All Questions
9,781 questions
7
votes
2
answers
988
views
Missing mass conjecture
Let $n,t$ be positive integers and $p_1,p_2,\ldots,p_n$ positive numbers summing to 1. Conjecture:
$$
\sum_{i=1}^n p_i (1-p_i)^t \le \frac{n(1-1/n)^n}{t}
$$
always holds.
The motivation comes from my ...
- 1,446
0
votes
0
answers
64
views
When is a symmetric block Toeplitz matrix invertible?
Let
$$
Q =
\begin{bmatrix}
Q_0 & Q_1 & Q_2 & \cdots\\
Q_{-1} & Q_{0} & Q_1 & \cdots\\
Q_{-2} & Q_{-1} & Q_0 & \cdots\\
\vdots & \vdots & \vdots & \ddots
...
- 63.9k
9
votes
1
answer
428
views
The cardinality of projections of subsets of the Hilbert cube by inner products
I have three related questions.
Question 1: Is there a subset $X$ of the Hilbert cube $[0,1]^{\Bbb N}$ of cardinality continuum, such that for each sequence $a\in [0,1]^{\Bbb N}$ with $\sum a_n$ ...
- 3,104
0
votes
1
answer
143
views
Is the space $C_0^{k}(\Omega)$ a Montel space?
I asked this question in the MathStackExchange, but I think I'm not get any answer.
I'm trying to find a reference for the following result:
Theorem: Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ ...
- 3,425
4
votes
2
answers
258
views
Simple proof that exactness implies strong mixing
Let $f$ be a continuous map defined on a compact metric space $X$. Suppose that $f$ preserves the Borel probability measure $\mu$ and that, for every positive-measure set $A\subseteq X$, we have $$\...
- 23.2k
0
votes
1
answer
50
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*}...
- 128k
2
votes
1
answer
201
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
76
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,\...
- 161
2
votes
1
answer
183
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, ...
- 128k
3
votes
1
answer
145
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_{...
- 16.2k
0
votes
1
answer
117
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}^...
- 12.9k
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 ...
- 12.9k
7
votes
2
answers
345
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 ...
- 41.1k
2
votes
1
answer
112
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. ...
- 24.2k
4
votes
1
answer
188
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 ...
- 24.2k
1
vote
0
answers
43
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 ...
- 93
5
votes
1
answer
351
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 ...
- 12.9k
1
vote
1
answer
53
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) ...
- 136
0
votes
0
answers
22
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 $X$ ...
- 661
-1
votes
2
answers
232
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 ...
- 6,486
0
votes
1
answer
121
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(\...
- 852
2
votes
0
answers
88
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
320
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
76
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
$$
H = H(\alpha_1,...
- 39.1k
3
votes
2
answers
393
views
Monotonicity of matrix conjugation
Let $A$ and $B$ be positive-definite matrices such that $A \le B.$
By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$
I am now curious under ...
- 6,351
0
votes
0
answers
80
views
Verifying the Cauchy behavior of a sequence
Let me consider the iteration $x_{n+1}=Tx_n$ and $T$ is a self-map from a non-empty subset $K$ of a smooth Banach space $X$ to itself, satisfying $W(Tx, Ty) \leq W(x, y)$, where $W(x, y)=\Vert x \Vert^...
- 85
0
votes
1
answer
118
views
Minimal norm problem whose unknown is an operator
Generally given an Hilbert space $X$ with and a bounded linear operator $H : X \to X$ given a vector $y \in X$ we seek an $x \in X$ such that
$$
f(x) = \frac{1}{2} \left\lVert Hx - y \right\rVert_2^2
$...
- 39.1k
5
votes
1
answer
188
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 ...
- 1,848
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-...
- 439
1
vote
1
answer
180
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:
$$f_i(x)=a_i*f(x+b_i)+c_i$$
...
CommunityBot
- 1
0
votes
3
answers
753
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 $I$...
- 6,367
0
votes
0
answers
32
views
reference request: mercer expansion and kernel underlying Sobolev spaces?
Let us define the periodic Sobolev spaces, for $s > n/2$ by
$$
H_{s}([0, 1]^n) = \{f : [0, 1]^n \to \mathbb{R} :\mbox{for}~j\leq s, f^{(j)} |_{\partial[0, 1]^n} \equiv 0, ~~
\int_{[0, 1]^d} (f^{(s)...
- 380
12
votes
2
answers
5k
views
Where was/is Compensated Compactness used?
This last summer, I read up on Tartar's so called Method of Compensated Compactness (or at least how it applied to scalar conservation laws). I used this theory to prove the existence of $L^{\infty}$ ...
- 4,713
5
votes
1
answer
696
views
Is any order bounded continuous linear functionals a difference of positive continuous functionals?
Let $B$ be a Banach space and $K$ a closed proper cone in $B$ such that the induced partial order makes $B$ a vector lattice. Let $K'=\{x^*\in B':\langle x^*, x\rangle\geq 0\ \forall x\in K\}$ the ...
- 4,219
0
votes
0
answers
55
views
Strong sub-differentiability of an equivalent strictly convex norm
First, we define the notion of strong sub-differentiability(SSD) of a norm on a Banach space $X$. The norm $\Vert \cdot \Vert$ of $X$ is said to be SSD if the one-sided limit $$\lim_{t \to 0+} \frac{\...
- 85
4
votes
1
answer
138
views
Fredholm property of linearization of Floer map
I am reading Audin and Damian's book "Morse theory and Floer homology". In Proposition 8.1.4 which reveals the transversality property of moduli space of solutions of Floer equation, the ...
- 9,580
1
vote
1
answer
67
views
Norm of differentiation operator with respect to Gaussian norm
Here is a problem from Luenberger's optimization by vector space methods. I would appreciate steps to proceed.
Let $\mathcal{P}_n\subset\mathbb{R}[x]$ be polynomials of degree at most $n\ge0$. Compute ...
- 24.2k
0
votes
1
answer
140
views
Approximating a sequence of tempered distributions "uniformly" by Schwartz functions
This question has been motivated by the post making sense of distributions on the diagonal.
Let $T$ be a tempered distribution on $\mathbb{R}^2$ and $\eta$ be a given mollifier on $\mathbb{R}$. For $f ...
- 3,584
2
votes
0
answers
103
views
What $1$-forms $\theta$ solve $\Delta \theta = f\theta$ for a smooth function $f$?
I have a seemingly basic question that I cannot find any literature on. Let $(M,g)$ be a smooth Riemannian manifold and let $\Delta:\Omega^1(M) \to \Omega^1(M)$ be the Laplace-De Rham operator on $1$-...
CommunityBot
- 1
2
votes
0
answers
35
views
Continuity of Kernel Mean Embeddings
Given some kernel $k: X \times X \to \mathbb{R}$ with RKHS $H_k$ we say that $k$ is characteristic on the space of signed Radon measures over $X$, denoted by $\mathcal{M}(X)$, if the kernel mean ...
- 161
1
vote
1
answer
182
views
Inequalities involving entropy: quantum discord and mutual information
My question is inspired by the following paper of Olivier and Żurek but for this question to be self-contained I will recall all the necessary definitions: for a quantum state $\rho$ we define the ...
- 9,340
4
votes
3
answers
482
views
Does the uniform boundedness principle holds for multilinear maps as well?
This question has been motivated by weak* completeness of distributions.
According to the answer in the above post, any barrelled locally convex topological vector space $E$ satisfies the uniform ...
- 3,584
2
votes
1
answer
128
views
On the existence of a complicated fractal-like set of finite perimeter
Let $f\in BV(\Bbb R^n)$ be an integer-valued function that maps into $\{0, 1\}$ and is identically $0$ outside some bounded set in $\Bbb R^n$. In particular, $f$ determines a bounded Caccioppoli set $...
- 28k
0
votes
0
answers
48
views
Computation of Brown measure of the shift operator on $\ell^2(\mathbb N)$?
This looks an extremely simple question - I am just trying to give an example of Brown measure, https://en.wikipedia.org/wiki/Brown_measure, so I try to compute it for the left/right-shift operator on ...
- 1,755
2
votes
1
answer
232
views
Banach spaces locally having a basis
The $\mathcal{L}_p$-spaces ($1\leq p \leq \infty$) are Banach spaces $X$ such that there exists a constant $\lambda$ so that every finite dimensional subspace $E$ of $X$ is contained in another ...
- 2,605
3
votes
1
answer
3k
views
Approximate point spectrum
I have a question concerning the relation between the approximate point spectrum and the spectrum of an operator.
Let $T$ be a bounded linear operator of a complex Hilbert space $H$. The approximate ...
CommunityBot
- 1
1
vote
0
answers
109
views
PDE coupled with the pronic numbers (related to triangular numbers)
I am studying the linear PDE:
$$ t^2\frac{\partial^3}{\partial t^3}\sum_{n=1}^\infty \Psi_n(t,s)=s^2\frac{\partial}{\partial s}\sum_{n=1}^\infty \Psi_n(t,s)+\sum_{n=2}^\infty b(n)\frac{\partial}{\...
- 383
3
votes
0
answers
94
views
Harmonic heat flow, formal and rigorous
Let $ (M,g) $ be a smooth Riemann manifold without boundary, $ S^{n-1} $ is an $ n $-dimensional sphere, and $ T>0 $. Consider a weak solution $ u:M\times[0,T]\to S^{n+1} $ of
$$
\partial_tu-\Delta ...
- 519
2
votes
2
answers
155
views
"Completeness" for weak convergence of unbounded closed operators on a separable Hilbert space $H$
Let $H$ be a separable Hilbert space with the inner product $\langle, \rangle$ and $\{ T_n \}$ be a sequence of unbounded closed linear operators with a common dense domain $D \subset H$ such that $...
- 24.2k
7
votes
1
answer
361
views
What is the analogue of the Jacobi theta function in the Weyl representation?
It is known (see for example the associated Wikipedia entry) that the Jacobi theta function
$$\vartheta(z; \tau) = \sum_{n\in\mathbb{Z}} \exp(\pi in^2\tau + 2\pi inz)$$
arises from a certain ...
- 6,367