All Questions
10,935 questions
2
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0
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157
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Why do von Neumann algebras possess identity?
My starting point is that a von Neumann algebra is a $C^*$-algebra with a predual. The usual approaches to showing the existence of identity involve spectral theory (for approximate identity), but ...
- 433
1
vote
1
answer
115
views
What is the socle of the $2\times 2$ matrix algebra over a Banach algebra?
$\DeclareMathOperator\soc{soc}$Let $\mathcal{A}$ be a unital semisimple Banach algebra. The socle of $\mathcal{A}$, $\soc(\mathcal{A})$ is defined as the sum of the minimal right ideals (which equals ...
- 515
0
votes
1
answer
242
views
When do the weak-star and compact convergence (compact-open) topology coincide on the dual of a Banach space?
In Measures Which Agree on Balls by Hoffmann-Jørgenson, it is claimed on page 323 that for an arbitrary Banach space $E$, if $\pi$ is the topology on $E^*$ of uniform convergence on compact subsets of ...
- 297
1
vote
1
answer
329
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
$$
R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)...
- 6,853
17
votes
1
answer
986
views
Uncountably many subsets of the natural numbers with certain natural density condition
Are there uncountably many $A_\alpha $ of subsets of $\mathbb{N}$ with the following two properties:
Each $A_\alpha$ has positive upper natural density
$A_\alpha \cap A_\beta$ is a finite set for $\...
- 356
2
votes
0
answers
180
views
Approximating $L^p$ functions by eigenfunctions of Laplacian
I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932.
In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
- 121
94
votes
6
answers
14k
views
Quasicrystals and the Riemann Hypothesis
Let $0 < k_1 < k_2 < k_3 < \cdots $ be all the zeros of the Riemann zeta function on the critical line:
$$ \zeta(\frac{1}{2} + i k_j) = 0 $$
Let $f$ be the Fourier transform of the sum ...
- 22.3k
3
votes
2
answers
382
views
Singular support: equivalent definition
Let $U\subset\mathbb{R}^{d}$ be an open set. The singular support of a distribution $u\in\mathcal{D}^{\prime}(U)$ is defined to be the compliment of the set of points, which have a neighbourhood in ...
- 1,171
5
votes
1
answer
311
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Maximal operator estimates for the Schrödinger equation
Let $a>0$ and consider the operator
$$Tf(t,x)= \int_{\mathbb{R}^{n}}e^{ i x\cdot \xi} e^{i t \lvert\xi\rvert^{a}} \widehat{f}(\xi) \, d\xi.$$
When $a=2$, the function $Tf$ solves the Cauchy problem ...
- 852
1
vote
0
answers
146
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Intuition behind right-inverse of map from Johnson-Lindenstrauss Lemma
The Johnson–Lindenstrauss lemma states that for every $n$-point subset $X$ of $\mathbb{R}^d$ and each $0<\varepsilon\le 1$, there is a linear map $f:\mathbb{R}^d\to\mathbb{R}^{O(\log(n)/\varepsilon^...
- 5,405
0
votes
0
answers
81
views
Ultraproduct reflexive
Hello I have the following construction: Let $(E,\|\cdot\|_E)$ be a Banach space, $E_n:=E^n$ and $\|x_n\|_n:=\frac{1}{n}\sum_{k=1}^n \|x_n(k)\|_E$ for all $x_n=(x_n(1),...,x_n(n))\in E^n$ and $n\in\...
- 1
2
votes
1
answer
547
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Shift-invariant spaces
We can define a shift-invariant space as
$$V_{\varphi}(\mathbb{Z}):=\left\{\sum_{k\in\mathbb{Z}}c_k\varphi({\cdot}-k):(c_k)\in \ell_2\right\},$$
where convergence of the series is taken to be in $L^2(\...
- 49
2
votes
1
answer
215
views
Asymptotics for oscillatory integral
Consider the following integral for $f \in C_c^{\infty}(\mathbb R^n)$, $x_0$ fixed (possibly zero), and $n \ge 3$
$$F(\lambda) = \int_{\mathbb R^n} e^{i\lambda \vert x-x_0 \vert^2} \frac{f(x)}{\vert x ...
1
vote
0
answers
123
views
Dependence of Sobolev embedding theorem constant on smoothness
Assume that $\Omega \subset \mathbb{R}^d$ is "nice" enough and $k$ is a positive real number. Using the Sobolev embedding theorem, we can get that
$$
\|f\|_{W^{0,2d/(d-2k)}\ \ \ \ \ (\Omega)}...
- 11
1
vote
0
answers
80
views
Weak$^\ast$ closure of a countably complete sublattice of the unit ball of $L^\infty(\Omega, \mu)$
This is a reframing of my previous question from a Banach lattice perspective: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity? The previous question ...
- 2,830
0
votes
0
answers
119
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About definition of stable solution. $Q_u(\phi) \ge 0$ for all $\phi \in C_c^1(\Omega)$ replaced by "for all $\phi \in W_0^{1,2}(\Omega)$"
I want to ask about a remark about the stable solution of elliptic PDE Remark 1.1.1.
We say $u$ is stable solution of $-\Delta u=f(u) \ \text { in } \Omega$ and $u=0$ on $\partial \Omega$ if it ...
- 809
1
vote
0
answers
103
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Cyclic representation isomorphic to L2 space
This question is also posted on Math Stack Exchange.
I need some help understanding a proof of the following claim: every cyclic representation is isomorphic to some $L^2$ space.
First, formal ...
- 111
2
votes
0
answers
144
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Positivity of the Fourier transform: prove or disprove that $\operatorname{Re}(\overline{\widehat{u}}(\xi) \widehat{F\circ u}(\xi))\geq0$
Let $F:[0,\infty) \to[0,\infty)$ be increasing, $C^1$ and $L-$Lipschitz with $F(0)=0$. Let $u\in L^1 (\Bbb R^d)$, $u\geq0$ so that $F\circ u\in L^1 (\Bbb R^d)$
I would like to prove (or disprove) ...
- 1,101
6
votes
1
answer
288
views
Sigma-weakly dense *-subalgebra of von Neumann algebra has increasing net of positive elements convergent to the identity
Let $M$ be a von Neumann algebra and $A\subseteq M$ a $\sigma$-weakly dense $*$-subalgebra of $M$. Does there exist an increasing net $\{a_i\}_{i\in I}\subseteq A\cap M^+$ such that $a_i\to 1$ in the $...
- 175
2
votes
1
answer
165
views
Continuity of an upper semi-continuous function over periodic points
Let $f: X \to \mathbb{R}$ be an upper semi-continuous function on $X$, which is a compact subspace of a vector space. Let sequence $x_n, n \in \mathbb{N}$, with positive elements - periodic: there ...
- 1,043
1
vote
2
answers
311
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Is there a bounded sequence $(e_n)$ such that $e_n \in E_n$ and that $(e_n)$ does not have any convergent subsequence?
Let $(E, |\cdot|)$ be an infinite-dimensional Banach space. Assume that
$T:E\to E$ is a compact (bounded linear) operator, and
$(\lambda_n)$ is a sequence of distinct eigenvalues of $T$.
Let $E_n$ ...
- 657
2
votes
0
answers
89
views
On dense subspaces of $L^p$-spaces of finitely additive measures
Let $\mu$ be a finite, finitely additive measure defined on the Borel $\sigma$-algebra of a real separable Hilbert space $\mathcal{H}$ with dual $\mathcal{H}^{*}$. Write $L^{p}(\mathcal{H},\mu)$ for ...
- 505
1
vote
0
answers
119
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Continuity of a minimizing measure w.r.t a parameter
Let $V_t(x)=x^2+t\phi(x)$ where $t>0$ and $\phi\in C^\infty_c(\mathbb{R})$.
My question is what can be said about the continuity of the (unique) minimizer (among probability measures) of the ...
- 41
0
votes
1
answer
165
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For $\mu$-a.e. $x \in X$, the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^1 (Y)$. Then $(f_n)$ is Cauchy in $L^1 (X \times Y)$
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the ...
- 825
1
vote
0
answers
64
views
The operator $D^{p}\colon \mathcal{S}\subset L^{1}(\gamma)\to L^{1}(\gamma)$ is closable for every integer $p =1,2,\dots$
I am reading Nourdin and Peccati’s textbook (Normal Approximations with Malliavin Calculus From Stein’s Method to Universality). My question is about Lemma 1.1.6. Which says
Lemma 1.1.6:
The operator $...
0
votes
0
answers
145
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$L_\infty([0,1], \mathbb{C})$ is it isomorphic to $\ell_\infty(\mathbb{N}, \mathbb{C})$?
By a result of Pełczyński, $L_\infty([0,1], \mathbb{R})$ is isomorphic to $\ell_\infty(\mathbb{N}, \mathbb{R})$. That is the case of real valued functions and sequences.
A natural question then is: ...
- 141
28
votes
1
answer
956
views
Grothendieck's in-spirit-category-theoretic functional analysis?
I heard several times (for instance in these general lectures) that Grothendieck did functional analysis before he started doing algebraic geometry and category theory. It is said that at the time he ...
- 333
2
votes
0
answers
73
views
Derivative of a functional involving integral and level set
Let $\Omega$ be a bounded smooth domain. For $u\colon \Omega \to \mathbb{R}$, define the functional
$$F(u) = \int_{\{u=0\}}g(x) \; \mathrm{d}x$$
where eg. $u \in H^2(\Omega) \cap C^0(\bar\Omega)$ and ...
- 29
5
votes
1
answer
340
views
How to give a counterexample of this estimate related to Paley-Littlewood theorem?
I am studying Paley-Littlewood theorem in Harmonic analysis, and I met an exercise. I would like to construct a function $f$ as a counterexample to show that the inequality
\begin{equation}
\| f \|^...
- 187
2
votes
1
answer
150
views
Sufficient conditions for the graph measurability of a multivalued function
I am currently working on a problem related to the measurability of multi-functions in the context of mathematical economics. Specifically, I am searching for sufficient conditions regarding the graph ...
- 79
8
votes
3
answers
1k
views
Ramanujan's Master Formula: A proof and relation to umbral calculus
The Ramanujan's master theorem states that:
$$
\int_0^{\infty}x^{s-1}\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}a_nx^ndx=\Gamma(s)a_{-s}
$$
I found a really strange proof recently on a personal blog:
Define
$...
- 302
1
vote
0
answers
127
views
an eigenvalue problem for Jacobi Forms
Assume $G(q,z)$ is a Jacobi form of a certain index k. It is known that $G$ can be expanded in a Taylor series with coefficients in the ring of quasi-modular forms (generators $E_2, E_4$ and $E_6$).
$\...
- 43.2k
2
votes
2
answers
167
views
LF or LB space that happens to be finite dimensional
Let $\{V_n\}_{n=1}^\infty$ be a collection of finite dimensional vector subspaces of $L^2[0,1]$ such that $V_n \subset V_{n+1}$ and $\bigcup_{n=1}^\infty V_n$ is dense in $L^2[0,1]$. Suppose further ...
- 3,477
2
votes
1
answer
389
views
Can we interpret fractional Sobolev spaces in terms of fractional derivatives?
Let $1 \leq p < \infty$, $0<s<1$, and $\Omega \subseteq R^n$ be a domain. The Banach space $W^{s,p}(\Omega)$ is defined as
$$W^{s,p}(\Omega) := \left\{ f \in L^p(\Omega) \colon \int_{\Omega \...
- 5,327
1
vote
1
answer
100
views
Does convergence of Radon transforms of a sequence of probability distributions implies convergence of the distributions themselves?
Let $P_1,P_2,\ldots $ be a sequence of absolutely continuous probability measures on $\mathbb R^n$, and let
$f_j:\mathbb R^n\to\mathbb R$ be their PDFs. Assume that $\operatorname{E}P_j = 0$ and $\...
- 13
1
vote
0
answers
81
views
Pre-positive definite functions?
A function $f(x,y)$ is positive definite if matrices $( f(x_i, x_j) )_{i, j \in F}$ are positive definite for all finite index sets $F$. This is frequently hard or impossible to check given some ...
- 620
4
votes
1
answer
259
views
The real and the imaginary part of a vector
In an infinite-dimensional Banah space $(X, \|\cdot\|)$ with a countable Schauder basis $\{x_n\}$, define:
$$
F_r: \operatorname{Span}(\{x_n\}) \rightarrow \operatorname{Span}(\{x_n\}), \hspace{0.3cm} ...
- 307
2
votes
1
answer
375
views
Radon-Nikodym derivative in a compact Hausdorff space
Let $X$ be a compact Hausdorff space where $X$ have infinitely many points and the topology is non-discrete, $m$ be a regular probability measure defined on the Borel $\sigma$-algebra of $X$, and $g$ ...
- 307
4
votes
1
answer
523
views
Is there any strengthened version of Rademacher's Theorem or any counterexample?
The following theorem is well-known in the ordinary analysis textbook:
Theorem: Assume the function $f:U\to\Bbb R^n$ is Lipschitz continuous on an open set $U\subset\Bbb R^m$, then $f$ is almost ...
- 300
3
votes
1
answer
499
views
Thin-Plate-Spline understanding and solution
This is a migrated question from: Thin-Plate-Spline understanding and solution.
If I need to delete one of the questions let me know. I was suggested to post it here as well.
As I understand it a Thin-...
- 115
11
votes
2
answers
1k
views
Is there a relation between Gelfand duality and the spectrum of a ring (with its Zariski topology)?
Compare the following two results:
Thm A) Let $A$ be a commutative $C^*$-algebra and let $X$ be its Gelfand spectrum. Gelfand duality says that there's a natural isometric $*$-isomorphism from $A$ to ...
- 721
1
vote
1
answer
74
views
Subspaces generated by the orbits of the group of isometries on $C(K)$
Let $X$ be an extremally disconnected compact Hausdorff space with no open points, and $f:X\to\mathbb{C}$ be a non-constant continuous function. Let $D_f$ be the linear span of the functions of the ...
- 2,605
3
votes
0
answers
74
views
Locally compact rings with reciprocals
A topological field is defined to be a topological ring $F$ with reciprocals such that the reciprocal function $F\setminus\{0\} \to F\setminus\{0\}$ is continuous. Locally compact topological fields ...
- 974
1
vote
1
answer
69
views
Local completion of bornological space
I recently stumbled across this old publication which defines the local completion of a Hausdorff locally convex space.
The construction is as follows:
A Hausdorff locally convex space $E$ is locally ...
- 651
3
votes
1
answer
353
views
Sequential separability on $C_p(X)$
Definition. Let $E$ be a topological space. Suppose that $E$ contains a sequence $\{x_n\}$ such that for every $x\in E$, there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ with $x=\lim x_{n_k}$. ...
- 4,058
3
votes
1
answer
190
views
Can the equation $1+z+z^q=z^n$ have multiple complex roots $z$?
It is proved here that the equation $1+z+z^2=z^n$ have no multiple complex roots.
Q. Let us consider the equation $1+z+z^q=z^n$ where $q$ and $n$ are natural numbers with $1<q<n$. Any ...
- 4,058
1
vote
0
answers
72
views
Almost everywhere convergence of the Fourier series of a continuous function
I want to know if there is a simple proof of the almost everywhere convergence of the Fourier series of a function in $C(\mathbb{T})$. I am aware of Carleson-Hunt Theorem. I was wondering if the proof ...
- 895
3
votes
1
answer
182
views
How to choose some $h$ so its Fourier transform supported in some set?
Suppose that $K=[-N, N]$ is some compact subset of $\mathbb R$, for some $N>2.$
Can we expect to choose $h$ such that $h=1$ on $K$ and the support of the Fourier transform of $\widehat{h}$ ...
- 325
0
votes
1
answer
218
views
Intersection of Hilbert spaces with Schauder basis
Let
$H$ be a infinite dimensional, separable, complex Hilbert space,
$\{v_{1_n}\}_{n \in \mathbb{N}}$ be a sequence in $H$,
$V_1=\operatorname{span}\{v_{1_n}\}_{n \in \mathbb{N}}$
$U_1=\overline{V_1}$...
- 433
1
vote
0
answers
125
views
Interpolating sequences are strongly separated
I am reading Agler and McCarthy's Pick Interpolation and Hilbert Function spaces. In Chapter 9, titled "Interpolating Sequences", the authors say that every interpolating sequence is ...
- 151