All Questions
10,241 questions
7
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Fourier transform surjective on $L^p(\mathbb{R}^n)$ for $p \in (1,2)$?
I know that $F_2:L^2 \rightarrow L^2$ is of course unitary, whereas $F_1:L^1 \rightarrow C_0$ is injective but not surjective. This can be seen by looking at the dual map.
Riesz-Thorin gives us that ...
- 85
7
votes
1
answer
2k
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Orthonormal bases on Reproducing Kernel Hilbert Spaces
Recall that a Hilbert space $\mathcal{H}$ is a reproducing kernel Hilbert space (RKHS) if the elements of $\mathcal{H}$ are functions on a certain set $X$ and for any $a\in X$, the linear functional $...
- 577
7
votes
1
answer
548
views
Spectrum of unitary elements of a Banach algebra
Unitary elements of a Banach space have been defined in this paper as follows:
Let $A$ be a Banach space and $a\in A, \|a\|=1$. Let $S_{a}=\{f\in A':\|f\|=1=f(a)\}$. Then $a$ is said to be (...
- 173
7
votes
2
answers
208
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Characterizing when matrices are 'dissipative'
An $n$ by $n$ matrix A is said to be dissipative with respect to a norm $\|\cdot \|$ if for all $x$ and $t\geq 0$, we have $\|e^{At}x\|\leq\|x\|$. Two matrices $A$ and $B$ are said to be jointly ...
- 291
7
votes
1
answer
907
views
Lebesgue differentiation theorem holds on locally doubling space?
It's known that for a metric space with doubling measure $(X,\mu)$, the Lebesgue differentiation theorem holds , i.e. If $f:X\to \mathbb{R}$ is a locally integrable function, then $\mu$-a.e. points ...
- 471
7
votes
2
answers
385
views
Can phase significantly concentrate a function's spectrum?
Let $F$ denote the Fourier transform over some group. What is known about the following quantity?
$$\gamma:=\inf_{x\neq 0}\frac{\|Fx\|_1}{\|F|x|\|_1}$$
Here, $|x|$ denotes the pointwise absolute ...
- 7,633
7
votes
2
answers
992
views
Is there a nice "synthetic" way for doing differential geometry on infinite dimensional vector spaces?
If $V$ is an infinite dimensional vector space, for example the space of smooth functions on $\mathbb{R}$, we can introduce some differential geometry concepts by choosing a topology on $V$ and doing ...
- 922
7
votes
1
answer
395
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$\mathcal S(\mathbb R^n) \hat \otimes_\pi \mathcal S(\mathbb R^m) \simeq \mathcal S(\mathbb R^{n+m})$?
If $S(\mathbb R^n)$ is the Scwartz space of smooth rapidly decaying functions equipped with the topology generated by the family of semi-norms
$$\mathcal N_p (\varphi)= \sum_{|\alpha|, |\beta| \leq p} ...
- 630
7
votes
1
answer
1k
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laplace equation on manifolds with boundary
in aubin's book on page 104 theorem 4.7 there is the theorem: Let $(M,g)$ be a compact $C^{\infty}$ Riemannian manifold. There exists a weak solution $\varphi \in H_{1}$ of $\Delta \varphi = f $ if ...
- 213
7
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1
answer
1k
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How to construct a scalar differential operator having the same spectrum as a non-scalar differential operator exploiting symmetries?
I am interested in eigenvalue problems for differential operators acting on one forms on closed two-dimensional manifolds and how they relate to eigenvalue problems of associated operators acting on ...
7
votes
2
answers
657
views
Subspaces isomorphic to $C[0, \omega_1]$
Let $\omega_1$ be smallest uncountable ordinal. I am trying to understand the possible "large" subspaces of $C[0,\omega_1]$, namely those which are isomorphic to the whole space. Therefore I have the ...
- 11.3k
7
votes
1
answer
2k
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A good reference for the wave front set
Hello,
I am wondering whether anyone know some good references for the theory of wave front set, microlocal analysis? I have some basic knowledge of distribution theory at the level of the Rudin's ...
- 1,649
7
votes
3
answers
1k
views
A Question concerning the Fourier Transform of $\mathbb{R}$
Consider the classical Schwartz space $\mathcal{S}(\mathbb{R})$ together with the Fourier transform $\mathcal{F} : \mathcal{S}(\mathbb{R}) \rightarrow \mathcal{S}( \mathbb{R})$.
Consider the subspace ...
- 11.2k
7
votes
2
answers
790
views
Question about von Neumann algebra generated by a complete algebra of projections
Hi all, sorry if this is a dumb question, I don't know much about von Neumann algebras except the definition and a few relevant facts I've managed to prove by myself so I expect the answer will turn ...
- 391
7
votes
1
answer
281
views
Norm in the minimal tensor product of C*-algebras
Let $A$ and $B$ be two $C^*$-algebras, and let $A \otimes B$ denote their minimal tensor product. Given positive, linear functionals $\varphi$ on $A$ and $\psi$ on $B$, we obtain a positive, linear ...
- 3,497
7
votes
1
answer
299
views
Intermediate spaces of test functions between $\mathcal{S}$ and $\mathcal{D}$?
On $\mathbb{R}^n$, let $\mathcal{S}(\mathbb{R}^n)$ be the Schwartz space and $\mathcal{D}(\mathbb{R}^n)$ be the space of smooth, compactly supported functions.
According to p.145 of the book by Reed &...
- 3,477
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)
$$
This is Mehler's kernel, and can be ...
- 90
7
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1
answer
635
views
Convergence of $\exp(tQ)$ in operator norm as $t\rightarrow\infty$
This is not a homework problem, so I am not sure whether this has a "good" answer or not. I came up with this question when I am now learning functional analysis and wonder whether my "...
- 715
7
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1
answer
319
views
Is every sequentially $\sigma(E',E)$-continuous linear functional on a dual Banach space $E'$ necessarily a point evaluation?
$\newcommand{\bf}[1]{\mathbb #1}\newcommand{\sc}[1]{\mathscr #1}$
A duality between two vector spaces $E$ and $F$ over $\bf K$ ($= {\bf R}$ of ${\bf C}$)
is, by definition, a bilinear form
$$
\...
- 2,263
7
votes
1
answer
3k
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Operator norm and spectrum
I am wondering about when an operator norm coincides with the maximum eigenvalue of an operator and there is one particular aspect that confuses me quite a lot.
Let's say we have a symmetric positive ...
- 71
7
votes
1
answer
754
views
Closed convex hull in infinite dimensions vs. continuous convex combinations
tl;dr: When is the closed convex hull of a set $K$ equal to the set of "continuous" convex combinations of $K$?
I am essentially asking for the most general, infinite-dimensional analogue of ...
- 71
7
votes
1
answer
1k
views
Properties of convolutions
Consider the function
$$f_{n}(x)=e^{-x^2}x^n.$$
and the function
$$h_p(x):=e^{-\vert x \vert^p}.$$
My goal is to analyze
$$ F_p(y):=\frac{(f_2*h_p)(y)}{(f_0*h_p)(y)}- \left(\frac{(f_1*h_p)(y) }{(f_0*...
- 173
7
votes
1
answer
593
views
Fractional powers of an operator
What is the large class of operators for which one can define fractional powers? For example, we can consider an operator $A: D(A) \subset X \rightarrow X$, generator of an analytic semigroup on a ...
- 395
7
votes
1
answer
200
views
If $\ $ $yx_n\to 0 $ for all $y$ in a C$^*$-algebra, Is it true that $x_n$ is weakly convergent to $0$?
$A$ is a C$^*\! $-algebra and $(x_n)_{n\in \mathbb{N}} \subseteq A $.
If $\ $ $yx_n\to 0 $ for all $y\in A$, Is it true that $x_n$ is weakly
convergent to $0$ ?
For unitals this is trivial. ...
- 327
7
votes
2
answers
427
views
On the Fourier-Laplace transform of compactly supported distributions
Let $\mathcal{E}'(\mathbb{R})$ be the space of all compactly supported distributions on $\mathbb{R}$.
For $f\in \mathcal{E}'(\mathbb{R})$, let $\widehat{f}$ denote the entire extension of the ...
- 191
7
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1
answer
337
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Flows in Hilbert spaces
Let $\varphi: [0,T] \rightarrow H$ be a Hilbert space valued $C^1$-function. Let $H = X \oplus X^{\perp}$ such that $\varphi(0) \in X$ and the implication $\varphi(t) \in X \Rightarrow \varphi'(t) \in ...
- 83
7
votes
1
answer
305
views
Reflexive subspaces of bidual Banach spaces
The answer to the question is almost surely negative (as almost always in Banach space theory) but I cannot find a relevant example.
Is there an example of an infinite-dimensional Banach space $X$ ...
- 75
7
votes
2
answers
521
views
Kazhdan constant and finite index subgroups
I am wondering if there is some general relation between Kazhdan constants of a group and it finite index subgroups?
Let $G$ be a finitely generated group with a generating set $\Sigma$ that ...
- 165
7
votes
1
answer
510
views
It is true that $\overline{\text{Im}(A)}\otimes \overline{\text{Im}(B)}\subset \overline{\text{Im}(A\otimes B)}$?
Let $H$ be a complex Hilbert space and $\mathcal{L}(H)$ be the algebra of all bounded linear operators on $E$.
If $A,B\in \mathcal{L}(H)$, It is true that $\overline{\text{Im}(A)}\otimes \overline{\...
- 1,154
7
votes
2
answers
449
views
Distribution that vanishes against approximated delta is zero
Suppose we have a Schwartz distribution $\phi$ on $\mathbb{R}^d$ such that $$ \forall x, \ \lim_{\lambda \to 0}| \langle\phi, \psi^{\lambda}_x \rangle| =0$$
where $\psi^{\lambda}_{x}=\lambda^{-d}{\...
- 387
7
votes
3
answers
442
views
Weak compactness in the James space and its dual
It is known that there are characterizations of weak compactness in most of classical non-reflexive spaces (e.g. $L_{1}$-spaces and $C(K)$-spaces). I wonder whether there are characterizations of weak ...
- 3,343
7
votes
3
answers
546
views
Do non-normal states exist in the Solovay model?
Let H be an infinite dimensional Hilbert space. Then there exist non-normal states on B(H) in ZFC (i.e. states that are not represented by a density operator).
Is this also true in the Solovay model ?...
- 2,753
7
votes
1
answer
497
views
Inequality of the norm of the convolution in $L^p(\mathbb{R}^n)$ with symmetric decreasing rearrangement?
Is it true that
$$
||f*g||_p \le ||\,|f|^* * |g|^*||_p\quad ?
$$
where $|f|^*$ and $|g|^*$ are the symmetric decreasing rearrangements of the functions $|f|$ and $|g|$. Under what conditions on $f$ ...
- 193
7
votes
1
answer
699
views
When $C(X)$ is an injective $C(X)$-module? Current answer is erroneous
It is an old question if every injective Banach space is isomorphic as Banach space to $C(X)$-space.
I would like to know if the weakened module version of this question is answered. More precisely: ...
- 1,697
7
votes
1
answer
433
views
Extending compact operators
Let $X$ be a separable, infinite-dimensional complex Banach space and $Y\subseteq X$ an infinite-dimensional closed subspace. Suppose $K:Y\to X$ is an arbitrary compact operator. I would like to ...
- 1,591
7
votes
1
answer
2k
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Proving that a specific kernel is positive definite
Most theoretical papers concerning kernels assume that they are given a positive definite kernel. In this question, we want to show that a specific kernel is positive definite.
We are interested in ...
- 981
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1
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439
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About the convergence rate for an approximation to the heat kernel
Let $G(t,x)$ be the heat kernel
$$
G(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}, \quad t>0, \:x\in\mathbb{R}.
$$
Here is one approximation to $G(t,x)$:
$$
G_\epsilon(t,x)=e^{-t/\epsilon} \...
- 1,649
7
votes
1
answer
703
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A Question About Pure States, Support Projections and Central Covers
I am trying to study the paper Consistency of a Counterexample to Naimark’s Problem by Charles Akemann and Nik Weaver, and there is a claim in Lemma 1 of the paper that I am stuck at, which is as ...
user36116
7
votes
2
answers
1k
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For what spaces is the Hardy-Littlewood maximal operator of strong type $(p,p)$ if and only if $p > p_0 > 1$?
(This is essentially a continuation of my previous question, here.)
Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. Further assume (though you ...
user14166
7
votes
1
answer
1k
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dependence of eigenvalues on parameters
Let $f$ be a positive real-analytic function on the closed unit disk. Consider the eigenvalue problem $\Delta \phi = \lambda f \phi$,
with $\phi = 0$ on the boundary. There exists a sequence of ...
- 1,233
7
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3
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1k
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If *Y* is weakly dense in *X*, is the unit ball in *Y* necessarily dense in the unit ball in *X*?
Let X be a normed space and denote by X* the space of all bounded linear functionals on X. Take a linear subspace G ≤ X* which separates the elements of X, i.e., for each x ∈ X, there is an f &...
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7
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3
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495
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Noninteger iterates of functions: How to get ODE from flow at a given time?
Suppose you have the autonomous ordinary differential equation $dx(t)/dt = f(x(t))$ with $x: \mathbb{R} \to \mathbb{R}$ and the initial condition $x(0)=x_0$. Then, assuming some regularity conditions, ...
- 4,014
7
votes
3
answers
1k
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Can Stein's maximal principle be strengthened?
Let $T$ be an operator on $S(G)$ where $G$ is the line $R$ or the circle $T$, and $S(G)$ denotes the Schwartz space of functions on $G$.
We can ask if the operator T is bounded (as an operator from $...
- 13k
7
votes
1
answer
371
views
Duality of $H^1$ and BMO
While proving that the dual of $H^1$ is $BMO$ in Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, page 143, Stein says that we have $\left\Vert g \right\Vert_{H^1} \...
- 71
7
votes
1
answer
284
views
A characterization of Hilbert spaces by norm one projections
Suppose a (separable) Banach space $X$ has the following property: If $P:X\to X$ is a bounded projection different from $I$ such that $\|P\|=1$, then $\|I-P\|=1$. Does this imply that $X$ is a Hilbert ...
- 1,361
7
votes
2
answers
248
views
Subspaces of $\ell_\infty^3$
Let $a,b\in\mathbb C$ be suc that $\max\{|a+b|,|a-b|\}\leq 1$ but $|a|+|b|>1.$ According to this paper by Arias, Figiel, Johnson and Schechtman https://www.jstor.org/stable/2155206?origin=crossref#...
- 1,879
7
votes
2
answers
508
views
A little problem in PDE or function analysis
Let
$E$ be the usual sobolev space $H^{1}_{0}(\Omega)$ on a smoothly bounded domain $\Omega$,
$E_{k}$ be its subspace spanned by the first $k$ eigenfunctions of the Laplace operator, i.e.
$$E_{k}:=\...
- 705
7
votes
1
answer
195
views
Self-dual Orlicz sequence spaces
Suppose that $\ell_\phi$ is a reflexive Orlicz sequence space such that its dual space $\ell_\phi^*$ is isomorphic to $\ell_\phi$.
Is $\ell_\phi$ isomorphic to $\ell_2$?
- 4,461
7
votes
1
answer
264
views
Can the intersection of a unitary and an irreducibly represented injective $C^*$-algebra be $\{0\}$?
Let $\mathcal{A}$ be an injective $C^*$-algebra irreducibly acting on a Hilbert space $\mathcal{H}$, and let $\phi$ be a completely positive idempotent from $\mathbb{B}(\mathcal{H})$ onto $\mathcal{A}$...
- 619
7
votes
1
answer
756
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Taylor expansion with remainder on locally convex spaces
It is usual to introduce Fréchet and Gâteaux derivatives in Banach spaces. In this context, the familiar Taylor expansion with remainder is also at hand, as you can see on the picture below taken from ...
- 3,535