All Questions
10,936 questions
1
vote
2
answers
307
views
Closed formula for this sum $\sum^\infty_{n=0}\frac{1}{n^4+n^2+1}.$ [closed]
How to calculate this sum $$\sum^\infty_{n=0}\frac{1}{n^4+n^2+1}.$$
Thank you in advance
- 531
2
votes
2
answers
921
views
Decay of the Fourier transform of a non-differentiable function
It is well known that if $\varphi$ is a Schwartz function on $\mathbb{R}$ (i.e. smooth and decaying at infinity faster than polynomials), then its Fourier transform decays faster than polynomials. ...
- 421
3
votes
1
answer
355
views
Extremely disconnected or extremally disconnected?
In the context of Banach space theory, what is the correct terminology: extremally disconnected or extremely disconnected. Looking through the internet I have met using both extremely and extremally ...
- 42k
5
votes
1
answer
213
views
States "absorbed" by a Haar idempotent on a compact quantum group
Firstly, a small question of nomenclature. If $(S,\bullet)$ is a magma, is there good terminology to relate $a$ to $b$ when
$$a\bullet b=b=b\bullet a?$$
Can we say that $b$ absorbs $a$? Can we say ...
- 1,037
2
votes
0
answers
148
views
Quotients of $c_0\mathbin{\hat{\otimes}_{\pi}}\ell^1$
Let $\hat{\otimes}_{\pi}$ denote the projective tensor product. Let $$\mathcal{S} = \{V\subseteq c_0\mathbin{\hat{\otimes}_{\pi}}\ell^1\textrm{ closed subspace}: {c_0\mathbin{\hat{\otimes}_{\pi}}\ell^...
- 2,605
11
votes
2
answers
8k
views
About the Fourier transform of the logarithm function
I want to calculate / simplify:
$$\mathcal{F} (\ln(|x|)\mathcal{F(f)}(x))=\mathcal{F} (\ln(|x|)) \star f$$
where $\mathcal{F}$ is the Fourier transform ($\mathcal[f](\xi)=\int_{\mathbb R}f(x)e^{ix\...
- 1,199
10
votes
2
answers
2k
views
Why does Riesz's Representation Theorem apply in quantum mechanics?
$\DeclareMathOperator\tr{tr}$One begins with a quantum mechanical system, i.e. a unital $C^*$-algebra $A$.
It is common to begin the discussion with embedding $A$ into the algebra of bounded operators ...
- 2,071
5
votes
1
answer
169
views
Hadamard factorization of a function in the Fock space
An entire function $F: \mathbb C \to \mathbb C$ belongs to the Fock space $\mathcal F^2$ if
$$
\int_{\mathbb C} |F(z)|^2e^{-|z|^2} \, dA(z) < \infty.
$$
It is well-known that every $F \in \mathcal ...
- 190
1
vote
1
answer
281
views
Continuous wavelet transform of a periodic function
I have a question regarding the Continuous Wavelet Transform (CWT) of non finite energy functions, such as $g(t) = a\exp(i\omega_0t)$. We know that the CWT is defined for functions in the Hilbert ...
3
votes
1
answer
229
views
A pexiderization of the sine addition law on semigroups
Can we solve the follwing functional equation
$$f(xy)=g(x)h(y)+g(y)h(x)$$
on semigroups for unknown complex valued functions $f,g,h$ ?
17
votes
3
answers
905
views
Existence of translation-invariant basis on $C_c(\mathbb R)$
Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
- 2,071
0
votes
0
answers
114
views
Norm distance in a Banach space
Consider the Hilbert space $l_2(\mathbb{N})$ under the square summable norm $\Vert \cdot \Vert_2.$ Let us define a new norm $||| \cdot ||| $ equivalent to $\Vert \cdot \Vert_2$ such that the closed ...
- 85
2
votes
1
answer
195
views
Spectrum of $(Jx)_n =i((2n+1)x_{n+1}-(2n-1)x_{n-1})$ on $\ell^2(\mathbb{Z})$
I've been working on the spectrum of the closure of the operator $J: \mathcal{D}(J)= \mbox{span}\{ e_n: n \in \mathbb{Z}\} \subseteq \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ defined for $x=(x_n)_{n \...
- 229
2
votes
0
answers
175
views
Banach space of vector measures
Let $S$ be a set and $\Sigma$ be a $\sigma$-algebra of subsets of $S$. Let $A$ be a Banach space over the field of complex numbers. A countably additive map $\mu:\Sigma\to A$ is called a vector ...
- 552
1
vote
0
answers
145
views
Operator norm of linear functional $\varphi \mapsto \int_\Omega f\varphi$ with respect to different norms
Let $\Omega \subseteq \mathbb{R}^n$ be open. For some $f \in L^2(\Omega)$ consider the continuous linear functional $$T \colon C^\infty_c(\Omega) \to \mathbb{R}, \qquad T(\varphi) := \int_\Omega f \...
- 548
3
votes
1
answer
466
views
Equivalence between two fractional Sobolev spaces
For $s \in (0,1)$, we consider the spectral fractional Laplacian
\begin{align}
(-\Delta)^{-s}u = \sum_{k=1}^{\infty}\lambda_k^{-s}(\phi_k,u)_{L^2}\phi_k
\end{align}
where
\begin{align*}
\begin{cases}
...
- 161
0
votes
0
answers
342
views
Main ideas behind the proof of the Carleson theorem
I tried to read a few years ago the book "Pointwise Convergence of Fourier Series" (Springer, Juan Arias De Reyna) which is a detailed proof of the Carleson theorem, but I was lost after a ...
- 587
5
votes
1
answer
410
views
Is there always a real $x$ such that $\cos n_1 x + \cos n_2 x + \cos n_3 x < -2$?
Problem: Given three positive integers $0 < n_1 < n_2 < n_3$. Is there always a real number $x$ such that
$$\cos n_1 x + \cos n_2 x + \cos n_3 x < -2?$$
- 1,053
9
votes
2
answers
394
views
Can one detect a cyclic and separating vector for a concrete $C^*$-algebra using a dense subalgebra?
Let $A$ be a $C^*$ algebra of operators acting on some Hilbert space $H$, and $A_0$ is a norm dense $*$-subalgebra of $A$. Suppose there exists some unit vector $\xi \in H$, such that (i) $A_0 \xi$ is ...
- 1,586
-1
votes
1
answer
354
views
The grail of functional analysis?
Does $g\in C([0,1],[0,1])=A$ exist such that $\{g^n ,n\in\mathbb N\}$ is dense in $A$ provided with the uniform norm?
with $g^2=g \circ g $
If we can find $g$ then $F$ a closed of $A$, $id \in F$ ...
- 4,074
2
votes
1
answer
77
views
Measurability of random function with values in $C(K,E)$
Let $K \subset \mathbb{R}$ be compact, and let $E$ be a separable Banach space. Further, let $(\Omega, \mathcal{F},\mathbb{P})$ a probability space. I would like to show that a certain a random ...
10
votes
2
answers
559
views
Can Birkhoff's ergodic theorem for integrable functions easily be deduced from Birkhoff's ergodic theorem for bounded functions?
It seems to me that a considerably simpler proof [see below] of Birkhoff's ergodic theorem can be obtained for bounded observables than for more general $L^1$ observables. Therefore, I feel like it ...
- 708
5
votes
0
answers
265
views
Automorphic Banach spaces
A Banach space $X$ is called automorphic if for every closed subspace $Y\subseteq X$ with $\dim X/Y=\infty$, every automorphism (= linear continuous isomorphism) of $Y$ can be extended to an ...
- 4,535
1
vote
0
answers
166
views
About a weak$^*$ convergent net
Let $G$ be a locally compact abelian group and $A$ be semisimple commutative Banach algebra such that $A^{**}$ has Radon-Nikodym property. Denote by $\Gamma$ and $M(G)$ the dual group and the measure ...
- 2,118
1
vote
1
answer
195
views
Concrete example of non-norm-attaining bounded linear operator on disc algebra
A bounded linear operator $T$ from a Banach space $X$ to a Banach space $Y$ is called norm-attaining, if there exists a vector $x\in X$ with $\|x\|=1$ such that
$$\|Tx\|=\|T\|.$$
Let $\mathbb{D}=\{z\...
- 149
2
votes
1
answer
190
views
Approximating a function by a convolution of given function?
Let $g:\mathbb{R}\to \mathbb{R}$ be a given differentiable function of exponential decay on both sides. Now let us be given a function $f:\mathbb{R}\to \mathbb{R}$, also of exponential decay, if you ...
- 20.2k
8
votes
2
answers
837
views
Strict topology between weak and norm topologies
I want to believe that this has an easy answer, but I’ve never considered it before and can’t seem to answer it now either.
Does every infinite-dimensional Banach space admit a locally convex vector ...
- 1,453
4
votes
2
answers
199
views
Is the conditional expectation of a Caratheodory function a Caratheodory function?
Let $(Y, \Sigma,\mu)$ be measure space and $X$ a Polish space endowed with its Borel $\sigma$-algebra. Suppose that $f:Y\times X\to \mathbb R$ is a Carathéodory function (i.e. continuous in $x\in X$ ...
- 165
1
vote
0
answers
37
views
Unique smallest degree algebraic solution to polynomial ODE
Let's assume we are given a degree $d$ polynomial VF as a map $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$
$$f_j(x)=\sum_{i_{1},\dots,i_{n}=1}^{d}a^j_{i_{1},\dots,i_{n}}x_{1}^{i_{1}}\dots x_{n}^{i_{n}}$$...
- 247
4
votes
0
answers
88
views
Positive definiteness with nonnegative weights
Is there a simple criterion to certify if some function $f: \mathbb{R} \to \mathbb{R}$ satisfies that $\sum_{i,j=1}^n c_ic_jf(x_i-x_j) \ge 0$ for all $x_i \in \mathbb{R}$ and $c_i \ge 0$?
Note that if ...
- 121
1
vote
0
answers
73
views
$L^p$ norm of Fourier transform of function composed with a diffeomorphism
Suppose $f$ is a compactly supported smooth function from $\mathbb{R}^n$ to $\mathbb{C}$ and $A$ is a diffeomorphism on $\mathbb{R}^n$, do we have any theorems relating the $L^p$ norm of $\hat{f}$ and ...
- 265
3
votes
2
answers
651
views
Express Dirichlet energy $E_\mu(f) := \int \|\nabla f(x)\|^2 d\mu(x)$ in terms of Fourier information alone
Let $\mathbb R^d$ and let $\mu = p(dx)$ be a probability distribution thereupon, with density $p$ (which maybe assumed bounded, etc.). For a continuously differentiable function $f:\mathbb R^d \to \...
- 6,853
1
vote
1
answer
210
views
On a property of complex exponentials
Does there exist a simple smooth closed curve $\gamma:S^1\to \mathbb C$ such that
$$ \int_{0}^{2\pi} e^{\gamma(e^{it})} \, |\gamma'(e^{it} )|\,dt =0?$$
- 4,115
5
votes
1
answer
164
views
Quotients of $c_0$ that are complemented in $c_0$
Suppose $X$ is a closed subspace of $c_0$ with an unconditional basis and suppose also that it is a quotient of $c_0$. Is $X$ also a complemented subspace of $c_0$?
An affirmative answer implies that $...
- 1,073
2
votes
0
answers
92
views
Is every compact convex set covered by a Choquet simplex?
Here is a natural question which I have been unable to find discussed in the literature. If $K$ is a compact convex set in a locally convex topological vector space, is there a Choquet simplex $\...
- 2,049
7
votes
1
answer
598
views
A lecture by Rudin
Is it available a written version of this lecture by Rudin on the relation between Fourier analysis and the birth of set theory?
https://youtu.be/hBcWRZMP6xs
If not Rudin himself, maybe someone else ...
- 4,459
3
votes
1
answer
577
views
A constant ratio of integrals? Part II
This question is a follow up on my latest MO post which was addressed kindly by Iosif Pinelis. What is new here is that I need to correct the assumption by including a missing hypothesis. The context ...
- 43.2k
25
votes
2
answers
4k
views
Understanding of rough path
A rough path is defined as an ordered pair
$ (X, \mathbb X)$, where $X$ is a path mapping from $[0,T]$ to some Banach space $V$
and $\mathbb X:[0,T]^2 \mapsto V^2$ is another mapping for additional ...
- 1,399
3
votes
1
answer
558
views
Conditions for chain rule for Gateaux derivatives
Let $X,Y,Z$ be locally convex topological vector spaces over $\mathbb R$ (not necessarily Banach), $D_X \subseteq X$, $D_Y \subseteq Y$ and let $f \colon D_X \to D_Y$, $g \colon D_Y \to Z$. Let us ...
- 335
0
votes
1
answer
209
views
Derivative with respect to a Hilbert-Schmidt operator
Let $X,Y$ be real separable Hilbert space, and let $HS(X,Y)$ be the space of Hilbert-Schmidt operators from $X$ to $Y$, endowed with the Hilbert-Schmidt norm. Let $x\in X$ and $y\in Y$. I am ...
- 503
2
votes
2
answers
228
views
Minimal conditions on random vector $X \in R^n$ to ensure that $\lim_{t\to 0^+}\sup_{\|w\|_p = 1}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t)=0$
Let $X$ be a random variable on $\mathbb R^n$ and let $S_p^n := \{w \in \mathbb R^n \mid \|w\|_p = 1\}$ be the unit-sphere w.r.t to the $\ell_p$-norm in $\mathbb R^n$. We will be particularly ...
- 6,853
0
votes
0
answers
52
views
Coupled Kazdan-Warner type equation
Famous work of Kazdan and Warner shows that given $u\geq 0$ and a constant $c>0,$ the following equation in $f$ has a unique solution:
\begin{align*}
\Delta f+ u e^f=c
\end{align*}
I am interested ...
- 954
40
votes
5
answers
5k
views
"Entropy" proof of Brunn-Minkowski Inequality?
I read in an information theory textbook the Brunn-Minkowski inequality follows from the Entropy Power inequality.
The first one says that if $A,B$ are convex polygons in $\mathbb{R}^d$, then
$$ m(...
- 22.8k
15
votes
1
answer
1k
views
Borel-Écalle re-summation and resurgence: criteria and results
This is about the theory of Borel-Écalle re-summation and resurgence, see Refs below.
This states that the perturbative series (say of the vacuum expectation value of an operator $\mathcal{O}$ in ...
- 10.5k
3
votes
1
answer
394
views
Approximations for real functions
Is there some set of real functions ($S$) that has precisely the cardinality of the continuum, but is dense in the space of all the real functions in a sense that every real function can be ...
- 39
5
votes
1
answer
325
views
The discrete Fourier transform's Gaussian-like eigenvector
I have the $N$x$N$ matrix below where $N$ is a power of 2 (usually 64 or 256) and $\omega = 2\pi/N$. What is its largest eigenvalue?
$\begin{bmatrix}
2 & 1 & 0 & 0 & \cdots & 0 &...
- 1,547
1
vote
0
answers
125
views
Probabilistic interpretation of von Neumann's approach to quantum mechanics
One of the basic postulates in the mathematical formalism of quantum mechanics is that the probability of a measurement of an observable $A$ in the state $\psi \in \mathscr{H}$ to return a value in a ...
- 1,305
4
votes
1
answer
336
views
Support projection vs closed support projection of a normal state in enveloping von Neumann algebra
I preface this by saying that I am fairly new to the enveloping von Neumann algebra scene, so there may be some gaps in my understanding.
Given a $C^*$-algebra $A$ and a state $\phi$ on $A$, one may ...
- 135
4
votes
2
answers
158
views
A ball with slit at the radius is not $W^{1,1}$-extension domain
Recall that: A domain $\Omega\subset \mathbb{R}^d$ is an $W^{1,1}$-extension domain if there exists an operator $E:W^{1,1}(\Omega)\to W^{1,1}(\mathbb{R^d})$ and a constant $c= c(d,\Omega)>0$ such ...
- 1,101
4
votes
0
answers
135
views
Reverse Sobolev inequality for family of holomorphic functions
Denote by $P_m$ the space of polynomials of degree $m$ in a single complex variable $x$. This is a "Reverse Sobolev inequality":
Theorem. Let $U \subset \mathbb{C}^2$ be an open domain and $...
- 981