votes
1 answer
330 views

Proving the inequality involving Hausdorff distance and Wasserstein infinity distance

Prove the inequality $$d_{H}(\mathrm{spt}(\mu),\mathrm{spt}(\nu))\leq W_{\infty}(\mu,\nu)$$ where $d_H$ denotes the Hausdorff distance between the supports of the measures $\mu$ and $\nu$, and $W_\...
-1 votes
2 answers
199 views

Determining if $\|f\|_\infty \leq C\, \|f\|_{2}^{2/3} $ holds under $f(0) = f(1) = 0$, $\|f'\|_2 \leq 1$

Suppose $f \colon [0, 1] \to \mathbb{R}$ is continuously differentiable, and satisfies $f(0) = f(1) = 0$ and $\|f'\|_2 \leq 1$. I am wondering if it there is a constant $C > 0$ such that for all ...
0 votes
1 answer
77 views

A simple bilinear estimate

Let $2\leq p,q <\infty$ and fix $0<\alpha<1$ such that $\frac{1}{p}+\frac{1}{q}\leq 2-\alpha$. Suppose that $f\in L^{p}([0,1])$ and $g\in L^{q}([0,1])$. What is the optimal value of $t=t(\...
3 votes
1 answer
1k views

Borel-Cantelli lemma for general measure spaces (those with infinite measure)

The Borel-Cantelli lemma is often stated for a probability space or spaces with finite measure. But it seems to me that it still holds if the space $X$ is of infinite measure. I seem to be able to ...
2 votes
0 answers
64 views

An example of an $\mathcal{L}_\infty$ Banach space with property p-(V) and without property (V)

Here are the definitions for property $p$-$(V)$ and property $(V)$. A Banach space $X$ has property $(V)$ if and only if every unconditionally converging operator $T$ from $X$ to any Banach space $Y$ ...
2 votes
1 answer
280 views

Fourier series but different waveform

Given a nondegenerate smooth simple closed convex curve $f: [0,2\pi]\to \mathbb C \setminus \{0\}$ with winding number (around origin) $1$, and $f$ have zero mean. Let $f_n: [0,2\pi]\to \mathbb C \...
-3 votes
1 answer
60 views

Minimal norm problem with linear combination of translation operator to be estimated

Follow up question from this one Suppose $X = L^2(G)$, where $G$ is some locally compact group. Let $x, y \in G$ I for fixed $n$ I am seeking for an operator $H \in B(X)$ of the form $$ H = H(\alpha_1,...
1 vote
1 answer
282 views

Hölder continuity of Radon transform of smooth function

Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by $$ R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)...
0 votes
1 answer
385 views

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

$\def\zbar{\smash{\overline z}\vphantom z}$I would like to construct an approximation of the product \begin{equation} f(z) = \frac{1}{\zbar-a} \frac{1}{z-b}, \end{equation} where $a, b \in \mathbb{C}$,...
3 votes
2 answers
376 views

Monotonicity of matrix conjugation

Let $A$ and $B$ be positive-definite matrices such that $A \le B.$ By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$ I am now curious under ...
0 votes
0 answers
77 views

Verifying the Cauchy behavior of a sequence

Let me consider the iteration $x_{n+1}=Tx_n$ and $T$ is a self-map from a non-empty subset $K$ of a smooth Banach space $X$ to itself, satisfying $W(Tx, Ty) \leq W(x, y)$, where $W(x, y)=\Vert x \Vert^...
0 votes
1 answer
97 views

Minimal norm problem whose unknown is an operator

Generally given an Hilbert space $X$ with and a bounded linear operator $H : X \to X$ given a vector $y \in X$ we seek an $x \in X$ such that $$ f(x) = \frac{1}{2} \left\lVert Hx - y \right\rVert_2^2 $...
5 votes
1 answer
175 views

On a property for normed spaces

I asked this question on Math Stackexchange, but I didn't get an answer: https://math.stackexchange.com/questions/4881155/on-a-property-for-normed-spaces?noredirect=1#comment10410489_4881155 I came ...
7 votes
0 answers
115 views

Stochastic analysis on nuclear Fréchet spaces

This is a reference request question, so to make it clear what I am after, I will give a quick outline of the area I am thinking in and some questions that arise. A lot of the time in infinite-...
1 vote
1 answer
179 views

Shape, shift and scaling retrieval of a sampled function

Let $f(x)$ be some unknown continuous square-integrable function defined on the interval $[0,1]$. Suppose we have $i=1,...,n$ samples $f_i$ of $f$ of the following form: $$f_i(x)=a_i*f(x+b_i)+c_i$$ ...
3 votes
2 answers
900 views

Extensions of Urysohn's inequality

A version of Urysohn's inequality states that for a symmetric convex body $K \subset \mathbb{R}^n$, one has $$ \left(\frac{\text{vol}(K)}{\text{vol}(B_2)} \right)^{1/n} \le \frac{1}{\sqrt{n}} E \; \| ...
0 votes
3 answers
710 views

center of the algebra of bounded operators [closed]

Suppose that $X$ is a Banach space. How to prove that the center of the algebra $B(X)$ (the algebra of bounded operators on $X$) consists only of operators of the form $aI$, where $a$ is scalar and $I$...
0 votes
0 answers
25 views

reference request: mercer expansion and kernel underlying Sobolev spaces?

Let us define the periodic Sobolev spaces, for $s > n/2$ by $$ H_{s}([0, 1]^n) = \{f : [0, 1]^n \to \mathbb{R} :\mbox{for}~j\leq s, f^{(j)} |_{\partial[0, 1]^n} \equiv 0, ~~ \int_{[0, 1]^d} (f^{(s)...
12 votes
2 answers
4k views

Where was/is Compensated Compactness used?

This last summer, I read up on Tartar's so called Method of Compensated Compactness (or at least how it applied to scalar conservation laws). I used this theory to prove the existence of $L^{\infty}$ ...

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