All Questions
3,842 questions with no upvoted or accepted answers
6
votes
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203
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Product of distributions under wavefront set condition is zero
Assume $u, v \in \mathcal{D}'(\mathbb{R}^n)$ are distributions with compact support. Denote by $\operatorname{WF}(\bullet) \subset T^*\mathbb{R}^n \setminus 0$ the wavefront set of a distribution $\...
- 501
6
votes
0
answers
213
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Equivalent forms of Fourier restriction conjecture
this question is posted in mathstackexchange, but it seems that no one answers it. Sorry to the administrator if this question is not appropriate on Mathoverflow.
I'm reading Pertti Maattila's book ...
- 196
6
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0
answers
529
views
Infinite-dimensional "algebraic varieties"
This question was also formerly posted on MSE but has not received any answer or comment.
Let $H$ be the infinite-dimensional seperable complex Hilbert space, and $P(H)$ denote its projectivization. ...
- 1,543
6
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0
answers
208
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Interpolation between (or: simultaneous Whitney extension for) $C^\alpha$ and $C^{1,\gamma}$ on a Lipschitz domain
I would like to know whether for a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$ (in the weak Lipschitz, so a "Lipschitz manifold", sense, not necessarily a Lipschitz graph domain), ...
- 2,670
6
votes
0
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318
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Is there any connection between deformation theory in algebraic geometry and perturbation theory in functional analysis/PDEs?
Particularly, is there any connection between formal/first-order/infinitesimal deformation theory and perturbation theory? Both subjects involve "perturbing" some structure at a point, so ...
- 1,094
6
votes
0
answers
182
views
Factorization of metric space-valued maps through vector-valued Sobolev spaces
Let $(X,d,m)$ and $(Y,\rho,n)$ be metric measure spaces and let $f:X\rightarrow Y$ be a Borel-measurable function for which there is some $y_0$ and some $p\geq 0$ such that
$$
\int_{x\in X}\,d(y_0,f(x)...
- 5,405
6
votes
0
answers
211
views
Regularity of $|u|^{\alpha}$ when $u$ is Schwartz
Let $0<\alpha<1$. Let $D_x^{\alpha}$ denote the Fourier multiplier given by $\xi\to |\xi|^{\alpha}$. Suppose $u:\mathbb{R}^d\to\mathbb{C}$ is Schwartz (or even just smooth with compact support). ...
6
votes
0
answers
120
views
Condition on the support of $f$ which ensure that $\widehat{f}$ has a zero-measure nodal region
Suppose that $f\in L^2(\mathbb{R})$ is non-zero and compactly supported. Then its Fourier transform $\widehat{f}\neq 0$ is analytic, and in particular the nodal set $\{\xi\in\mathbb{R}\,s.t.\,\widehat{...
- 282
6
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1k
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Condensed/liquid vector spaces and path integrals
[Edited to take into account comments.]
Background
One approach to the problem of making rigorous various measures on spaces of paths (for example, the Wiener or Feynman measure) is the time-slicing ...
- 61
6
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0
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208
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Can every weakly converging sequence be made to converge strongly after taking a subsequence and rearranging?
Let $f_i: [0, 1] \to \mathbb R$ be functions in $L^1 \cap L^\infty$ with $\sup_i \|f_i\|_{L^\infty} < M$ for some $M > 0$.
Suppose $f_i$ converge weakly in $L^1$ to some $L^1$ function $f$ - ...
- 6,215
6
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0
answers
290
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Two questions about Fock spaces
Let $\mathscr{H}$ be a complex Hilbert space and denote $\mathscr{H}_{n}$ the tensor product $\overbrace{\mathscr{H}\otimes\cdots\otimes\mathscr{H}}^{\text{n}}$. Denote by $\Pi_{\pm}$ the projection ...
- 3,515
6
votes
0
answers
124
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Meagre sets of bounded operators
Let $H$ be a separable, infinite-dimensional Hilbert space and let $\mathbb{B}(H)$ be the algebra of bounded operators on $H$. The norm topolology on $\mathbb{B}(H)$ is stricly finer, hence the ...
- 9,853
6
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0
answers
113
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A continuity argument for a dispersive $gKdV$ estimate
I'm learning about the gKdV equation, following Schlag & Muscalu vol II. We're looking at
$$\begin{cases} u_t + u_{xxx} + F(u)_x = 0 \\ u_0 = g\end{cases}$$
where $F(u) = u^5$ (for example). The ...
- 163
6
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0
answers
129
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Weak-type inequality for the partial Fourier sum operator
I'm studying harmonic analysis by myself. One of the online notes gives the following claim as a remark:
For any $N \in \mathbb{Z}^{+}$, let's use $S_{N}$ to denote the partial ($N$ terms) Fourier sum ...
- 349
6
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132
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Mazur-Ulam bases in finite-dimensional Banach spaces
Definition. A basis $e_1,\dots,e_n$ of a finite-dimensional Banach space $X$ is called Mazur-Ulam if all vectors $e_1,\dots,e_n$ have norm one and every self-isometry $f:S_X\to S_X$ of the unit sphere ...
- 4,535
6
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0
answers
107
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Eigenvalues of splitting scheme
In numerical analysis it is common to approximate a solution to a PDE
$$u'(t) = (A+B) u(t), \quad u(0)=u_0$$
which is just given by $e^{t(A+B)}u_0$ by the splitting $e^{tB/2} e^{tA} e^{tB/2}u_0.$ Here,...
- 536
6
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0
answers
92
views
Does every compact abelian group contain a Kronecker set generating a dense subgroup?
Let $G$ be a compact metrizable abelian group with infinite exponent.
Let $S^1 = \left\{z \in \mathbb{C} : |z| = 1 \right\}$. A set $K \subset G$ is a Kronecker set if, for every continuous function $...
- 198
6
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0
answers
533
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Hamiltonian dynamics on cotangent bundle
I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that $N$ is a differential manifold and $S_0 ,S_1\subseteq N$ two ...
- 341
6
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153
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How absolute is NIP in a model?
The following question is motivated by a model theoretic question but doesn't really involve any model theory per se. That said, I don't know the appropriate keywords for the relevant functional ...
- 12.5k
6
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241
views
Extension of positive functionals
Let $X$ be a function space as $C(K)$ or $L^p$, with its usual norm and order, that is $f \le g$ if and only if $f(x) \le g(x)$ for a.e. $x$. If $M$ is a subspace of $X$ and $L:M \to \bf R$ is a ...
- 6,035
6
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445
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Vector-valued interpolation for sublinear operators
Grafakos in his $\textit{Classical Fourier Analysis}$ formulates (see Exercise 4.5.2 therein) the following vector-valued version of the Riesz-Thorin interpolation theorem.
$\textbf{Theorem}$
Let $1\...
- 421
6
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0
answers
107
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Real-world example of a Banach *-algebra with a nonzero *-radical
Is there a real-world example of a Banach *-algebra with a nonzero *-radical (intersection of kernels of all *-representations)? Textbooks give examples of finite-dimensional algebras with degenerate ...
- 3,816
6
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0
answers
158
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Quotients of subspaces of $C(\alpha)$
A well known problem, attributed to H. P. Rosenthal, asks whether or not every quotient of $C(\alpha)$, $\alpha$ countable ordinal, is $c_0$-saturated. As it is known, $C(\alpha)$ are $c_0$-saturated ...
- 986
6
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0
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348
views
Recent work on Pseudo-Laplacian and Pseudo-cuspform in the spirit of Riemann Hypothesis after the work of Bombieri and Garrett
( This is my first MO question . I'm totally inexperienced on MO so, forgive me for my mistakes .)
Paul Garrett and Enrico Bombieri were (are?) Secretly Working on Pseudo-Laplacians and Pseudo-...
user153636
6
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0
answers
144
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Does every locally convex space with a Schauder basis have the approximation property?
For Banach spaces, the existence of a Schauder basis implies that this space has the approximation property.
Since both the notion of Schauder bases and of the approximation property are well ...
- 799
6
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answers
99
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Is every separable Banach space with the MAP 1-complemented in a space with a monotone basis?
The question, already phrased in the title, looks like a classical problem from Banach space theory from the 1970s. Hence, my question is more of a reference request in its nature.
Can every ...
- 11.3k
6
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answers
227
views
Poking into a Lie group with your finger
I consider this as a differential geometry problem. I have asked some
of my classmates who are more interested in that, and also looked into
some literature, but none of what I've found seems to help.
...
- 5,230
6
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0
answers
315
views
Question on operator topologies convergence
Let $H$ be a complex Hilbert space, and let $\mathcal{B}(H)$ denote the algebra of bounded operators on $H$. It is known that the strong operator topology and the norm topology on $\mathcal{B}(H)$ ...
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6
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321
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Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?
Let $T:\ell^2\to\ell^2$ be a compact linear operator. Let $[T]=(a_{i,j})_{i,j=1}^{\infty}$ be the representing infinite matrix of $T$ with respect to the canonical base. Let $T_n$ be the finite rank ...
- 596
6
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181
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Blocksum induces a unital H-space structure on the space of Fredholm operators
Fix a complex separable infinite-dimensional Hilbert space $H$. It is well known that the space of (bounded) Fredholm operators $Fred(H)$ with the norm topology is a classifying space for the ...
- 386
6
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239
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Sheaves on Rectifiable Sets
Basic question: are there (co)homological or sheaf-based tools which might be useful in geometric measure theory?
Background: The jumping off point here is a simple analogy - geometric measure ...
6
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0
answers
113
views
Interpolation of some Sobolev spaces
Let $X_0=L^2(0,1)$, $X_1=H^4(0,1)$, $X_2=H^4(0,1)\cap H^2_0(0,1)$. We know the interpolation space $$(X_0,X_1)_{1/2,2}=H^2(0,1).$$
I am wondering what is
$$(X_0,X_2)_{1/2,2}=?$$
Would it be $H^2_0(0,...
- 395
6
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0
answers
117
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Homomorphisms from BV
Denote by $\mathsf{BV}(\mathbb T)$ the Banach space of functions on the circle with bounded variation which is a Banach algebra under the pointwise product. Is there a surjective homomorphism from $\...
6
votes
0
answers
214
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Divisor bound for $r_2$ off the origin
If $r_2(n)$ denotes the number of integer solutions to $a^2+b^2=n$, we have the "divisor bound" $r_2(n) = O(n^{\epsilon})$ for any $\epsilon>0$. Another way to state this is that the number of ...
- 1,235
6
votes
0
answers
237
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A characterisation of certain $C^*$-algebras
I was wondering if there is a characterisation for $C^*$-algebras (unital) for which the bidual does not have any central atoms. It is not sufficient for example to demand that the $C^*$-algebra does ...
- 633
6
votes
0
answers
921
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Direct proof of Closed Graph Theorem (or Bounded Inverse Theorem) from Uniform Boundedness Principle
I'm looking for a direct proof of the Closed Graph Theorem (or Bounded Inverse Theorem) from the Uniform Boundedness Principle. But I can't find one in the literature.
I'm hoping there's a nice ...
- 469
6
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0
answers
219
views
Extremal polynomial majorants of $\log{|f|}$: a multivariate extension of a theorem of Carneiro and Vaaler
Carneiro and Vaaler have proved, as an application of their work on Beurling-Selberg extremal majorants, that for any non-zero complex polynomial $f(z) \in \mathbb{C}[z]$, the infimum value of the ...
- 13.8k
6
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answers
83
views
Are invertible measures strictly dense?
Let $L_1(\mathbb T)$ be considered as a closed ideal of $M(\mathbb T)$, the Banach algebra of measures on the circle. Then $M(\mathbb T)$ can be identified with the multiplier algebra of $L_1(\mathbb ...
- 113
6
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0
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88
views
Density of squares of radial eigenfunctions
The eigenfunctions of the Laplace operator on the disc can be written in polar coordinates as $f(r,\theta)=R_{nk}(r)e^{ik\theta}$, where $k\in\mathbb Z$ and $n\in\mathbb N$ and the radial function is $...
- 8,086
6
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0
answers
282
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Spectral properties of Non-local Differential operators on real line
I am encountering non-local (and nonlinear) PDEs in my work. To compute stability, I am trying to numerically estimate the spectrum of linearized-but-nonlocal version of the said PDEs.
Definition: A ...
- 318
6
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0
answers
4k
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Interchange of supremum and integral
Let $f : X \to Y$, $X \subset R^n$, $Y$ Banach space, $g : X \times Y \to R \cup \{ \infty \}$, $L^n$ the n-dimensional Lebesgue measure.
Are there some results under which the following interchange ...
- 303
6
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0
answers
240
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Unitary representations of finite dimensional Lie groups on infinite-dimensional Hilbert spaces
I am interested in the proofs of continuity of some standard unitary representations appearing in Physics. Additionally, I am interested in the integration of finite-dimensional Lie algebras of skew-...
- 493
6
votes
0
answers
203
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Uniform estimates of Fourier transform of tempered functions with parameters
Consider the following function in $\mathbb{R}^3$:
$$
f_t(x)=(1+|x|^2)^{-\alpha}e^{-g(x)t},\,\,\,\,\, \text{where}\,\, g(x)=\frac{x^2_1\cdot x^2_2}{1+|x|^2},
$$
where $\frac{1}{2}<\alpha<1$, and ...
- 879
6
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0
answers
388
views
Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance
Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...
- 1,095
6
votes
0
answers
137
views
Spectrum of perturbed differential operators
I am looking for a reference that could help me with the following two questions:
Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with Lipschitz boundary. Consider a sequence of differential ...
- 61
6
votes
0
answers
486
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On Fourier coefficients of nonnegative function
Let $N$ be nonnegative integer, $F(x)$ be a nonnegative real
Lebesgue integrable function defined on $[0,1]$. Suppose that all Fourier coefficients $c(\lambda)=\int_0^1F(x)e^{-2\pi i \lambda ...
- 12.3k
6
votes
0
answers
281
views
Covariance operator analogue for manifolds and respective measure manifolds
Assume $E$ is a connected riemannian manifold with geodesic metric space structure given by $d$ and $P$ is a probability measure over $E$ with Borel sigma-algebra given by this metric structure. Also ...
- 519
6
votes
0
answers
798
views
What is the Banach dual of the Bochner space $L^\infty(\Omega;X)$?
Suppose $\Omega$ is a $\sigma$-finite measure space (I'm happy to take $\Omega = \mathbb{N}$) and let $X$ be a Banach space. It's pretty well known that the Banach dual of $L^\infty(\Omega)$ can be ...
user14166
6
votes
0
answers
210
views
Generalized singular numbers and the Haagerup $L^p$ spaces
Let $M$ be a semi-finite von Neumann algebra with a trace $\tau$.Let $S(M)$ be the algebra of all affiliated operators measurable with respect to $M$.
The $L^p$ norm on $M$ is given by
\begin{...
- 361
6
votes
0
answers
774
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Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term
Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs.
I'm reading Stochastic Differential Equations in ...
- 167