All Questions
3,599 questions with no upvoted or accepted answers
2
votes
0
answers
43
views
Selecting some linearly independent columns of a particular matrix
Let us consider the matrix $C=A_1+A_2$ where :
$A_1=(a_{k,l})_{k,l=0}^{n-1}$ is the $n$ by $n$ matrix given by $a_{k,l}=\frac{2}{\sqrt{n}}(\cos\frac{2kl\pi}{n})$
$A_2$ is the the $n$ by $n$ block ...
- 4,058
2
votes
0
answers
61
views
Translation request: Boundedness of Cauchy integral on Lipschitz boundary
The reference: "L'intégrale de Cauchy définit un opérateur borné sur $L^2$ pour les courbes lipschitziennes" (https://annals.math.princeton.edu/1982/116-2/p04) is written in French. Can we ...
- 269
2
votes
0
answers
201
views
Green function of a 2D exterior domain
Consider solutions of the laplace equation
\begin{equation}
\begin{split}
-\Delta u=f, \ \ u|_{\partial D}=0,
\end{split}
\end{equation}
where the domain $D\subset \mathbb{R}^2$. If $D$ is bounded ...
- 379
2
votes
0
answers
88
views
Explicit estimates on summability kernels
A "summability kernel" is a sequence of functions $k_n:[0,1)\to \mathbb C$ such that
$$ \int_0^1 k_n(t) \mathrm d t =1,$$
$$ \int_0^1 |k_n(t)| \mathrm d t =O(1),$$ with an implied constant ...
- 3,062
2
votes
0
answers
193
views
Commutative Banach algebras with zero-dimensional maximal ideal space and disjoint supports of Gelfand transforms
Let $A$ be a commutative semi-simple unital Banach algebra and let $\Delta$ be the maximal ideal space of $A$. Denote by $\widehat{\cdot}\colon A\to C(\Delta)$ the Gelfand transform.
If $\Delta$ is ...
- 21
2
votes
0
answers
77
views
Why is the essential numerical range defined as $W_e(T) = \bigcap_{K\in \mathcal K(H)} \overline{W(T+K)}$?
I have been introduced to the following definition of the essential numerical range of a bounded, linear operator on a separable, infinite-dimensional Hilbert space:
$$W_e(T) := \bigcap_{K\in \mathcal ...
- 165
2
votes
0
answers
65
views
Is it possible to extend Borel's lemma to the case of functional derivatives?
Let us think of a collection of tempered distributions $\{ T(x_1, \cdots, x_n)\}_{n=0}^\infty$. Here I will specifically set $x_i \in \mathbb{R}^4$ since I am considering quantum field theory and ...
- 3,477
2
votes
0
answers
99
views
Anisotropic Calderon-Zygmund decomposition
I am looking for the following version of Calderon-Zygmund decomposition, consider an function $f \in L^1(R^{d+1})$ and cylinders of the form $Q_{R,R^p}$ for some fixed $p \in (0,\infty)$, The ...
- 455
2
votes
0
answers
52
views
Can we decompose an increasing net of functions into two increasing nets with prescribed supports?
Let $K$ be a compact Hausdorff space and let $U,V\subset K$ be open. Let $\left(f_{i}\right)_{i\in I}$ be an increasing net of continuous non-negative functions such that $f_{i}\le 1$ and $f_{i}$ ...
- 5,529
2
votes
0
answers
57
views
Right spectral gap of vector of two independent Markov chains
Let $(X_i)$ be a stationary Markov chain on $S$ (a potentially uncountable space with a Borel sigma algebra) with stationary distribution $\pi$ and transition kernel $P$. Let $(Y_i)$ be a stationary ...
- 203
2
votes
0
answers
56
views
Existence of a suitable smooth kernel
Denote by $H=L^2([0,1])$ the Hilbert space of square integrable real valued functions on the interval $[0,1]$. Does there exist a nontrivial smooth real valued function $k$ on the unit interval such ...
- 4,115
2
votes
0
answers
57
views
Does the snowflake $X^\alpha$ allows isometric embeddings into $L_1$ if $X$ does?
Question: Suppose that finite metric space $X$ allows isometric embedding to $L_1$. Does it mean that a snowflake space $X^\alpha$ allows isometric embedding to $L_1$ for every $0 < \alpha < 1$?
...
- 1,058
2
votes
0
answers
104
views
Schwarz space on the upper half-plane
Let us think of the Schwartz space $\mathcal{S}(\mathbb{R}^2_+)$ on the upper half-plane $\mathbb{R}^2_+=\mathbb{R}\times(0,+\infty)$ defined as
$$
\mathcal{S}(\mathbb{R}^2_+)=\left\{f\in C^\infty(\...
- 1,959
2
votes
0
answers
172
views
Fourier transform harmonic oscillator eigenstates
The normalized eigenfunctions of the quantum harmonic oscillator are
$$\psi_{n}(x)= \frac{1}{\sqrt{2^n n!}} e^{-x^2/2}H_n(x),$$
where $n \in \mathbb N_0$ and $H_n$ is the $n$-th Hermite polynomial, ...
- 124
2
votes
0
answers
72
views
Semilinear elliptic equations in complex plane
Let $D$ denote the closed unit disk centered at the origin in the complex plane. Let $F: D \times \mathbb C \to \mathbb C$ be a smooth function. Is there any theory for well-posedness (in the sense of ...
- 4,115
2
votes
0
answers
202
views
Prove or disprove that $u=0$ a.e. on $\Bbb R^d$
Let $\Omega\subset\Bbb R^d$ be an open set. Let $k:\Bbb R^d\to [0,\infty)$ be measurable such that $0\in \operatorname{supp}k$. This implies that $\Omega\subset \Omega_k=\Omega+\operatorname{supp}k$. ...
- 1,101
2
votes
0
answers
64
views
Scaling limit of ODE with double-well potential
Let us consider the ODE
$$
\frac{d}{dt}x_\epsilon(t) = -\frac{1}{\epsilon} W'(x_\epsilon(t))
$$
where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the ODE reads
$$...
- 839
2
votes
0
answers
66
views
Fourier transform of the hyperboloid
Equip $\mathbb{R}^{d+1}$ with the Lorentzian form $\langle x, y\rangle=-x^0y^0+{\bf x}\cdot{\bf y}$ where $x=(x^0,{\bf x})$ and $\cdot$ is the usual Euclidean dot product. We define the hyperboloid $\...
- 439
2
votes
0
answers
115
views
Application of Bishop-Phelps Theorem
Consider a real Banach space $X$ and its continuous dual $X^*$. The Bishop-Phelps Theorem states that the set
$$A^*=\{x^*\in X^* \mid x^* \text{ attains its supremum on } \text{ the unit ball } \...
- 193
2
votes
0
answers
40
views
Properties of infinite matrices of the form $A= \left ( \int_0^1 f(x)(x+k)^n \, dt \right )_{k,n \in \mathbb N_0}$
Let $f \in L^1(\mathbb{R})$ be an integrable function which does not vanish identically and let $n,k \in \mathbb N_0$. I'm working on a problem where an infinite matrix of the following form appears:
$...
- 173
2
votes
0
answers
170
views
Equivalence of implicit function theorem and Peano existence theorem in ODEs?
I was recently reading a book about the implicit function theorem (IFT): The implicit function theorem: history, theory, and applications, and before that I learned that Peano's existence theorem can ...
- 181
2
votes
0
answers
122
views
Comparing the truncated $\ell^{1}$-norm of polynomial coefficients with the supremum norm on the unit disc
Let $p=a_{0}+a_{1}z+\ldots+a_{n}z^{n}$ be a polynomial. Consider the following truncated $\ell^{1}$-seminorm of the coefficients of $p$:
$$\|p\|_{\ell^{1},\text{trun.}}:=\sum_{k=1}^{n}|a_{k}|=\|p-a_{0}...
- 321
2
votes
0
answers
76
views
Fractional integration in Orlicz spaces
I am reading the paper "Fractional integration in Orlicz spaces" by R. Sharpley.
And I would like to understand one question:
Let $A,B, C$ are Young's functions. The spaces $L_A, L_B$ are ...
- 343
2
votes
0
answers
112
views
Is there a way to detect instantaneous states of a Feller Process from its infinitesimal generator?
I’m working with generators of Feller processes. If $C(E)$ is the space of continuous functions over $E$; with $E$ a compact metric space, I proved that an operator $G$ over $C(E)$ is the ...
2
votes
0
answers
216
views
Fourier transform of Dirac delta distribution
Let $f,g$ be Schwartz functions on $\mathbb R^4$, we denote them as $\mathcal S(\mathbb R^4)$, one can then define the transform $V$ mapping $f,g$ to a Schwartz function $\mathcal S(\mathbb R^8)$
$$ V(...
- 73
2
votes
0
answers
74
views
Distributions taking values in a TVS which is not locally convex
It seems to me that Schwartz's two seminal papers on vector-valued distributions only deals with distributions taking values in a locally convex Hausdorff topological vector space (LCS). Most other ...
- 21
2
votes
0
answers
85
views
Pullbacks of LCS-valued distributions
Suppose $X$ is a locally convex space. Since the distributions $\mathcal{D}'\!(M)$ ($M$ a manifold) are a nuclear space, there is a canonical meaning to the topological tensor product $X\,\widehat{\...
- 439
2
votes
0
answers
53
views
Orthonormal eigenspinors of the gauge-covariant dirac operator on $\mathbb{R}^4$, with extra conditions are possible?
Let $G$ be a simple Lie group, and $V$ a representation.
Consider $\mathbb{R}^4$ with its flat Euclidean metric. Let $P$ be the trivial $G$-bundle on $\mathbb R^4$ equipped with some (non-trivial) ...
- 3,477
2
votes
0
answers
268
views
Compactness of a nonlinear operator
Let $H^{1}_{0}(0;\pi)=\{f\in L^{2}(0; \pi): f^{\prime}\in L^{2}(0; \pi)\ \text{and}\ f(0)=f(\pi)=0 \} .$ equipped with the following norm $$\|f\|=\Big(\int_{0}^{\pi}|f'(x)|^2dx \Big)^{\frac{1}{2}}$$
...
- 39
2
votes
0
answers
140
views
Extracting block sequences from $\ell_p^n$-like sequences in isomorphic copies of $\ell_p$
One of the characterizing properties of the canonical bases of the $\ell_p/c_0$ spaces is their perfect homogeneity (Zippin), so that every normalized block sequence in these spaces behaves like the ...
- 81
2
votes
0
answers
117
views
Bounding integral expression with BV norm of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\...
user139844
2
votes
0
answers
292
views
Covering/Bracketing number of monotone functions on $\mathbb{R}$ with uniformly bounded derivatives
I am interested in the $\| \cdot \|_{\infty}$-norm bracketing number or covering number of some collection of distribution functions on $\mathbb{R}$.
Let $\mathcal{F}$ consist of all distribution ...
- 93
2
votes
0
answers
81
views
An square root of the multiplicative operator on $\ell^1(\mathbb{Z}_n)$
Let us consider the finite group algebra $\ell^1(\mathbb{Z}_n)$. Let $x=(x_0,\cdots,x_{n-1})$ in $\ell^1(\mathbb{Z}_n)$ and define
$$M_x: \ell^1(\mathbb{Z}_n)\to \ell^1(\mathbb{Z}_n) : M_x(a)=a*x$$
...
- 4,058
2
votes
0
answers
39
views
Example when Lorentz-Shimogaki condition satisfied with a specific Young's function
Let $\Psi(t)=\int_0^t\psi(s)$ be a Young's function. Then, $\Psi$ satisfies the Lorentz-Shimogaki condition if
$$
\int_0^{\infty}\frac{\Psi(st)}{v(t)^2}\psi(t)dt< \infty.
$$
Denote $\rho_{\Psi}=\...
- 343
2
votes
0
answers
75
views
Dual space induced by a finer topology
Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two seminorms on a space $E$ such that $\|\cdot\|_2\geq\|\cdot\|_1$. Let further $E_i:=(E,\|\cdot\|_i)$ and
$$C_b(E_i):=\{f : E\rightarrow\mathbb{R}\mid f \ \...
- 463
2
votes
0
answers
155
views
Second differential of total variation
I am trying to give meaning to the notion of second differential of total variation.
For sufficiently regular $u:\Omega \subset \mathbb{R}^2 \to \mathbb{R}$ let the total variation be given by
$$TV(u)=...
- 319
2
votes
0
answers
94
views
Existing work on $\Delta u=c-h e^{u}$ on compact manifold with dimension n, I have read J.Kazdan's work, the condition $c > 0, n \ge 3$ is not solved
I'm reading Prof. Kazdan's lectures
At page 69, Prof. Kazdan describes the research on the $\Delta u=c-h e^{u}$ PDE on a compact $n$-dimensional manifold before 1983. (Here $c$ is a constant while $h$ ...
- 809
2
votes
0
answers
104
views
Relations between different "propagation of chaos" type results?
My questions come from the paper Logarithmic Sobolev inequalities for some
nonlinear PDE’s written by F. Malrieu (May 2001). The basic set-up is that we have a $N$-particle system $(X^{i,N}_t)_{1\leq ...
- 730
2
votes
0
answers
206
views
Fourier transform of unbounded linear operator
I am trying to construct Fourier transform of a family of unbounded linear operators.
Here is the construction.
Fix $H$ a Hilbert space. Let $D\subset H$ be a fixed dense subset.
Denote by $L(H)$ some ...
- 523
2
votes
0
answers
150
views
Reference for weighted Sobolev spaces
I'm looking for a comprehensive reference illustrating, from the ground up, the basics of weighted Sobolev Spaces on Lipschitz domains (this case should be included, but I don't need less than it). ...
- 235
2
votes
0
answers
131
views
Eigenvalues of Witten Laplacian induced by log-concave probability measure on manifold
Let $M$ be a closed $n$-dimensional Riemannian manifold and let $\mu=e^{-V}d\mathrm{vol}_M$ be a log-concave probability measure on $M$, such that the pair $(M,\mu)$ verifies the so-called Bakry-Emery ...
- 6,853
2
votes
0
answers
173
views
Product of Heavisides: calculus vs Fourier transform vs wavefront set
I decided to ask this question here, since I did not get any answer from MSE and perhaps this topic is somewhat far from MSE's topics. I am following the paper here. I am trying to understand how to ...
2
votes
0
answers
196
views
Gauge invariance of a QFT path integral
If we consider the usual formal construction of a path integral over fields with gauge symmetries e.g as in Weinbergs "The Quantum Theory of Fields - Volume 2" the notion of gauge invariance ...
- 661
2
votes
0
answers
161
views
The Laplace transform and the Lagrange compositional inversion formula
I'm looking for references which derive the Lagrange inversion formula, given below (in bold), for the Taylor series coefficients of the compositional inverse of a function $f$ analytic at the origin ...
- 10.5k
2
votes
0
answers
68
views
What about the structure theory in Baer *-rings?
In the literature, Baer *-rings are called as the algebraic analogue of von Neumann alegars.
It is well-known that
Theorem. Every von-Neumann algebra is decomposed into a direct sum of the algebras of ...
- 4,058
2
votes
0
answers
130
views
Smoothness of Radon transform
Let $f:\mathbb R^n \to \mathbb R$ be density function (i.e nonnegative function which integrates to $1$), and consider its Radon transform $R[f]$ defined by
$$
R[f](w,b) := \int_{\mathbb R^n}\delta(x^\...
- 6,853
2
votes
0
answers
70
views
A distribution $\pi \propto \exp(-f)$ satisfies log-Sobolev inequality, does $\exp(-af)$ also satisfy LSI?
Assume a distribution $\pi \propto e^{-f}$ satisfies log-Sobolev inequality (LSI)
$$\forall \rho \in P(\mathbb{R}^n), \quad KL(\rho\| \pi) \le \frac{1}{2\lambda} I(\rho \| \pi)$$
with LSI constant $\...
- 21
2
votes
0
answers
57
views
spilt the sum of singular values of matrices
Let $A_{i} \in GL(d, \mathbb{R})$ for $i=1, 2, 3.$ For $q>0$, we denote $t_{3}^{q}=\sum_{i=0}^{3} \sigma_{1}^{q}(A_{i})\sigma_{2}^{q}(A_{i})\sigma_{3}^{q}(A_{i})$, $t_{2}^{q}=\sum_{i=0}^{3} \sigma_{...
- 1,043
2
votes
0
answers
119
views
What is the justification for using Wiener integrals to integrate over a space of differentiable functions?
In the literature on stiff/semiflexible polymer chains modelled as continuous chains rather than as discrete links, the partition function (among other things) is taken to be an integral over the ...
- 33
2
votes
0
answers
191
views
Approximation of zonoids
I have a question regarding the papers on approximating zonoids with zonotopes. I'll first write down the approximation problem and then state what my question is.
Problem of Approximating Zonoids. ...
- 640