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$L^2$ norms of Whittaker vectors and zeros of Intertwining operators

For $\mu,\nu\in \mathbb{C}^2$ we denote $I(\mu,\nu)$ to be the principal series of $\mathrm{GL}_2(\mathbb{Q}_p)$ induced from $|.|^\mu\otimes |.|^\nu$. For $s=\mu-\nu$ one defines the standard ...
Subhajit Jana's user avatar
8 votes
0 answers
686 views

The function space defined by deep neural nets

Given a deep net graph and the activation functions on the hidden vertices do we have a description of the function space spanned by it? (even if for some specific architectures and activation ...
gradstudent's user avatar
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8 votes
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384 views

What is the name for a Banach space property closed under ultraproducts?

In Banach space theory, a super-property is a property of a Banach space that is preserved under ultrapowers. (Update (2015-09-28): The property must also be closed under isometric embeddings.) (...
Jason Rute's user avatar
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8 votes
0 answers
421 views

Approximate singular value decomposition in Banach spaces

I am interested in generalisations to Banach spaces of the following construction, which relates to the singular value decomposition of a finite-dimensional linear map. If $V$, $W$ are finite-...
Ian Morris's user avatar
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8 votes
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208 views

(Un)bounded Geometry and Sobolev Spaces

This post is related to this and this post. It is known that on a complete Riemannian manifold, the space $C^\infty_c(M)$ is generally not dense in the Sobolev spaces $W^{k, p}(M)$ ($1 \leq p < \...
Matthias Ludewig's user avatar
8 votes
0 answers
278 views

Pseudodifferential operators on compact manifolds with boundary

I have heard that the square root of the Dirichlet (or the Neumann) Laplacian is not a pseudodifferential operator on compact manifolds with boundary. The context in which this was said was that ...
student's user avatar
  • 81
8 votes
0 answers
1k views

On the classification of injective Banach spaces

A Banach space $E$ is injective when it is complemented in each Banach space $X$ that contains it as a closed subspace. The space $E$ is $1$-injective if the copy of $E$ in $X$ is the range of a norm-...
M.González's user avatar
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8 votes
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6k views

Convex hulls of compact sets

Let $A$ be a compact set in a separable Hilbert space $H$, and let $\bar A$ denote its convex hull. Is $\bar A$ compact?
Tom LaGatta's user avatar
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8 votes
0 answers
512 views

Proving a proposition which leads the irrationality of $\frac{\zeta(5)}{\zeta(2)\zeta(3)}$

Question : Is the following $(\star)$ true for $a,b,c\in\mathbb Z$ ? $$\begin{align}\int_{0}^{\frac{\pi}{2}}(ax^4+b\pi x^3+c{\pi}^{2}x^2)\log(\sin x)dx=0\Rightarrow a=b=c=0\qquad(\star)\end{align}$$ ...
mathlove's user avatar
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8 votes
0 answers
221 views

Density of odd and even eigenstates of an integral operator

Consider an integral operator $(Kf)(x)=\int_{-1}^{1}K(x-y)f(y)dy$, where the kernel $K(-x)=K(x)$ is an even function. Let $\lambda_n$ be the ordered eigenvalues of $K$ and $f_n(x)$ the ...
Alex's user avatar
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8 votes
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357 views

Ultrapowers of Banach spaces without the continuum hypothesis

Let $\mathcal{U}$ be a non-trivial ultrafilter on the set of integers $\mathbb{N}$, and let $C(K)$ denote the Banach space of continuous functions on a compact $K$. Under the continuum hypothesis CH, ...
M.González's user avatar
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8 votes
0 answers
952 views

About generator and isomorphism problems for free groups operator algebras

Let $H$ be an infinite dimensional separable Hilbert space. The $C^{*}$-algebras and von Neumann here unital and subalgebras of $B(H)$. Definition : Let $\mathcal{A}$ be $C^{*}$-algebra (resp. a von ...
Sebastien Palcoux's user avatar
8 votes
0 answers
221 views

Inertia group vs. differential equations

The tame quotient of the inertia group of $\mathbf Q_p$, say, is the profinite group generated by the Frobenius $\sigma$ and the monodromy $\tau$, subject to the relation $\tau^{p-1} [\tau, \sigma] = ...
Jakob's user avatar
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8 votes
0 answers
452 views

Preduals of $\ell_1$

The space $\ell_1$ has loads of (isomorphic) predulas. They can be as weird as possible but I am interested in Banach lattices. Question: Let $X$ be a Banach lattice with dual isomorphic to $\ell_1$. ...
Jan Vardøen's user avatar
8 votes
0 answers
1k views

Strictly singular operators and their adjoints

This is a question I thought about a while back and figured I'd throw it out there to see if anyone has some insight that I am missing. Let $X$ and $Y$ be infinite dimensional separable Banach ...
Kevin Beanland's user avatar
8 votes
0 answers
751 views

The log kernel and Bochner Theorem

I was wondering if it possible to find a measure $\eta$ on $\mathbb{R}$ such that $$ L(x):=\log\frac{1}{|x|}=\int e^{itx}\;d\eta(t) $$ for every $x\in [0,1/2]$. On a structural ground, this ...
Adrien Hardy's user avatar
  • 2,135
8 votes
0 answers
196 views

Parametrizing derivations from the algebra of smooth functions on a manifold to its dual

$\newcommand{\Der}{\operatorname{Der}}$ $\newcommand{\Real}{{\mathbb R}}$ (Disclaimer: I fear this question may be a bit too basic for MO, but in my defence I have essentially zero differential ...
Yemon Choi's user avatar
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8 votes
0 answers
349 views

Finding a dimension-free bound for a certain multiplier on Euclidean space

The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...
Steven Heilman's user avatar
8 votes
0 answers
525 views

Phase perturbations in oscillatory integrals

I am interested in learning about quantitative refinements of the method of stationary phase which allow to treat small perturbations in the phase of an oscillatory integral (of the first kind, in ...
user17240's user avatar
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1k views

G-delta of measure 0 containig the rationals.

It is well known that $\mathbb{Q}$ is not a $G_\delta$. In fact no countable dense subset of $\mathbb{R}$ is a $G_\delta$. We order the rationals in a sequence: $\mathbb{Q}=\lbrace r_k\rbrace$, and ...
Julián Aguirre's user avatar
8 votes
0 answers
605 views

convergence rate in Wiener's approximation theorem

Wiener has the following fantastic results about approximations using translation families: Given a function $h: \mathbb{R} \to \mathbb{R}$, the set $\{\sum a_i h(\cdot - x_i): a_i, x_i \in \mathbb{...
gondolier's user avatar
  • 1,839
8 votes
1 answer
207 views

Subspaces of $L_p([0,1])$ whose unit ball is compact for the topology of convergence in measure

Any information about the following questions would be welcome. I wonder whether there are (well-known or easy) closed and infinite dimensional subspaces of $L_p([0,1])$ ($1<p<\infty$) whose ...
user159631's user avatar
7 votes
0 answers
269 views

Looking for the eigenfunctions of the operator $T$ on $L_2(\mathbb R^+)$ defined by $Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy$

I'm looking to find a basis of eigenfunctions (and the corresponding eigenvectors) for the operator $T$ on $L_2(\mathbb R^+)$ defined by: $$ Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy $$ This operator ...
martin tassy's user avatar
7 votes
0 answers
249 views

Proving this function is convex

Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
Tom Solberg's user avatar
  • 4,049
7 votes
0 answers
131 views

Approximation of a continuous curve on commuting matrices

I have a continuous curve $A:\mathbf{R}_+\rightarrow \text{M}_N(\mathbf{R})$ such that $[A(t),A(s)] \operatorname*{\longrightarrow}_{t,s\rightarrow +\infty} 0$, where $[A(t),A(s)] = A(t)A(s)-A(s)A(t)$....
Ayman Moussa's user avatar
  • 3,425
7 votes
0 answers
294 views

Applications of Banach space homology

There is a well-developed theory of Banach space homology. What are some of its useful applications to Banach space theory and which important questions can one answer using it? In other words, how ...
Andromeda's user avatar
  • 175
7 votes
0 answers
151 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-...
J_P's user avatar
  • 439
7 votes
0 answers
164 views

Nontrivial examples of locally compact quantum groups

What are some families of locally compact quantum groups that are neither groups, duals of groups, compact, nor discrete?
Cameron Zwarich's user avatar
7 votes
0 answers
177 views

What is the current status of research on the von Neumann's inequality for $n \ge 3$?

Problem Let $(T_1, \ldots, T_n)$ be a tuple of commuting contractions in Hilbert space $H$. Does a constant $C_n \ge 1$ exist, for which it would be true, that: $$\forall_{p \in \mathbb{C}[x_1, \ldots,...
S-F's user avatar
  • 63
7 votes
0 answers
150 views

The space of analytic associative operations

This question is a follow-up to this old one of mine. Let $\mathcal{A}$ be the set of functions $\star:\mathbb{R}^2\rightarrow\mathbb{R}$ which are associative and $C^\omega$ (real analytic entire) in ...
Noah Schweber's user avatar
7 votes
0 answers
254 views

$C^0$-limit of volume-preserving maps on $\mathbb R^n$

Let $f_k:B_1\rightarrow \mathbb R^n$ be a sequence of injective differentiable volume-preserving maps (i.e. $\mu(f_k(A))=\mu(A)$ for any measurable $A\subset B_1$) that converges uniformly to $f:B_1\...
Tian LAN's user avatar
  • 435
7 votes
0 answers
162 views

Relation between the additive Haar measure on $(K,+)$ and the multiplicative Haar measure on $K^{*}$ for a global field $K$

The following question comes from my studying of Alain Connes's paper Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function. In it, on p. 11, Connes notes that if $K$ is ...
The Thin Whistler's user avatar
7 votes
0 answers
123 views

Steklov eigenvalue for circle valued functions

Let $(M,g)$ be a compact Riemannian manifold with boundary. It is well known that the first positive Steklov eigenvalue $\sigma_1$ of $M$ has the following variational characterization: $$\sigma_1(M,g)...
Eduardo Longa's user avatar
7 votes
0 answers
80 views

Given composition rules, determining whether a continuous map between smooth functions is a pseudodifferential operator

Let $M$ be a closed manifold, and let $P:C^{\infty}(M)\rightarrow C^{\infty}(M)$ be a continuous linear map in the smooth Fréchet topologies. In what is to come, if it helps, one can assume further ...
MyShepherd's user avatar
7 votes
0 answers
198 views

The spectrum of the Banach algebra of certain arithmetic functions under Dirichlet convolution

I was thinking about using the tools of functional analysis to study some subring of arithmetic functions under Dirichlet convolution. If I let $D_s$ be the ring of arithmetic functions with finite ...
Aareyan Manzoor's user avatar
7 votes
0 answers
120 views

What is the closed cone generated by constant and coordinate functions and closed under taking $f\mapsto\max(f,0)$?

Let $C$ be the smallest closed convex cone of functions from $\mathbb{R}^n$ to $\mathbb{R}$ that contains all constant functions, all coordinate functions, and such that $\max(f,0)\in C$ whenever $f\...
alesia's user avatar
  • 2,772
7 votes
0 answers
481 views

A seemingly trivial property of continuous functions differentiable at the origin (PART 2)

Let $F:\mathbb{R}^n\to\mathbb{R}^n$ be a continuous function such that $F(0)=0$, $F$ is differentiable at $0$ and $DF(0)$ is invertible. Is there an elementary way to show that for all $\epsilon>0$ ...
No-one's user avatar
  • 1,149
7 votes
0 answers
193 views

Reduced group C*-algebra $C^*_r(\mathbb{Z}/2*\mathbb{Z}/2)$: norm of specific elements

Consider the free product of $\mathbb{Z}/2$ with itself with generators $$ \mathbb{Z}/2*\mathbb{Z}/2=\langle u,v\mid u^2=1=v^2\rangle $$ and regard its group $C^*$-algebra $$ C^*(\mathbb{Z}/2*\mathbb{...
C-star-W-star's user avatar
7 votes
0 answers
215 views

Helmholtz decomposition of compactly supported fields

Let $\mathbf F$ be a smooth vector field in $\mathbb R^3$, which is null outside a finite compact domain $V$. By the Helmholtz decomposition theorem, there exist a scalar field $\Phi$ and a vector ...
MikeTeX's user avatar
  • 687
7 votes
0 answers
2k views

Algebraizing topology and analysis via condensed mathematics

I asked this question on Mathematics Stackexchange, but one of the users suggested that I ask this question at MathOverflow. I've just come across a Twitter thread by Laurent Fargues explaining a work ...
Ythyb's user avatar
  • 79
7 votes
0 answers
317 views

Multiple Fourier series

In the book by Elias M.Stein and Guido Weiss "Introduction to Fourier Analysis on Euclidean Spaces" one states in page 268 the following theorem: Theorem 1: The trigonometric series $$\...
Elmustapha NADIR's user avatar
7 votes
0 answers
369 views

On the solvability of a nonlinear differential system

A nonlinear formulation of differential Galois theory was discussed here and here for three dimensional nonlinear systems (proof is on pages 6 – 10). For a two dimensional system, the following system ...
UNOwen's user avatar
  • 79
7 votes
0 answers
207 views

Is the derivative the unique operation on points in the plane that preserves convexity?

Let $C(n)$ be the space of multisets of size $n$ of points in the Euclidean plane, topologised appropriately, and consider a surjective continuous map: $$D:C(n)\rightarrow C(n-1)$$ Such that the ...
Chris H's user avatar
  • 1,949
7 votes
0 answers
419 views

How would have Bezout proved Bezout's theorem?

How would Bezout have proved Bezout's theorem bounding the number of points in the intersection of two plane (polynomial) curves in $\mathbb{R}^2$? I have looked at a couple of modern algebraic ...
K Hughes's user avatar
  • 679
7 votes
0 answers
242 views

Has this Banach algebra been studied?

Given $\Omega$ as $[0,1]^n$ or the closed unit ball in $\mathbb{R}^n$, we can consider the algebra of complex valued polynomials with pointwise multiplication and its closure with respect to the norm ...
Alan's user avatar
  • 71
7 votes
0 answers
132 views

Smoothing property of a certain singular integral operator of non-convolution type

For simplicity, suppose that the dimension $d=2$, and let $g_s(x)$ be the Coulomb or Riesz potential defined by $$g_s(x) := \begin{cases} -\frac{1}{2\pi}\ln|x|, & {s=0} \\ c_s|x|^{-s},& {0<...
Matt Rosenzweig's user avatar
7 votes
0 answers
351 views

Fractional Laplacian and chain rule

For the classical Laplacian, we have $$\Delta (h(u)) = h'\Delta u + h''(u)|\nabla u|^2$$ for smooth functions $h$ and $u$. Does a similar chain rule hold (up to a reminder term) also for the ...
Zac's user avatar
  • 161
7 votes
0 answers
294 views

Weaker version of the Borel lemma for vector-valued functions

Borel's lemma for Frechét-spaces $V$ says: (i) For every $(v_j)_{j \in \mathbb{N}} \in V^\mathbb{N}$ there exists a smooth $f: \mathbb{R} \to V$ such that $$f^{(j)}(0) = v_j.$$ For general locally ...
Jannik Pitt's user avatar
  • 1,474
7 votes
0 answers
158 views

$C^*$ algebras whose nontrivial projections form a non empty compact connected set

Apart from $M_2(\mathbb{C})$. what is an example of a $C^*$ algebra $A$ whose set of non trivial projections form a non empty compact connected set? Is there an example of this situation such that ...
Ali Taghavi's user avatar
7 votes
0 answers
3k views

Definition of homogeneous Sobolev spaces

As we know the inhomogeneous Sobolev space (we only consider $s>0$) $${H}^{s}\left(\mathbb{R}^{n}\right)=\left\{f \in L^2(\mathbb{R}^n):\int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} ...
Slm2004's user avatar
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