All Questions
10,935 questions
1
vote
1
answer
249
views
Sufficient condition such that the set of zeros of an analytic function $f:\mathbb{R}^n \to \mathbb{R}$ contains only isolated points
Consider a real- analytic function $f: \mathbb{R}^n \to \mathbb{R}$. We know that zeros of $f$, roughly speaking, live in the low dimensional manifolds.
My question: Does a 'reasonable' sufficient ...
- 671
1
vote
2
answers
271
views
An integral inequality?
Let $v \in C^\infty(\mathbb R)$ such that $1 \ge v \ge 0$ and $\int_{\mathbb R} v \, dx = 1$.
I want to show that if
$$\int_{\mathbb R} v |v''|^2 \, dx < + \infty. \tag{$\star$}$$
then
$$ \int_{\...
- 83
5
votes
2
answers
342
views
Projections in atomless von Neumann algebras
Let $\varepsilon>0$.
If we consider a sequence $\{f_n\}$ in $L_\infty(0,1)$, then there exists a very small subset $A$ of $(0,1)$ with $m(A)<\varepsilon$ such that $$\|f_n \chi_A\|_\infty =\|...
- 617
2
votes
2
answers
329
views
$L^1$ norm for a product of cosines
Let $k$ be an integer and consider the function
$$
f(t)=\prod_{i=1}^{k} \cos(3^{i-1}\pi t).
$$
I'm interested in finding bounds for $\int_{0}^{1}|f(t)|dt$ in terms of $k$. The first idea that comes to ...
- 178
3
votes
2
answers
429
views
Functional equations based on composition
I have asked this question here (*), but there are no answer.
Let $n \in \mathbb N^*$, $\{a_0,\ldots,a_n\} \subset \left] 0,+\infty\right]$. We suppose $Eq : \sum\limits_{k=0}^n a_k f^k(x)=0$ have no ...
- 4,074
1
vote
1
answer
142
views
Complemented C*-algebras
Let $A$ and $B$ be unital separable commutative $C^*$ algebras, with $A\subset B$. Is it true that $A$ is complemented in $B$?
- 149
1
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0
answers
171
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Fast algorithm for computing certain signal transformations
Let $f,g,h:\mathbb Z\to\mathbb C$ supported on $[-n,n]$. For $\tau\in \mathbb Z$, let $\operatorname{sh}_\tau f$ be the shift of $f$ by $\tau$ (i.e. $(\operatorname{sh}_\tau f)(t) = f(t-\tau)$). ...
- 2,649
3
votes
0
answers
132
views
Is the Schwartz space a tame Frechet space?
I ran into the following definition of tame Frechet spaces and Nash-Moser therem.
It says that the space of smooth functions on a compact manifold is tame Frechet.
However, I wonder if
The Schwartz ...
- 3,477
1
vote
1
answer
113
views
An integrable estimate of the Hölder constant of the map $x \mapsto \int_{\mathbb R^d} f(y) \partial_1 \partial_1 g_t (x-y) \, \mathrm d y$
Let $(g_t)_{t>0}$ be the Gaussian heat kernel on $\mathbb R^d$, i.e.,
$$
g_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, x \in \mathbb R^d.
$$
Let $f : \mathbb R^d \to \...
- 825
0
votes
0
answers
44
views
Integral of gradient of a function times a vector fields, null whatever the function, implies null divergence and tangential limits conditions
I'm reposting this question from math.stackexchange, as I haven't got answers so far.
At the beginning of "Brenier, Y. (1987) Décomposition polaire et réarrangement monotone des champs de
...
- 101
4
votes
2
answers
904
views
Does every Banach space admit a continuous (not necessarily equivalent) strictly convex norm?
Trying to find and answer to this question, I have encountered two more-studied problems.
The first is to find when a Banach space admits an equivalent uniformly convex norm. The answer is that for ...
- 1,955
14
votes
5
answers
1k
views
Emergence of the discrete from the continuum
An almost eternal theme in Mathematics is the approximation of the Continuum by the Discrete. This core idea goes back at least to Archimedes, and remains active to these very days (and quite likely ...
- 7,853
0
votes
0
answers
205
views
Relationship between Fourier inversion theorem and convergence of "nested" Fourier series representations of $f(x)$
$\DeclareMathOperator\erf{erf}\DeclareMathOperator\sech{sech}\DeclareMathOperator\sgn{sgn}\DeclareMathOperator\sinc{sinc}$This is a cross-post of a question I posted on MSE a couple of weeks ago which ...
- 1,126
3
votes
0
answers
105
views
Finitely generated Banach lattice $C(K)$ and partitions of unity
Let $E$ be a Banach lattice. A Banach sublattice $L$ of $E$ is called finitely generated if there exists a finite subset $F \subseteq E$ such that
$$L = \bigcap \{ \hat{L} \mid F \subseteq \hat L, \, \...
- 467
16
votes
6
answers
2k
views
Finding closed subspaces whose sum isn't closed
Let $V_0$ be a closed infinite-dimensional subspace of a Banach space $V$ such that the quotient $V/V_0$ is also infinite-dimensional. Is it always possible to find a closed subspace of $V$ whose sum ...
- 42.8k
0
votes
0
answers
53
views
A problem about how to understand the existence of derivative of level set in Mountain-pass theorem
I'm confused about the Mountain pass theorem in Lemma4 of here.
Background :
$$
\begin{gathered}
I_\lambda(u)=\frac{1}{2} \int_M\left|\Delta_g^{\frac{m}{2}} u\right|^2 d \mu_g-\frac{\lambda}{2 m} \log ...
- 809
0
votes
0
answers
70
views
Multiplication with dilations of nonzero measurable function is injective
Denote $f_s(x):=f(sx)$ as the dilation of a function $f$. I want to know whether the following statement is true:
Suppose $f$ and $g$ are measurable functions on $\mathbb{R}$, and $f$ is not almost ...
- 807
0
votes
0
answers
40
views
Iterating partially-unconstrained optimization with projection
Let $f:H\to \mathbb{R}$ be a strictly convex Fréchet differentiable, coercive function on a separable Hilbert space $H$ and let $C_1,C_2\subseteq H$ be closed and convex.
I want to optimize
$$
\tag{(A)...
- 5,405
4
votes
2
answers
228
views
(Reference request) higher order Hölder spaces on riemannian manifolds
I am looking for a reference regarding the higher (than the first) order Hölder spaces on Riemannian manifolds. I am aware that defining Hölder spaces of form $C^{0,\alpha}$ is not an issue even ...
- 820
5
votes
1
answer
542
views
If $f$ is bounded, decays fast enough at infinity and $\int f=0$, does this imply that $f$ is in the Hardy space $\mathcal H^1(\mathbb R^n)$?
Let $\mathcal H^1(\mathbb R^n)$ be the real Hardy space (as in Stein's "Harmonic Analysis", Chapter 3). It is well known that $\mathcal H^1(\mathbb R^n)\subset L^1(\mathbb R^n)$ and its ...
- 1,075
2
votes
1
answer
231
views
Fourier coefficients of the logarithm of a given function
Let $f$ be a $1$-periodic real function that I know is bounded away from zero:
$$
f(x) = \sum_{n = -\infty}^\infty c_n e^{2\pi i n x}
$$
Let me also assume that $f$ is analytic with Fourier ...
- 1,263
5
votes
2
answers
625
views
Reconstruction of second-order elliptic operator from spectrum
Let $M$ be a compact smooth manifold, $(\lambda_n)_{n=1}^{\infty}$ be a square-summable monotonically increasing sequence of non-negative numbers, and let $(f_k)_{k=1}^{\infty}$ be continuous ...
- 364
3
votes
0
answers
153
views
Quasimode construction on a compact Riemannian manifold
Let $M$ be a closed Riemannian manifold, $\Delta$ be the usual Laplace-Betrami operator on $M$ and $\gamma : [0, L] \to M$ be a stable elliptic periodic geodesic of length $L$. I have heard in several ...
- 131
0
votes
0
answers
36
views
Sufficient condition for interpolation
If we have a couple of two compatible banach spaces (in this sense) $(X,Y)$ and a sequence of Banach spaces $\{Z\}_{\theta\in[0,1]}$ which are intermediate between $X$ and $Y$ satisfying:
$Z_0=X$, $...
- 101
2
votes
1
answer
223
views
Sobolev regularity via Laplace spectrum
Fix a positive integer $n$ and let $\mu$ be the uniform measure on the sphere $\mathbb{S}^n$, with respect to its usual Riemannian metric $g$. Let $\nabla$ be the Laplacian on $(\mathbb{S}^n,g)$ and ...
- 5,405
1
vote
1
answer
150
views
Unable to understand an application of Minkowski's inequality
Consider the following exerpt from the paper "Non-linear Quantum Processes" by Segal:
with the norm $\|F\|=\left(\int\|F(x)\|^p \, d x\right)^{1 / p}$, then the operator $T_1^{\prime}: F \...
- 90
0
votes
0
answers
60
views
The size of super level sets and the symmetry on a sphere
Let $u$ be a smooth function defined on the sphere $\mathbb{S}^2$, and let $R \in \mathrm{SO}(3)$ be a three-dimensional rotation. Define
$$
S_R = \{x \in \mathbb{S}^2 : u(x) \neq u(Rx)\}.
$$
Suppose ...
- 434
5
votes
0
answers
94
views
When a compact subset of a TVS can be continuously projected on a closed linear subspace?
Let $V$ be a (Hausdorff) topological vector space, $W\subset V$ a closed linear subspace, $X\subset V $ a compact.
(Q):
When there is a continuous map $P:X\to W$ such that $P(x)=x$ for every $x\in X\...
- 60.6k
3
votes
1
answer
120
views
Spectra of products variously permutated
Let $x,y$ be elements of a Banach algebra $A$ and $\lambda\in\mathbb C\setminus\{0\}$. If $\lambda-xy $ is invertible, then $\frac1{\lambda}\big[1+y(\lambda-xy)^{-1}x \big]$ is clearly an inverse of $...
- 60.6k
0
votes
0
answers
64
views
When is a symmetric block Toeplitz matrix invertible?
Let
$$
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
...
4
votes
0
answers
116
views
Weakly null sequences in projective tensor products II
The question in this post is the question below from an article by Rodriguez & Rueda Zoca [1].
Below is a complimentary salad/side dish that accompanies the main course.
Let $B^2(X,Y)$ denote ...
- 2,605
0
votes
0
answers
126
views
Building representation of an arbitrary umbral calculus
Consider a set of integrable functions on the interval $(0,1)$.
Let's introduce an operation $\operatorname{eval}f=\int_0^1 f(x)\,dx$ (which is the mean value of the function).
In such system the ...
- 10.1k
2
votes
0
answers
60
views
Basis vectors using anti-commuting operators?
Let $V$ be a finite-dimensional inner product space. Suppose $A_{1},...,A_{N}$ are anti-commuting operators, meaning that these are linear operators on $V$ that satisfy:
$$A_{i}A_{j}+A_{j}A_{i} = A_{i}...
- 1,305
0
votes
0
answers
44
views
Are there probability densities $\rho, f_n$ such that $\lim_n \frac{[\rho * f_n]_\alpha}{\|\rho * f_n\|_\infty} = \infty$?
We fix $\alpha \in (0, 1)$. Let $[f]_\alpha$ be the best $\alpha$-Hölder constant of $f: \mathbb R^d \to \mathbb R^k \otimes \mathbb R^m$, i.e., $[f]_\alpha := \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x-...
- 825
2
votes
1
answer
264
views
Continuous path of unitary matrices with prescribed first column?
Consider a continuous curve $u \colon [0,1] \to \mathbb{C}^n$ where $u(t)$ is always a unit vector, $u(t)^* u(t) = 1$.
Question 1: Does there exist a continuous curve $U \colon [0,1] \to \mathbb{C}^{n ...
- 637
3
votes
1
answer
243
views
Can a non-free Whitehead group embed as a discrete subgroup of a normed space?
Every countable discrete subgroup of a normed space is isomorphic to the direct sum of the group of integers. I wonder whether it is possible to push this beyond such direct-sum (free abelian) groups ...
- 11.3k
2
votes
0
answers
193
views
If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dotsb+ B_n'$ also satisfies the same inequality
Related: On a deceptively tricky calculus problem.
The way that Leonard Gross proves the log Sobolev inequality is in the following stages:
He proves that for any operator $B$ that satisfies the log ...
- 90
5
votes
1
answer
886
views
Showing that decay results of Fourier coefficients are sharp
I originally post the question here but the it seems too advanced for undergraduate level. At least, an example is very hard to find, so I am putting it here.
It is well-known that if $f$ is ...
- 1,755
2
votes
0
answers
111
views
Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}$
Previously, I asked the same question for a non-linear PDE, but I have got no answer. Below, I consider the linear counterpart it.
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb ...
- 825
2
votes
2
answers
290
views
Making sense of the limit $\lim\limits_{x \to y} T(x,y) $ for a tempered distribution $T$ on $\mathbb{R}^{2n}$
I already posted a similar question on MO and looked into the references therein.
However, I cannot find a satisfactory answer for my question..So I ask here again in a more refined form.
Let $T \in \...
- 3,477
3
votes
0
answers
132
views
Takesaki's duality in representation theory of $C^*$-algebras
In M.Takesaki's 1967 article titled A Duality in the Representation Theory of C-Algebras*, admissible operator fields are defined in order to generalize Gelfand transform to a non-abelian setting.
...
- 139
2
votes
0
answers
30
views
Dual of homogeneous Triebel-Lizorkin
Let $ p, q \in (1,\infty)$ and consider the homogeneous Triebel- Lizorkin space $\dot{F}^{s}_{p,q}$ to be the space of all tempered distributions (modulo polynomials) with
$$
[f]^{p}_{\dot{F}^{s}_{p,q}...
- 31
0
votes
0
answers
34
views
Support of a function acting on an algebra?
Quick: for a measurable function $f$ its support on Euclidean space is clearly just the subset where $f$ does not vanish.
Now, let’s have $f$ acting on an finite Lie algebra, f.e. $\mathfrak{gl}$ as $...
2
votes
1
answer
184
views
Prove if the fractional Laplacian of a function is bounded
Take $s\in (0, 1)$. I am trying to understand if $(-\Delta)^s (\log(1+x^2))$ is bounded, that is if there exists $R>0$ such that $|(-\Delta)^s (\log(1+x^2))|\le R$.
Here $(-\Delta)^s$ is the ...
- 123
1
vote
1
answer
109
views
Solution to $a=e^t (t-r_1)(t-r_2)$ with Lambert $W$ function, where $r_1, r_2 $ are complex
Lambert $W$ works when $r_1$, and $r_2$ are real. However, I am trying to solve the equation when $r_1$, and $r_2$ are complex numbers.
1
vote
0
answers
94
views
positive invertible maps which are not *-automorphisms
Let $A$ be a unital C*-algebra. Is there a unital positive self-map $F:A\to A$ which is invertible (i.e. injective and surjective) but not a $*$-automorphism? If yes, how does appear its Gelfand-...
- 41
5
votes
1
answer
594
views
Exponential sum vs. exponential integral via Poisson summation
When we want to estimate an exponential sum
$$
\sum_{M<m\le M'}e(f(m))
\quad\text{with}\quad
1\le M\le M'\le 2M
\quad\text{and}\quad
e(x):=\exp(2\pi ix)
$$
where $e(x):=\exp(2\pi ix)$
and the phase ...
- 363
3
votes
1
answer
163
views
Operator Semigroup: Resolvent estimates and stabilization, a detail in the paper of Nicoulas Burq and Patrick Gerard
In Appendix A of the paper Stabilization of wave equations on the torus with rough dampings https://msp.org/paa/2020/2-3/p04.xhtml or https://arxiv.org/abs/1801.00983 by Nicoulas Burq and Patrick ...
3
votes
0
answers
84
views
About the naturality of Krasnoselskii genus on Variational Methods
I have recently watched a seminar about Variational Methods from Mónica Clapp and she gave a very interesting motivation of why the Lusternik–Schnirelmann category (click on the link for the ...
- 131
0
votes
0
answers
49
views
On the $L^p$ estimate and Weyl's law of Eigenfunctions in Sogge's Book
I have recently started to study the book "Fourier integrals in classical analysis " by Sogge mainly oscillatory integral decay methods. I have a question from the chapters 4 and 5. Mainly ...
- 11