All Questions
10,936 questions
1
vote
2
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590
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Inequality between coefficients of a polynomial and its supremum
For $d, m \in\mathbb{N}$ fixed, let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$. (That is, the above sum ranges over all ...
- 463
1
vote
0
answers
82
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Commutator of self-adjoint operators and $C^1$-type formula
Let $\mathcal{H}$ be a (complex) Hilbert space.
Let $H$ be a self-adjoint operator on $\mathcal{H}$ with dense domain $\mathcal{D}(H) \subset \mathcal{H}$, generating the unitary one-parameter ...
- 71
4
votes
1
answer
379
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A constant ratio of integrals? Part I
Let $u(x)$ be a harmonic polynomial in the unit ball $B_1(0)\subset\mathbb{R}^n$ with $u(0)=0$.
For $0<r\leq1$, consider the average of its Dirichlet integral
$$A(r):=\frac1{\vert B_r(0)\vert}\int_{...
- 43.2k
1
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0
answers
165
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Question about stationary phase with Hessian close to $0$
Let $\phi$ be a smooth real function in one variable and say $w$ is a smooth function with compact support say $[- 1, 1]$. Let me define
$$
I_{\lambda} = \int_{\mathbb{R}} w(t) e^{i \lambda \phi(t)} ...
- 3,625
1
vote
1
answer
189
views
Complemented subspaces in a dual Banach space
Let $Y$ be a complemented subspace in a dual Banach space $X$. Is it true that $Y$ is itself isomorphic to a dual?
This is the case of a $w^*$-closed subspace $Y$, but a complemented subspace of $X^*...
- 60.6k
2
votes
1
answer
117
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Vertical Fourier decomposition for skew-Hermitian 1-forms
In an arXiv preprint [2108.05125v1], the authors use the following vertical Fourier decomposition (page 7 therein).
Let $(M,g)$ be a Riemannian surface and $SM$ be its unit tangent bundle. Denote by $...
- 257
1
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1
answer
161
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Spectral theorem for unital $C^{*}$-algebras
Let $A$ be a unital $C^{*}$-algebra and $a \in A$ be normal, with spectrum $\sigma(a)$. Let $B = C^{*}(a)$ be the $C^{*}$-algebra generated by $1$ and $a$, which is abelian. Let $\hat{B}$ be the space ...
- 1,305
6
votes
0
answers
217
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Detailed examples of induction on scale
I'm trying to understand the induction on scale argument in harmonic analysis. On this abstract it's mentioned that induction on scale can be used to prove Cauchy Schwartz inequality, Beckner's tight ...
- 265
5
votes
1
answer
151
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On existence of a concave function
Let $a$ be a strictly positive $C^\infty$ smooth function on the unit interval. Does there exist a strictly positive $C^\infty$ smooth function $f$ on $I$ such that
$$ f’’(x) \leq 0\quad \text{and} \...
- 4,115
7
votes
3
answers
521
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Count of non-trivial ergodic measures of a topological dynamical system
Given a compact Hausdorff space $X$ and a continuous mapping $\varphi: X \to X$. We denote by $C(X)$ the space of continuous functions $f: X \to \mathbb{C}$. A probability measure $\mu$ on the Borel-$\...
- 467
1
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1
answer
217
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Uniqueness of Fourier–Stieltjes transform for finite complex valued measures
Let $\mu$ be a finite complex valued measure on $\mathbb{R}$ and let $\hat{\mu}$ be it's Fourier–Stieltjes transform
$$
\hat{\mu}(\omega)= \int_{\mathbb{R}} e^{it\omega} d \mu(t)
$$
Question: Does $\...
- 671
4
votes
1
answer
663
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The decay of Fourier coefficients and the continuity of functions
Let $ f $ be a function on $ \mathbb{T}=[0,1] $ ($ 1 $-periodic) with bounded variation. Prove that if $ \widehat{f}(k)=\int_0^1f(x)e^{-2\pi ikx}dx=o(1/|k|) $, then $ f\in C(\mathbb{T}) $. I do not ...
- 1,219
1
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0
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105
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Friedrichs extension of the Laplacian from a smooth subspace and density of its eigenbasis in the Frechet topology of the subspace as well?
Let $C^\infty_\text{div}(\mathbb{T}^3)$ be the "real" Frechet space of periodic, divergence-free smooth vector fields on $\mathbb{R}^3$. That is, $\mathbb{T}^3$ is the $3$-dimensional torus.
...
- 3,477
5
votes
1
answer
487
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Fokker-Planck: equivalence between linear spectral gap and nonlinear displacement convexity?
In a smooth, bounded and convex domain $\Omega\subset \mathbb R^d$ consider the usual linear Fokker-Planck equation with Neumann (some would say Robin) boundary conditions
\begin{equation}
\label{FP}
\...
- 5,380
3
votes
1
answer
451
views
Image of a quadratic form is a closed cone
Let $Q : E \to F$ be a quadratic form induced by a symmetric bilinear form $B : E \times E \to F$ defined in a finite dimensional real normed vector space $E$, with values in the normed vector space $...
- 2,095
3
votes
1
answer
425
views
Fourier series of $e^{(\cos(\pi x) - m)^2}$
I'm looking for the Fourier coefficient of a "periodic Gaussian", which writes
$$
f(x) = e^{-\frac{1}{2s}(\cos(\pi x) - m)^2}
$$
It is a real even 2-periodic function, so its Fourier ...
- 31
2
votes
1
answer
176
views
Is the measure density condition a necessary condition for bounding the Sobolev norm $W^{n,p}(\Omega)$ by the extremal terms?
Let $\Omega \subseteq \mathbb{R}^M$ be a measurable subset of positive measure. R. A. Adams and J. Fournier in their article have proven that if $\Omega$ satisfies the so-called weak cone property, ...
- 820
0
votes
0
answers
127
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Closure of BV paths in space of paths of finite $p$-variation
Let $p\ge1$ and $T>0$. Define $\mathscr D([0,T])$to be the space of partitions of $[0,T]$, where each partition is a finite collection of distinct points of $[0,T]$. Consider a continuous path $X:[...
- 177
7
votes
1
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290
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Square-root lattices: where do they appear?
As an experimental physicist working on crystallography I'm often dealing with the reconstruction of an object from intensity data that emerge from an imaging device. In mathematics the problem is ...
user491917
-1
votes
1
answer
213
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Building a smooth function from a rapidly decreasing sequence
Is it possible to build a 1-priodic smooth function from a rapidly decreasing sequence such that the sequence be the Fourier coefficients of the function?
More precisely:
Let $\lbrace c_k\rbrace _{k \...
- 581
2
votes
1
answer
241
views
Strategy of the proof of the "minimal entropy condition" for scalar conservation laws
Combining Theorem 2.3 and Corollary 2.5 of this paper gives that, for a strictly convex conservation law
$$u_t + f(u)_x = 0,$$ satisfying the entropy condition
$$\eta(u)_t + q(u)_x \le 0$$ in the ...
user139844
1
vote
0
answers
273
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Characterization of the Hamiltonian's spectrum in quantum mechanics
This is basically the same question I made on physics stack exchange The spectrum of the Hamiltonian in quantum mechanics, but I got no answers so far and decided to move it to mathoverflow with some ...
- 1,305
1
vote
1
answer
150
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Show that a certain convergence of measures is equivalent to a certain convergence of integrals
Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of a measures. We know, by the Portmanteau Theorem, that:
$$\int f d \mu_n \to \int f d\mu, \quad \forall \, f \in C_b \hbox{(class of continuous and ...
- 13
10
votes
1
answer
349
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On a variant of Carlson’s theorem
My question is on whether or not there exists some monotone strictly decreasing sequence of positive numbers $c_1>c_2>\ldots$ such that given any $f$ which is a uniformly bounded holomorphic ...
- 4,115
10
votes
1
answer
368
views
Group of isometries of Banach spaces a topological group?
Let $X$ be a Banach space and let $\mathrm{Iso}(X)$ be its group of isometries, i.e., the set of surjective linear maps $T: X \to X$ with $\|Tx\| = \|x\|$.
Q: Is $\mathrm{Iso}(X)$ a topological group ...
- 9,853
4
votes
1
answer
800
views
Bound the operator norm of the Fréchet derivative of a Lipschitz function in this setting
I want to find a bound for the operator norm of the Fréchet derivative of a Lipschitz continuous function in the following setting:
Let
$E$ be a $\mathbb R$-Banach space;
$v:E\to[1,\infty)$ be ...
- 167
3
votes
1
answer
251
views
Asymptotic behavior of a double oscillatory integral
Let $0<\theta_1,\theta_2<\pi/2$. Suppose $\psi$ is a smooth real-valued function with compact support.
Consider the oscillatory integral
$$I(t):=\int_{0}^{1}\frac{1}{(y-e^{\dot{\imath}\theta_1})
...
- 852
1
vote
1
answer
150
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eigenvalues of matrices (with positive entries)
I am reading an old paper by Kawpien and Pelczynski, Studia Math. 1970. It claims that singular values of a matrix (with positive entries? I am not sure) is given by $t_i=\sqrt{\sum_{j\ge 1}a(i,j)^2}$....
- 617
0
votes
0
answers
47
views
Is the embedding $i: (L^p_\text{loc} (Y), \| \cdot \|_{L^p_\text{loc}}) \to (L^0(Y), \hat \rho)$ continuous or Borel measurable?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...
- 657
6
votes
1
answer
491
views
Harmonic analysis for a beginner
I am currently dealing with discrete Fourier transform and correlation technique to construct the spectrum of a broad band signal. It's already known that if I have enough observations of the signal, ...
- 159
6
votes
2
answers
1k
views
In what precise sense is quantum (i.e., non-commutative) probability not expressable in terms of classical probability?
The quantum set-up has many settings, so let's fix some definitions. I will be taking the Hilbert space approach with a minor modification that I will make explicit.
We begin with a Hilbert space $\...
- 323
2
votes
1
answer
154
views
Function monotony between [0,T] and $L^2$
Let $\Omega\subseteq\mathbb{R}^N$ be a bounded and smooth domain. If $z:[0,T]\to L^2(\Omega)$ is a function in $H^1([0,T],L^2(\Omega))$ with the property that $z'(t)(x):=z'(t,x)>0$ a.e. on $\Omega$ ...
- 1,759
3
votes
2
answers
307
views
Smoothing a map $f:X\to \mathbb{R}$ while fixing it over a closed $C\subset X$
$\newcommand{\R}{\mathbb{R}}$I have a map $f\in C^0(X,\mathbb{R})$, where $X$ is a compact and Hausdorff topological space, which is a manifold outside of a compact subset $K\subset X$.
I would like ...
- 2,533
0
votes
0
answers
143
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Is $C^\infty(\Omega) \cap W^{s_1,p_1}(\Omega)$ dense in $W^{s_2,p_2}(\Omega)$ if $W^{s_1,p_1}(\Omega) \subset W^{s_2,p_2}(\Omega)$?
Background: The proof of Theorem 6.4 in http://mate.dm.uba.ar/~jrossi/Fractional-1-lapla-07_02_2015.pdf, I want to use the density that $C^\infty(\Omega) \cap W^{r_0,q_0}(\Omega) \cap L^2(\Omega)$ is ...
- 1
13
votes
1
answer
777
views
Large sieve inequality for sparse trigonometric polynomials
Let $S(\alpha) = \sum_{n\leq N}f(n) e^{2\pi i \alpha n}$ for some arithmetic function $f$. Suppose $\alpha_1, \ldots, \alpha_R$ are real numbers that are $\delta$-spaced modulo $1$, for some $0 < \...
- 253
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0
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119
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Weak convergence in $W^{ 1,q}(\Omega) $ $ (1 \leq q < \infty)$ space
Let $\Omega \subset \mathbb { R } ^ n $be a bounded domain, and suppose that in the space $W^{ 1,q}(\Omega) $ $ (1 \leq q < \infty)$, the sequence $\{ u_j \} $ converges weakly in $W^{ 1,q}(\Omega)...
- 471
1
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0
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220
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Reference for Schwartz kernel theorem on vector bundles
In this notes Linear Analysis on Manifolds by Chris Kottke at page 20 he has
Theorem $1.16$ (Schwartz kernel theorem, c.f. [Hör85] Thm. 5.2.1). Let $M$ and $N$ be a compact Riemannian manifolds with ...
3
votes
3
answers
1k
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Unbounded operators vs compact operators
The operator $L:\operatorname{dom}(L)\subset C[0,1]\to C[0,1]$ given by $Lx=x'$
a) is closed, unbounded and densely defined
b) also has a compact right inverse, namely $K:C[0,1] \to C[0,1]$ given by $...
- 31
0
votes
0
answers
78
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Sobolev trace inequality with exterior domains
Let $x_1\in \mathbb{R}^n$, $n\geq 3$, $\Omega=\mathbb{R}^n\backslash B_1(x_1)$, define $D_{\Omega}$ by taking the closure of $C_{c}^{\infty}(\overline{\Omega})$ under the norm
\begin{align*}
\|u\|_{D_{...
- 471
20
votes
1
answer
1k
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Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$
Can one show that Fourier transform of
$$ f_a(x) = a^{-2} \exp(-|x|^a), \qquad a \in (0,2)$$
is decreasing in $a$?
I have a solution for $a \in (0,1]$ which cannot be used for $a\in (1,2)$.
- 178
9
votes
0
answers
321
views
Best smoothing for the Prime Number Theorem?
There are plenty of proofs of the Prime Number Theorem with explicit error terms - it actually looks like a rather competitive field (see Remark 1.4 in https://arxiv.org/pdf/2204.02588.pdf). Several ...
- 20.2k
1
vote
1
answer
80
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Prove that $N_1(A_1,A_2)= N_2(A_1,A_2)$ when $A_1A_2=A_2A_1$ and $A_1$ and $A_2$ are normal operators
Let $\mathcal{L}(E)$ be the algebra of all bounded linear operators on a complex Hilbert space $E$.
On $\mathcal{L}(E)^2$, we have two equivalent norms:
\begin{eqnarray*}
N_1(A_1,A_2)
&=&\sup\...
- 1,154
7
votes
1
answer
253
views
Are all quasi-regular points on Polish spaces generic points?
Let $X$ be a Polish space and $T\colon X\to X$ be a continuous map. We say that a point $x\in X$ is quasi-regular if for every bounded continous function $\varphi\colon X\to\mathbb{R}$ the sequence $...
- 1,658
1
vote
1
answer
103
views
Are there PDEs in which Hessian appears in the weak formulation
Before stating the question, I would like to first use an example for the type of formulation that I'm interested in.
Suppose we consider the continuity equation $\partial_t \rho + \mathrm{div}( \rho ...
- 820
4
votes
1
answer
133
views
A $C^*$ algebraic analogy of the concept of complemented subspace in the particular case of $\ell^\infty$
Let $A$ be a $C^*$ algebra. A $C^*$ subalgebra $C\subset A$ is said to be $C^*$ algebraic complemented of $A$ if there exist a $C^*$ subalgebra $D\subset A$ with $A=C\oplus D$ and the obvios mapping $...
- 366
1
vote
1
answer
259
views
Are separable, continuous, monotonic and scale invariant real-valued functions everywhere differentiable?
Consider a function $f:\mathbb{R}_+^2\rightarrow\mathbb{R}$ of two non-negative real variables (or more generally of several real variables) that is increasing in each argument, continuous, additively ...
- 399
2
votes
1
answer
262
views
An observation of Nazarov on $\Lambda(p)$-systems
I have been reading this note of F. Nazarov "Summability of large powers of logarithm of classic lacunary series and its simplest consequences (1995)", available here https://users.math.msu....
- 21
1
vote
0
answers
106
views
Question on the existence of a certain decomposition method for real square matrices
I was working around with the decomposition of the multidimensional linear canonical transform (which is not even continuous w.r.t. the parameters) into a few fractional Fourier transforms (and other ...
- 286
5
votes
1
answer
631
views
Uniqueness of Kantorovich potentials?
$\newcommand{\R}{\mathbb R}$Take $\Omega\subset \R^d$ bounded, convex, and smooth.
Consider the optimal transport problem with cost $c(x,y)=\lvert x-y\rvert^2$, leading to the quadratic Wassersein ...
- 5,380
0
votes
2
answers
101
views
Minimal set of functions to characterize a distribution
In probability theory, there are a number of equivalent ways to characterize a distribution on $\mathbb R^n$. For example, the distribution of a random vector $X\in\mathbb R^n$ may be characterized by:...
- 109