All Questions
10,934 questions
6
votes
1
answer
331
views
If $t \to \lVert f(\cdot,t) \rVert_{L^2_x}^2$ is absolutely continuous, can we interchange the spatial integral and time derivative? (from MSE)
I originally posted this question on MSE. But it seems more nontrivial than expected, so I guess MO is a more appropriate place to ask.
I repeat the question for the sake of completeness:
Let $f(x,t) ...
- 3,477
1
vote
0
answers
63
views
Extension of meromorphic distribution
Let $W$ be a topological vector space (e.g. Frechet) with a dense subspace $V$. Let $D_s$ be a distribution on $V$ that is meromorphic in $s\in\mathbb C$ and extends continuously to $W$ with respect ...
- 3,799
0
votes
2
answers
167
views
Is a signed measure $\mu$ on $\mathbb{R}^d$ characterized by the transform $\mathcal{L}_\mu (\lambda ):=\int e^{\langle \lambda,x\rangle }\mu (dx)$?
In the book "Probability Theory" by Achim Klenke there's the following theorem: a finite measure $\mu$ on $[0,\infty )$ is characterized by its Laplace transform $\mathcal{L}_\mu(\lambda):=\...
- 15
4
votes
1
answer
170
views
When is a solution $P(f'(x)) = Q(f(x))$ periodic or double periodic?
$\newcommand{\cl}{\operatorname{cl}}\newcommand{\sl}{\operatorname{sl}}\newcommand{\cm}{\operatorname{cm}}\newcommand{\sm}{\operatorname{cm}}$Consider the differential equation
$$P(f '(x)) = Q(f(x))$$
...
- 763
9
votes
1
answer
639
views
Prove J.L. Lions’s Lemma without using Fourier transform
When I read the book Linear and Nonlinear Functional Analysis with Applications, I came across J.L. Lions's Lemma (the book doesn't give a proof), which states
Let $\Omega \subset \mathbb R^n$ be a ...
- 807
0
votes
3
answers
278
views
A generalisation of Tchebychev inequality
Let $f,g \in C(\mathbb R)$ with $\exists M \in \mathbb R^*, \forall (x,y) \in \mathbb R^2, M\times (f(x)-f(y))(g(x)-g(y)) \geq 0$.
Is it true that exists $ u$ any real function, and $a,b$ monotone ...
- 4,074
0
votes
1
answer
80
views
Orthogonal space of polynomials
Let $f \colon [0,+\infty) \to \mathbb R$ be a continuous function. Assume that for any non-negative integer $n$, the function $f(t) t^n$ in integrable in $(0,+\infty)$ and
$$
\int_0^{+\infty} f(t) t^n ...
1
vote
0
answers
105
views
Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$. Schwartz space is dense in $A$ wrt $\|f\|:= \|\hat{f}\|_1+\|\hat{f'}\|_1$?
Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$, where $\hat{f}$ is the Fourier transform of $f$. Then is it true that Schwartz space $\mathcal{S}(\mathbb{R})$ is dense in $A$ ...
- 29
2
votes
1
answer
126
views
Subspaces of $C_0$ on which $p$-norm are equivalent?
I have a question concerning the generalization of the following fact.
Let $E = C^0([0,1],\mathbb{R})$ endowed with the $\|.\|_\infty$ norm. One can show that if $F$ is a subspace of $E$ for which ...
- 125
-1
votes
1
answer
286
views
Check an equation on the Heisenberg group $H_1$
The Heisenberg group $H_1$ is the set $\mathbb C\times \mathbb R$ endowed with the group law
$$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right); \quad \forall z,w \in \mathbb C\,...
- 650
0
votes
1
answer
96
views
Extracting each field operator as Wightman fields from a set of time-ordered products satisfying Eckmann-Epstein axioms
The paper by Eckmann-Epstein proves that Schwinger functions at "coinciding points" uniquely defines "time-ordered products".
In physics, these "time-ordered products" ...
- 3,477
19
votes
3
answers
1k
views
What standard Banach space is isomorphic to the completion of this different normed structure on $\ell^1$?
A colleague asked me the following question:
"What can one do with the following norm on $\ell^1$: $|x|=\int_1^2 |x|_pdp$ where $| \;\; |_p$ is the standard norm on $\ell_p$?"
This ...
- 356
7
votes
0
answers
132
views
Relation between Fourier series and Schur polynomials
Asked initially at MSE.
I would like to know how to express the Fourier series of a symmetric function, $f(\theta_1,...,\theta_N)$, in terms of Schur polynomials $s_\lambda(x_1,...,x_N)$ in the ...
- 1,549
2
votes
0
answers
107
views
Finite dimensional manifolds as subspace of $\mathbb{R}^\mathbb{N}$
For embedded submanifold, specifically with ambient space being $\mathbb{R}^{n}$, there are many nice properties and results. Specifically there are many examples of matrix manifolds such as the ...
- 275
7
votes
2
answers
592
views
Prove that the following function is positive
Consider the following function:
$$K(x, y; t) = \sum_{n \geq 0} \frac{e^{-(2n+1)t}}{\sqrt{\pi} 2^n n!} H_n(x) H_n(y) \exp\left(-\frac{(x^2 + y^2)}{2}\right)
$$
This is Mehler's kernel, and can be ...
- 90
1
vote
1
answer
121
views
An asymptotic integral with complex phase
Suppose that $D\subset \mathbb R^2$ is the closed unit disk and that $f\in C^{\infty}(D)$. Assume that for all $\lambda \in (1,\infty)$ there holds
$$ \left|\int_D f(x^1,x^2)\, e^{\lambda (x^1+ix^2)}\,...
- 4,143
0
votes
0
answers
45
views
Mean value property for fractional laplacian
I just started reading about fractional Laplacian. I am curious on the following questions
Does fractional laplacian i.e., $(-\Delta)^su=0$ in $\mathbb{R}^n$ this equation satisfies any mean value ...
- 41
2
votes
0
answers
56
views
Convergence of conformal metrics with prescribed curvature
We know that for any function $K: \mathbb{D} \to \left[-a, -b\right]$, where $a, b > 0$, there is a unique metric $h$ on the disk $\mathbb{D}$ which is conformal to $dz^{2}$, and has curvature ...
- 63
4
votes
0
answers
262
views
Spectrum of ring in algebraic geometry vs spectrum of Banach algebra
For a commutative unital Banach algebra $A,$ and $x\in A,$ we have $\lambda \in \sigma_A(x)$ if and only if $\phi(x) = \lambda$ for some algebra homomorphism $\phi:A \to \mathbb C.$ The set of all ...
- 1,755
2
votes
0
answers
71
views
Complemented subspaces of $\mathcal s$
Crossposted from Math Stack Exchange
It is well known that a nuclear Fréchet space $X$ is isomorphic to a complemented subspace of $\mathcal s$ (the space of rapidly decreasing sequences) if and only ...
- 655
4
votes
0
answers
158
views
Measurability of $L^{p}(L^{q})$ integrable functions
Let $ F: \mathbb{R}^n \times (0,\infty) \to \mathbb{R}$ be a function with the property that
$
\int_{\mathbb{R}^n} \big[ \int_0^\infty |F(x,r) |^q \, dr \big]^{p/q} \, dx < \infty
$
In addition we ...
- 41
2
votes
1
answer
321
views
A strange functional inequality
Let $f,g \in C([-2,2],\mathbb R_+^*)$ even and concave real functions.
Is it true that
$$
\int_0^1 f\big(\cos(x^{-1})+\sin(x^{-1})\big) \cdot g\big(\cos(x^{-1})-\sin(x^{-1})\big) \mathrm{d}x\\ \leq f(...
- 4,074
2
votes
0
answers
97
views
On the second order analog of the upper 1-Lipschitz envelope of a function
Let $u: \mathbb R \to \mathbb R$ be a given function. Then we can consider its upper 1-Lip envelope
$$
\hat u(x) \doteq \inf\{g(x) \, \mid\, g \, \text{has Lipschitz constant 1 and}\, g(y) \geq u(y) \,...
- 355
69
votes
3
answers
12k
views
Nonconvexity and discretization
Edit: Here's a more down-to-earth, and somewhat weakened, but I believe still nontrivial, version of the main theorem.
Prototypical nonconvex spaces are $\ell^p$-spaces for $0<p<1$, say $\ell^p(\...
- 21.3k
6
votes
2
answers
290
views
If a Banach / Fréchet manifold $M$ happens to be a topological vector space, is $M$ just a Banach / Fréchet space?
In finite dimensions, if $M$ is a smooth manifold that happens to be a vector space, then it is indeed just the Euclidean space.
I wonder if the same result holds valid in infinite dimensions. More ...
- 3,477
11
votes
1
answer
309
views
Which closed subsets $Y$ of a compact space $X$ admit a linear extensor $C(Y)\to C(X)$?
In the following $X$ is a Hausdorff compact topological space. Let $Y$ be a closed subset of $X$.
The restriction operator $R_Y:C(X)\to C(Y)$ is surjective (Tietze), so it admits a continuous right ...
- 60.5k
3
votes
0
answers
79
views
Continuity of disintegrations in non locally compact spaces
Let $X$ and $Y$ be Radon spaces, $\mu$ a Borel probability measure on $X$, $F\colon X\to Y$ measurable. Then the disintegration theorem gives us a disintegration $\{\mu^y\}_{y\in Y}$ of $\mu$ with ...
- 63
2
votes
0
answers
137
views
Holder-Besov space and time continuity
Let $\mathbb{T}^d$ be the $d$-dimensional torus, $\mathscr{S}:=C^\infty(\mathbb{T}^d)$ the Schwartz space, $\mathscr{S}'$ the space of tempered distributions.
We consider a dyadic partition of unity $(...
- 573
1
vote
0
answers
63
views
$\operatorname{ker}(q_I \otimes^{\text{min}} q_J) $ is a primal ideal of $\mathcal{A} \otimes^{\text{min}} \mathcal{B}$
In the proof of Theorem $4.1$ of the paper titled continuous bundles of $C^{\ast}$-algebras and tensor products following result is mention with a reference to Proposition $3.3$ of the paper "A. ...
- 1,115
6
votes
2
answers
644
views
Explicit form of this unitary transformation
Disclaimer: This question has its motivation from physics. It is probably not entirely rigorous at the moment. I just want to clarify some steps and try to make the arguments rigorous afterwards, if ...
- 3,515
2
votes
1
answer
272
views
Decompose a function into a bounded part and a Lipschitz part
Let $f: \mathbb R^d \to \mathbb R^d$ be a measurable function such that
$$
\sup_{x,y \in \mathbb R^d} \frac{|f(x) - f(y)|}{\max \{1, |x-y| \}} < \infty.
$$
Are there functions $g,h: \mathbb R^d \...
- 825
7
votes
0
answers
294
views
Applications of Banach space homology
There is a well-developed theory of Banach space homology. What are some of its useful applications to Banach space theory and which important questions can one answer using it? In other words, how ...
- 175
4
votes
1
answer
96
views
Is there any example of linear operator which is bounded on all Besov spaces but not on Triebel-Lizorkin spaces
Is there any linear operator $T:S'(\mathbb R^n)\to S'(\mathbb R^n)$ such that $T:B_{pq}^s(\mathbb R^n)\to B_{pq}^s(\mathbb R^n)$ for all $0<p,q\le\infty$ and $s\in\mathbb R$, but there exist a $F_{...
- 938
0
votes
0
answers
89
views
Weakly compact set
I want to show that if the set
$$
\big\{u \in L^{q}([0, n] ; X): u(t) \in \phi(t, x(t)), t \in[0, n]\big\}
$$
is weakly compact, then the set
$$
\mathcal{S}_{\phi}(x)=\Big\{u\in L_{loc}^{q}(\mathbb{R}...
- 39
23
votes
5
answers
8k
views
Why do we have two theorems when one implies the other?
Why do we have two theorems one for the density of $C^{\infty}_c(\mathbb{R}^n)$ in $L^p(\mathbb{R}^n)$ and one for the density of $C^{\infty}_c(\Omega)$ in $L^p(\Omega)$? with $\Omega$ an open subset ...
0
votes
1
answer
114
views
Ball in separable Banach space has positive Gaussian measure
I have (presumably non-degenerate) Gaussian $\mu$ over separable Banach space $X$. I would like to prove that for any ball of radius $r$ centered at $x$, $\mu(B_r(x))$. I know how to prove this in ...
- 379
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, ...
- 65
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 ...
- 4,461
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
59
views
Gaussian Hypercontractivity of Chaos based on Gaussian with value in Hilbert spaces?
The classical Gaussian hypercontractivity is stated as following: Suppose $\xi$ is a Gaussian variable and $H_n(\xi)$ is the space of n-th homogeneous Wiener chaos constructed from $\xi$, then for any ...
- 253
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 \...
- 807
6
votes
1
answer
319
views
How are coordinate charts constructed in noncommutative geometry?
In noncommutative geometry, one is given a triple $(A,D,H)$, where $A$ is a commutative C* algebra, $H$ is a Hilbert space, and $D$ is an operator. There is a somewhat long list of conditions that ...
- 593
3
votes
0
answers
124
views
Estimating a solution to Euler-type ODE #2
This is a similar question to this but with a different ODE.
Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer, $R&...
- 969
13
votes
0
answers
573
views
Classical (i.e. commutative) spaces with quantum symmetry but no classical symmetry
In a recent preprint (arXiv:2311.04889), my coauthors and I constructed a sequence of graphs with no classical symmetry which nevertheless have quantum symmetry.
For graphs this had been an open ...
- 2,339
3
votes
1
answer
169
views
Can homomorphisms be Borel in some weaker topology?
Let $X$ be a Banach space with separable dual and let $A$ be a Banach algebra. Consider a norm continuous homomorphism $h$ from $L(X)$, the Banach algebra of bounded operators on $X$ onto $A$. In $L(X)...
- 31
2
votes
0
answers
151
views
Does the complex interpolation space $(L^1(\mathbb{R}),W^{2,1}(\mathbb{R}))_{\frac{1}{2}}$ continuously embed into $L^\infty(\mathbb{R})$?
The complex interpolation space between $(L^p(\mathbb{R}),W^{2,p}(\mathbb{R}))_\theta$ with interpolation parameter $\theta=\frac{1}{2}$ is known to be $W^{1,p}(\mathbb{R})$ for $1<p<\infty$. As ...
- 989
4
votes
1
answer
279
views
Schroedinger operator in 2 dimensions with singular potential
Consider the Schroedinger operator
$$H = -\Delta + \frac{c}{\vert x \vert^2}$$
in two dimensions with $c >0$
This operator has a self-adjoint realization, since it is a positive symmetric operator ...
0
votes
1
answer
126
views
Clarification on the Interpretation of Fourier Coefficients in the Context of Fourier Projections
I am currently studying a paper (Section 3.4.3 of Lanthaler, Mishra, and Karniadakis - Error estimates for DeepONets: a deep learning framework in infinite dimensions) where the authors define an ...
2
votes
2
answers
365
views
Is there a compactly supported differentiable function whose Fourier transform is not in L1?
In my MSE answer here, I discussed the example of compactly supported continuous function
$$g(x)=
\begin{cases}
\dfrac{\frac12 -x}{\log(x)},&0<x\leq1/2\\
0,&\text{otherwise}
\end{cases}$$
...
- 833
2
votes
1
answer
158
views
Are the coefficients in the stationary phase approximation computed explicitly somewhere
In Stein's "Harmonic analysis" book, page 334, one can find
the asymptotic expansion
An instructive proof is given for the case $k=2$. It is clear enough to generalize to the cases $k\geq ...
- 852