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10 votes
1 answer
635 views

What's the nearest algebraic theory to inner product spaces?

Following the references to the accepted answer to Is the category of Banach spaces with contractions an algebraic theory? one discovers that there is an algebraic theory (infinitary) which is closely ...
Andrew Stacey's user avatar
10 votes
1 answer
901 views

Approximation of a compactly supported function by Gaussians

Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function whose support is a closed interval, e.g. $\text{supp}(f)=[a,b]$. Then $f$ can be approximated (e.g. in $L^2$) by a linear combination of Gaussian ...
JohnA's user avatar
  • 710
10 votes
1 answer
274 views

Cutting a Gaussian in two pieces that are maximally separated in the Wasserstein metric

Denote the standard Gaussian probability measure on $\mathbb R^n$ by $\gamma$. We partition $\mathbb R^n$ into two sets $A$ and $A^c$ such that $\gamma(A) = \gamma(A^c) = 1/2$. Denote by $\gamma_{A}$...
VSJ's user avatar
  • 1,034
10 votes
1 answer
1k views

Basic examples of induction on scales arguments

An important ingredient in recent progress on Euclidean harmonic analysis has been that of "inductions on scales". A few examples are the papers of Wolff, Tao, and Bourgain and Guth. Here is a (vague)...
Mark Lewko's user avatar
10 votes
1 answer
337 views

What are the predictive implications of conditional non-commutative probability?

To simplify things, let's consider the Hilbert approach to quantum probability over a finite dimensional vector space $V$ of dimension $n$. In this context a state $S$ is a positive semi-definite ...
Mehmet Coen's user avatar
10 votes
1 answer
330 views

(Sharp) Bounds on $E(XYZ)$ given all the bivariate marginals

Suppose $X,Y,Z$ are all real-valued random variables. Suppose I know the joint marginal distributions of $(X,Y)$, $(Y,Z)$ and $(X,Z)$. I want to find bounds on $E(XYZ)$. In the case of bounding $E(XY)$...
Atom Vayalinkal's user avatar
10 votes
1 answer
486 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, ...
C M's user avatar
  • 381
10 votes
1 answer
586 views

Nonlinear Schrödinger equation with discrete Laplacian

In the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$" by Colliander, Keel, Staffilani, Takaoka and Tao it is argued in the beginning ...
user avatar
10 votes
2 answers
2k views

Pull-back of generalized functions

Let $f\colon X\to Y$ be a smooth map between smooth manifolds. Then the pull-back operation $f^*\colon C^\infty(Y)\to C^\infty(X)$ is a linear continuous operator when $C^\infty$ is equipped with the ...
asv's user avatar
  • 21.8k
10 votes
1 answer
869 views

Completeness of Borel measure

Let $X$ be a compact Hausdorff space and $\mu$ a finite Borel measure without atoms which is outer regular with respect to open sets and inner regular with respect to compact sets. Can such measure be ...
arc's user avatar
  • 277
10 votes
0 answers
225 views

Can the trace be computed in any Schauder basis?

I'm cross-posting this question from Math.SE, as it didn't get much attention there. Let $H$ be a separable Hilbert space and $T \in L(H)$ a trace-class operator. It is well known that the trace of $T$...
WillG's user avatar
  • 233
10 votes
1 answer
518 views

Inverse function theorem for $W^{2,n}\cap W^{1,\infty}$ functions

Let $n\ge 2$, $f:B_1\subset \mathbb R^n\rightarrow \mathbb R^n$, $f\in W^{2,n}\cap W^{1,\infty}(B_1)$, $\text{det}(Df)>c>0$, where $B_1$ is the unit ball. Can we show that $f$ is a homeomorphism ...
Tian LAN's user avatar
  • 435
10 votes
0 answers
422 views

Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (u (t, x))|^2 u(t, x) \}$

We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R \to \mathbb R$ belong to the Hölder space $C^{1, \alpha}_b (\mathbb R)$ for some $\alpha \in (0, 1)$. Let $u : \mathbb T \times \...
Akira's user avatar
  • 825
10 votes
0 answers
652 views

Eigenfunctions of the integral kernel $1/(x^2 + x'^2)$

My question seems elementary, yet I could not find the solution after working on and searching for several days... I'd like to find the eigenfunctions of a simple integral kernel: \begin{equation} \...
Yuli Nazarov's user avatar
10 votes
0 answers
657 views

“Taylor series” is to “Volterra series” as “Laurent series” is to _________?

Preamble My question is similar to an earlier MathOverflow question: “Taylor series” is to “Volterra series” as “Padé approximant” is to _________? which I just answered (hopefully my first ever ...
Nike Dattani's user avatar
10 votes
0 answers
226 views

Extremal bases in finite-dimensional Banach spaces

Definition. A basis $e_1,\dots,e_n$ for a Banach space $X$ is called extremal if there exists a point $s$ in the unit sphere $S_X=\{x\in X:\|x\|=1\}$ such that for every $i\in\{1,\dots,n\}$ the ...
Lviv Scottish Book's user avatar
10 votes
0 answers
230 views

Norm-attaining operators with values in a 2-dimensional Hilbert space

Is the set $N\!A(X,\ell_2^2)$ of norm-attaining operators from a Banach space $X$ onto the $2$-dimensional Hilbert space $\ell^2_2$ dense in the Banach space $L(X,\ell_2^2)$ of all linear continuous ...
Lviv Scottish Book's user avatar
10 votes
0 answers
266 views

Are biduals of spaces of differentiable functions on hypercubes Grothendieck?

Consider the space $E_n = C^1([0,1]^n)$ of continuously differentiable functions with the usual norm $$\max\{ \|f\|_\infty, \|f^\prime_{x_1}\|_\infty, \ldots, \|f^\prime_{x_n}\|_\infty\}.$$ making it ...
Tomasz Kania's user avatar
  • 11.3k
10 votes
0 answers
845 views

Witt's proof of Gelfand-Mazur / Ostrowski's Theorem

Previously asked on Math Stackexchange without answers. Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
Torsten Schoeneberg's user avatar
10 votes
0 answers
175 views

A combinatorial proof of the Harrow--Kolla--Schulman theorem

Let $Q^n := \{0,1\}^n$ be the Hamming cube with the Hamming metric. (Recall that the Hamming is defined by the distance $d(x,y) := \# \{ i : x_i \neq y_i \}$. For integers $0 \leq k \leq n$, define a ...
K Hughes's user avatar
  • 679
10 votes
0 answers
251 views

Do sufficiently large Banach spaces admit non-compact operators with not too large range?

As in the title, does there exist a cardinal number $\lambda$ such that for every Banach space $X$ of density/cardinality at least $\lambda$ there exists a non-compact bounded, linear operator $T\...
Tomasz Kania's user avatar
  • 11.3k
10 votes
0 answers
201 views

Masas in SAW*-algebras

I asked this question three years ago at MSe but it has no response; let me try here. Pedersen distilled the following class of C*-algebras which he termed SAW*-algebras (Journal of Operator Theory, ...
Tomasz Kania's user avatar
  • 11.3k
10 votes
0 answers
325 views

Are ideals in separable C*-algebras complemented subspaces?

Let $A$ be a separable C*-algebra and $J\subseteq A$ a closed two-sided ideal. Does this make $J$ into a complemented subspace of $A$? In other words, does the quotient map $A\to A/J$ have a ...
Tobias Fritz's user avatar
  • 6,406
10 votes
0 answers
207 views

Projective tensor squares of uniform algebras

In discussion with a colleague recently (Jan 2017), $\newcommand{\AD}{A({\bf D})}\newcommand{\CT}{C({\bf T})}$ I was reminded that if $A(D)$ denotes the disc algebra and $\iota: \AD\to \CT$ is the ...
Yemon Choi's user avatar
  • 25.8k
10 votes
0 answers
761 views

Reference request : Grothendieck's topological space valued integral

As I am learning the different kind of Banach space valued integrals (Pettis, Bochner), I know that Grothendieck made a "mémoire" in his youth about this topic, but I don't know if it is available ...
Paul-Benjamin's user avatar
10 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) = ...
mathias_l's user avatar
  • 145
10 votes
0 answers
2k views

Questions on de Branges' work on the Riemann hypothesis

According to Wikipedia, Louis de Branges de Bourcia has obtained some notable results, such as a proof of the Bieberbach conjecture in 1985, which is now known as de Branges' theorem. Initially, his ...
mayank's user avatar
  • 163
10 votes
0 answers
508 views

Tensorial decomposition of $B(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
TrzyTrypy's user avatar
  • 101
10 votes
0 answers
744 views

Is the set of real-valued lower semi-continuous functions measurable in epigraph topology (= topology of Gamma convergence)?

Let LSC = LSC([0,1]) be the set of non-negative, lower semi-continuous functions on the unit interval which take values in $\mathbb{R}_+ \cup \{\infty\}$. We use epigraph topology on LSC, i.e. a ...
Wolfgang Loehr's user avatar
10 votes
0 answers
509 views

Lacunary hyperbolic groups and weak amenability

In the paper called Lacunary Hyperbolic group, Y. Ol'shanskii, D. Osin and M. Sapir define and characterize the lacunary hyperbolic groups, which contains the hyperbolic groups but also Tarski's ...
Denis Poulin's user avatar
10 votes
0 answers
609 views

Asymptotic non-distortion of the separable Hilbert space

By the work of E. Odell and Th. Schlumprecht, we know that the separable Hilbert space $\ell_2$ is arbitrarily distortable. But I don't know if an "asymptotic" version of their result is true. To ...
Pandelis Dodos's user avatar
9 votes
4 answers
2k views

How may I find all continuous and bounded functions g with the following property?

Find all continuous and bounded functions $g$ with : $$\forall x \in \mathbb R, 4g(x)=g(x+1)+g(x-1)+g(x+\pi)+g(x-\pi).$$ I have posted this question here, but received no answer.
Dattier's user avatar
  • 4,074
9 votes
3 answers
1k views

Does there exist a notion of discrete riemannian metric on graph?

I would like to know if there is any notion of a discrete Riemannian metric on graphs. C. Mercat has worked on discrete Riemann Surfaces, but that's not exactly what I am working on. To be more ...
Laurent.C's user avatar
9 votes
2 answers
1k views

Density of restrictions of harmonic functions inside a ball

Let $B$ be the closed unit ball in $\mathbb R^3$ centered at the origin and let $U= \{x\in \mathbb R^3\,:\, \frac{1}{2}\leq |x| \leq 1\}.$ Let $$ S_U= \{u \in C^{\infty}(U)\,:\, \Delta u =0 \quad\text{...
Ali's user avatar
  • 4,143
9 votes
3 answers
601 views

Associative mean

Can there be a function $m(a,b)$ that is both associative and a mean, i.e., $\min (a,b) \leq m(a,b) \leq \max (a,b)$? The obvious solutions are $m(a,b) = \max(a,b)$ or $\min(a,b)$, but are there ...
Hauke Reddmann's user avatar
9 votes
5 answers
2k views

Convexity of distance-to-boundary function

Let $\Omega\subset\mathbb{R}^{n}$ be an open, bounded convex domain. Denote $d_{\Omega}:\Omega\rightarrow\mathbb{R}$ the distance-to-boundary function, that is, $$ d_{\Omega}\left(x\right):=\inf\left\...
Hadarmad's user avatar
9 votes
2 answers
758 views

Number of critical points of smooth functions on $S^1$

Let $u$ be a smooth function on the unit circle $S^1$ such that $\int_{S^1}ux_j=0$, for $j=1,2$. Is the number of critical points of $u$ strictly bigger than 2?
A random mathematician's user avatar
9 votes
4 answers
4k views

Is the space of Radon measures a Polish space or at least separable?

Background: I work on a SPDE problem where in order to apply Prokhorov's theorem I need that some measure space is Polish space. And additionaly it would be good if that space is Banach space. Earlier ...
Mark's user avatar
  • 657
9 votes
1 answer
2k views

Is a manifold generically real analytic (with generic real analytic metric)?

I have heard it said in some differential geometry talks that "the generic situation in such and such case is real analytic". My question is, is the generic smooth manifold also real analytic in some ...
Guest's user avatar
  • 123
9 votes
3 answers
568 views

Defining the abstract tensor product of W*-algebras via a universal property

I am playing around a bit with $W^*$-algebras, and I'm trying to come up with a definition for the $W^*$-algebraic tensor product. Here is my first attempt: It is easy to show that such an object ...
Andromeda's user avatar
  • 175
9 votes
4 answers
2k views

Books about capacity theory

While I was studying the book Variation et Optimisation de formes by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, which is defined for ...
Beni Bogosel's user avatar
  • 2,222
9 votes
2 answers
334 views

Does $\pm A \leq B$ imply that $B^{-1} A$ is bounded?

Lately I have to use a lot of functional calculus. A question that keeps popping up and that I don't manage to resolve is the following: Let $A,B$ be self-adjoint (not necessarily bounded) operators ...
Severin Schraven's user avatar
9 votes
4 answers
905 views

Defining the value of a distribution at a point

Let $\omega\in D'(\mathbb R^n)$ be a distribution and $p\in \mathbb R^n$. If there is an open set $U\subset \mathbb R^n$ containing $p$ such that $\omega|_U$ is given by a continuous function $f\in C(...
B K's user avatar
  • 1,942
9 votes
3 answers
736 views

Cyclic vectors for the shift operator

Let $S:\ell^2\to\ell^2$ be the shift operator $$ S(x_1,x_2,\dots)=(0,x_1,x_2,\dots). $$ Let $x\in \ell^2$ with $x_1=1$. Is $x$ cyclic for $S$? In other words, is the span of the vectors $x,Sx,S^2x,\...
user avatar
9 votes
2 answers
298 views

Two inequalities in $C^*$ algebras

Under what conditions on a $C^*$ algebra $A$ we have the following inequality: $$x^*a^*ax+a^*x^*xa\leq x^*x+a^*x^*ax+x^*a^*xa\;\;\; \forall x,a\in A$$ The second identity which I am looking for is ...
Ali Taghavi's user avatar
9 votes
2 answers
8k views

The Paley-Wiener theorem and exponential decay.

Consider a function whose Fourier transform is supported on a half-ray: $$ A(t)=\int_0^\infty \omega(E) e^{-iEt}d E, $$ where I can suppose $\omega(E)\geq 0$ and any suitable regularity conditions on $...
Emilio Pisanty's user avatar
9 votes
3 answers
3k views

The classical Krein-Rutman theorem

The classical Krein-Rutman theorem states that any positive compact linear endomorphism $T:X \to X$ on a Banach space $X$ with positive spectral radius $r(T)$ has an eigenvalue $r(T)$ with a positive ...
Mark Kim-Mulgrew's user avatar
9 votes
2 answers
611 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{\...
Henry's user avatar
  • 93
9 votes
1 answer
564 views

$L^1$ norm of exponential sum of $n^2 x$

What is the asymptotic order of $$ \int_0^1 \left| \sum_{n=1}^N e^{2 \pi i n^2 x} \right| ~dx $$ as $N \to \infty$. This should be known, but I cannot find it in the literature.
Kurisuto Asutora's user avatar
9 votes
3 answers
4k views

Projections in Banach spaces

Dear All, I am absolutely lost in the following problem: Let $P_s, \: s \in [0,1],$ be a uniformly bounded family of projections (idempotents) in a Banach space $X$ such that $P_s P_t = P_{{\rm min}...
zoltan.leka's user avatar

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