All Questions
10,828 questions
6
votes
2
answers
2k
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What is the translation in Fourier transform for a function to have exp. decay at $x\to -\infty$
It is known that smooth functions with exponential decay at $\pm\infty$ are functions whose Fourier transform have analytic continuation in some suited complex strip. I was wondering what happens if ...
- 319
6
votes
4
answers
1k
views
Existence of dominating measure for weak*-compact set of measures
I have posted the following question also here a longer time ago, but due to no answers I thought it might fit better to MO.
Let $(\Omega,\mathcal F)$ be a measurable space and $\mathcal P$ a weak*-...
- 215
6
votes
1
answer
754
views
Banach Manifold
Let $M$ and $N$ be closed manifolds. Is it true that
$C^{k}(N,M)$, which is the space of functions $f: N\to M$ such that $f\in C^{k}$, is a $C^{\infty}$ Banach manifold? If so, can you help me to ...
- 225
6
votes
6
answers
1k
views
Proving continuity on spaces of distributions?
Let $\mathcal{D}'(\Omega)$ be the space of distributions on an open set $\Omega$, and $\mathcal{E}'(\Omega)$ the compactly supported ones.
When you have a linear operator $T:\mathcal{D}'(\Omega)\...
- 61
6
votes
2
answers
4k
views
Is there dual space of the distributions $\mathcal{D}'(R)$?
Dear MOs,
Let $\mathcal{D}(R):=C_c^\infty(R)$ be the smooth functions with compact support. Its dual space is the space $\mathcal{D}'(R)$ of distributions. This space $\mathcal{D}(R)$ has its weak *-...
- 1,649
6
votes
4
answers
8k
views
Characterization of the non-negative definite functions $f(x,y)$
The common definition of the non-negative definite functions is as follows:
Definition 1: A continuous complex-valued function $f(x)$ is called non-negative definite, if for any real numbers $x_1,\...
- 1,649
6
votes
2
answers
4k
views
Bounded and weakly bounded sets in top. vector spaces
Consider a locally convex topological vector space V over the complex numbers. Is it true that every weakly bounded subset of V is indeed bounded? If not, what additional requirements are needed for ...
- 61
6
votes
1
answer
288
views
Sigma-weakly dense *-subalgebra of von Neumann algebra has increasing net of positive elements convergent to the identity
Let $M$ be a von Neumann algebra and $A\subseteq M$ a $\sigma$-weakly dense $*$-subalgebra of $M$. Does there exist an increasing net $\{a_i\}_{i\in I}\subseteq A\cap M^+$ such that $a_i\to 1$ in the $...
- 175
6
votes
2
answers
463
views
Spectrum of operator involving ladder operators
The ladder operator in quantum mechanics are the operators
$$a^\dagger \ = \ \frac{1}{\sqrt{2}} \left(-\frac{d}{dq} + q\right)$$
and
$$a \ \ = \ \frac{1}{\sqrt{2}} \left(\ \ \ \!\frac{d}{dq} + q\...
6
votes
2
answers
348
views
Does there exist a framework for determining if a power series is "differentially algebraic"
It is a well studied problem to take a function $f$ expressed (usually expressed as a solution to a differential equation w/ some initial conditions) and ask if it has an "elementary closed form&...
- 2,689
6
votes
2
answers
333
views
Is there a way to reconstruct the convolution $(f * g)(x)$ of $f$ with a Gaussian $g$ from sampled values, $(f*g)(a), a \in A$?
Suppose that $f: \mathbb{R} \to \mathbb{C}$ is a function which has support in $[-1,1]$. Let $g = g_\sigma$ be a centered Gaussian with variance $\sigma^2$. Is there a way to reconstruct the ...
- 437
6
votes
1
answer
338
views
Atiyah-Singer for Riemannian and Kaehler manifolds
I am trying to understand the proof of the Atiyah--Singer index theorem, and would like to see how it works for two "simple" examples. Could somebody direct me to a proof for the special ...
- 203
6
votes
1
answer
214
views
Are lattice operations in a Lipschitz space sequentially continuous in the weak* topology?
This is a follow-up on this (answered) question on math.SE, but involves a different topology. I think this time it is more appropriate for MO. I will repeat the background from the question cited ...
- 437
6
votes
1
answer
271
views
Approximation property counterexamples? (Also: relation to tensor products)
I remember reading somewhere (but unfortunately, I've forgotten where it was) that the canonical map from the (completed) projective tensor product of two Banach spaces to the (completed) injective ...
- 508
6
votes
2
answers
486
views
Equivalence classes of norms on $R^n$ under symmetries
Let $G \leq {\bf GL}_n$ be a symmetry group on $\mathbb{R}^n$. For simplicity, we can consider the case $G = {\bf GL}_n$.
Define two norms $\|\cdot\|_1$ and $\| \cdot\|_2$ to be equivalent under $G$ ...
- 252
6
votes
2
answers
735
views
Tensor product space with projective norm is incomplete
Ryan says in his book "Introduction to Tensor Products of Banach Spaces"(pg. 17) that for Banach spaces $X$ and $Y$, $X\otimes Y$ equipped with projective norm is not complete unless $X$ and $Y$ are ...
- 163
6
votes
3
answers
266
views
Approximating dense subspaces of Fréchet spaces
If $H$, $H_0$ are two separable Hilbert spaces and $H$ is continuously and densly embedded in $H_0$, it is possible to construct a sequence of linear operators
$$ P_n : H_0 \to H $$
such that for all $...
- 570
6
votes
3
answers
601
views
Differential calculus of functions of self-adjoint operators
Let $H$ be a Hilbert space over $\mathbb{C}$. Fix a self-adjoint operator $A:D(A)\rightarrow H$ and a Borel function $f:\mathbb{R}\rightarrow\mathbb{C}$. The operator $f(A)$ is defined by the spectral ...
- 61
6
votes
2
answers
3k
views
Closed convex bounded sets are weakly compact for which spaces?
It is known that for all reflexive Banach spaces, closed convex bounded sets are weakly compact (compact for the weak topology).
What is the general class of topological vector spaces for which this ...
- 549
6
votes
3
answers
855
views
Fundamental solution of Discrete Laplace in the plane
We consider a discretization of the Laplace operator on $\mathbb Z^2$, https://en.wikipedia.org/wiki/Discrete_Laplace_operator
Then, it is natural to consider its fundamental solution $u$, i.e. $|u(x)...
- 5,063
6
votes
1
answer
277
views
Diagonalization of the matrix $(1/(i+j+\rm{const}))_{i,j}$
Consider the following infinite matrix: $A_{i,j}=\frac1{i+j+\gamma}$, $0\leq i,j<\infty$, $\gamma>0$ is a constant. Is it known how to diagonalize $A$, or, say, calculate $(I+tA)^{-1}$ for ...
- 109k
6
votes
2
answers
929
views
Regularity of random Fourier series
The following two statements appear to be true (but do correct me if I am wrong):
The coefficients of a $C^k$ function on the torus $T^n$ decay at least as fast as $x^{-k}$ (where $x$ is some norm on ...
- 96.4k
6
votes
1
answer
760
views
Example of an infinite dimensional reflexive Banach algebra
If a $C^\ast$-algebra is reflexive (as a Banach space) then it is finite dimensional. Can anyone provide (or give a reference to) a nice example of an infinite dimensional non-commutative Banach ...
- 777
6
votes
4
answers
1k
views
How does one show the existence of discrete and complementary series for SL(2,R)?
In his book on $\mathrm{SL}(2,\mathbb{R})$, Lang shows that any nontrivial irreducible unitary representation of this group is infinitesimally isomorphic to an irreducible admissible subrepresentation ...
- 486
6
votes
1
answer
428
views
Poincaré lemma in infinite dimensions
Hi everyone,
Is the Poincaré lemma true in infinite dimensions?
Here's a precise statement:
Let $X$ be a Banach (or maybe Hilbert) vector space, $U$ a simply connected open set in $X$. Is it true ...
- 1,347
6
votes
2
answers
1k
views
Commuting Linear Operators In Hilbert Spaces
Let $V$ be a finite dimensional vector space over the complex field $\mathbb C$. Let $L:V\rightarrow V$ be a linear operator. Using the matrix of $L$ and the Jordan canonical form it is easy to find ...
- 545
6
votes
2
answers
605
views
$\beta\mathbb{N}$ vs $\beta\mathbb{Z}$
Just started learning the Stone-Cech compactification of discrete groups this week. My motivation comes from a question on $\beta\mathbb{Z}$. Surprisingly, I realized there are muchhhh more literature ...
6
votes
2
answers
979
views
Literature on behaviour of eigenfunctions under multiplication?
Dear community,
I would be happy about any literature or comments on the behaviour of the pointwise product of eigenfunctions of a self-adjoint operator with discrete spectrum, acting on a separable ...
- 199
6
votes
2
answers
909
views
Do maps have flows?
In A New Kind of Science: Open Problems and Projects(pg. 36).
How can one extend recursive function definitions to continuous numbers? What is the continuous analog of the Ackermann function? The ...
user37691
6
votes
1
answer
444
views
When does a matrix define a convolution operator on a hypergroup?
Let $H$ be a discrete hypergroup. Suppose I have a matrix $A=(A_{x,y})$ indexed over $H$ with nonnegative entries which defines a bounded operator on $\ell^2(H)$. When does there exist $f\in\ell^1(H)$ ...
- 5,425
6
votes
3
answers
282
views
Extreme points of the dual unit ball of a Banach algebra
Let $A$ be a unital Banach algebra. Let $f\in A^*$, $\Vert f\Vert=1$ satisfy that there exists a maximal left ideal $L\subset A$ such that $L\subseteq\ker{f}$.
Question: Is $f$ an extreme point of ...
- 2,605
6
votes
1
answer
528
views
A functional equation
I am working on some physics problem and got stuck with the following equation: Let $a$ be a very small positive number. Is there a bounded function $F$, $0 \leq F \leq 1$, such that for all $x \in \...
- 63
6
votes
1
answer
436
views
Diagonalizing the ‘restricted’ Hilbert transform on $L^2(0,1)$, $f(z_1) \mapsto \mathrm{p.v.} \int_0^1 \frac{i}{z_1-z_2}f(z_2) dz_2$
Consider the following operator on functions $\mathcal{T}: L^2(0,1) \to L^2(0,1)$ over the complex numbers.
\begin{equation}
(\mathcal{T} f)(z_1) = \mathrm{p.v.} \int_0^1 \frac{i}{z_1-z_2}f(z_2) dz_2
...
- 545
6
votes
1
answer
397
views
Absolute values of two functions and absolute values of their Fourier transform coincides
Let $f, g \in L^2(\mathbb{R})$.
Is it true that if both $|f|=|g|$ and $|\hat f|=|\hat g|$ hold, then there exists $\theta \in \mathbb{R}$ such that $f=ge^{i\theta}$?
I am not able to prove it or ...
- 489
6
votes
3
answers
832
views
Representation theorem for quadratic form on Hilbert space
I think my question is more suitable for Mathematics Stack Exchange than to MathOverflow but I've already posted two related questions there and I got even more confused, so maybe I can clarify things ...
- 1,305
6
votes
2
answers
282
views
The Calkin representation for Banach spaces
Let $X$ be an infinite dimensional Banach space. Let $\Lambda_{0}$ be the set of all finite dimensional subspaces of $X$ directed by the inclusion $\subseteq$. For each $\alpha\in \Lambda_{0}$, let $...
- 3,343
6
votes
1
answer
693
views
Do multiplicative Banach limits exist?
Let $(D, \succeq)$ be a directed set, and let $B$ be the space of real-valued bounded functions on $D$. A Banach limit $\ell$ on $D$ is a linear functional that satisfies
$$\sup_{d \in D} \inf_{c \...
- 869
6
votes
3
answers
1k
views
Additional conditions under which separately continuous implies jointly continuous
Let $X,Y$ be compact metric spaces and consider $f:X\times Y\rightarrow X$ a separately continuous function.
I am wondering if there could be some additional conditions on $f$ (for example $f(\cdot,y):...
- 61
6
votes
1
answer
741
views
Is the following integral nonzero?
Recently I met an integral as follow:
$$\int_0^{2\pi}\cdots\int_0^{2\pi}\left(\prod\limits_{1\leq i<j\leq9}\sin\frac{\theta_i-\theta_j}{2}\right)\left(\prod\limits_{i=1}^9(1+\cos(\theta_i-\theta_{i+...
- 1,997
6
votes
2
answers
2k
views
Equivalent Norms on Sobolev Spaces
When $k$ is a positive integer and $1<p<\infty$, we know that there is some
$C>0$ such that for all $u\in W^{k,p}\left(\mathbb{R}^{N}\right) :$
$$
\left\Vert \left( -I+\Delta\right) ^{\...
- 233
6
votes
1
answer
517
views
Monge–Ampère with drift
Let $I\subseteq \mathbb{R}$ be an interval.
Let smooth $M(x,y):I\times(0,\infty) \to \mathbb{R}$ satisfies PDE:
$$
M_{xx}M_{yy}-M_{xy}^{2}+\frac{M_{y}M_{yy}}{y}=0.
$$
My question is to describe/...
- 3,946
6
votes
3
answers
2k
views
Generalized Hardy-Littlewood-Sobolev Inequality
The Hardy-Littlewood-Sobolev Inequality says that
$$\text{for $p,q,r\in (1,+\infty)$ such that }\quad
1-\frac1p+1-\frac1q=1-\frac1r,\tag {$\sharp$}
$$
$$
\exists C, \forall u\in L^p(\mathbb R^n),\...
- 16.2k
6
votes
3
answers
2k
views
Estimating the variance of a discrete normal distribution
Let $f(x; \sigma) = \frac{1}{\sigma\sqrt{2\pi}}\cdot e^{-\frac{x^2}{2\sigma^2}}$ be the probability density function of a normal distribution $\mathcal{N}(0, \sigma^2)$. We consider a discrete normal ...
- 167
6
votes
2
answers
476
views
function that is the average of affine transformations of itself
Consider the function $f : \mathbb{R} \to [-1,1]$ with
$$
f(x) = \begin{cases}
-1 & x \le -1 \\
+1 & x \ge +1 \\
\frac{f(\frac32 (x-\frac13)) + f(\frac32 (x+\...
- 165
6
votes
1
answer
509
views
closed subspaces of locally convex inductive limits
It's a duplicate of this question, since I really want to get an explanation.
Let $\left(V_{n},\phi_{n,n+1}\right)_{n\in\mathbb{N}}$ be an inductive sequence of LCTV spaces. A locally convex ...
- 173
6
votes
3
answers
1k
views
Is the set of the absolutely continuous functions a Borel set of the space of the continuous functions?
Does anyone knows whether the set of the absolutely continous functions $F :[0,1]\to \mathbb{R}^d$ of the form $$F(t)= a + \int_0^tf(s) ds$$ where $f$ is an integrable function is a Borel set of the ...
- 125
6
votes
1
answer
2k
views
Polynomials are dense in weighted $L^2$ space
Hi,
It seems to be a common knowledge that the polynomials $x^n$ are dense in $L^2$ spaces with various probability weights, such as the gamma distribution weight $x^{\alpha-1}e^{-x}/\Gamma(\alpha)\;...
- 1,765
6
votes
3
answers
1k
views
Zero sets of harmonic fucntions
Can a two variable Harmonic function f(x,y) be zero on a curve with a cusp?
- 365
6
votes
1
answer
404
views
Unique preduals up to (nonisometric) isomorphism?
It's well known that there are Banach spaces which has a unique isometric predual-- for example, any von Neumann algebra. As other questions on here (for example, Isomorphisms of Banach Spaces ) ...
- 18.7k
6
votes
2
answers
3k
views
Dense inclusions of Banach spaces and their duals
This seems like a really simple question, but I'm struggling with it. Let $X$ be a separable Banach space, $H$ be a separable Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, ...
- 8,512