All Questions
10,936 questions
1
vote
0
answers
260
views
Riemann hypothesis and zero of Mellin transform of eigenfunction of Schrödinger equation
The paper [1] shows that the eigenfunction of the Schrödinger equation
$$
\left(x^2-\frac{1}{4\pi}\frac{d^2}{dx^2}\right)f_n=\frac{2n+1}{2\pi}f_n
$$
satisfies the same functional equation as the ...
- 328
2
votes
2
answers
160
views
Schauder bases in Banach spaces with a symmetric $k$-FDD
The Kalton-Peck Banach space $Z_2$ (see Section 6 in this paper) does not admit an unconditional basis, but it admits an unconditional, even symmetric, FDD (finite dimensional decomposition) into ...
- 4,461
0
votes
0
answers
28
views
Show that functional $J$ satisfies Palais-Smale condition iff every $P-S$ sequence is bounded
Let $H$ be a Hilbert space and let $K: H \rightarrow \mathbb{R}$ be $C^1$ and such that $\nabla K: H \rightarrow H$ sends bounded sets into precompact sets. Consider the functional $J: H \rightarrow \...
- 415
5
votes
1
answer
199
views
Is the unit ball of $B(H)$ a Baire space (with the SOT)?
Let $H$ be a Hilbert space, and let $B(H)$ be the set of bounded linear operators $t \colon H \to H$. Recall that we say $t_i \to t$ in the strong operator topology if $t_i \xi \to t \xi$ for every $\...
- 733
3
votes
1
answer
166
views
A limit arising from Mellin Inversion: How to compute a specific term of an asymptotic series?
So I am wondering if there exists a general procedure for the following problem:
given a monotonically increasing function $f(n)$ which is nonegative on the interval $[0,\infty)$ and grows faster than ...
- 2,689
1
vote
2
answers
188
views
Does a measurable $F :[0, T] \to L^p (\mathbb R^d; \mathbb R_{\ge 0})$ have a "flattened" measurable version?
Let $d \in \mathbb N^*,p \in [1, \infty]$ and $T>0$. Let
$$
F :[0, T] \to L^p (\mathbb R^d; \mathbb R_{\ge 0}), t \mapsto F_t
$$
be measurable. I would like to ask if there is a measurable function ...
- 825
10
votes
1
answer
695
views
Rigorous proof of the pentagon identity
I briefly recall the statement of the pentagon identity in quantum dilogarithm and cluster algebra.
For $b\in\mathbb{C}$ with $\operatorname{Re}(b)>0,\operatorname{Im}(b)\geq0$, Faddeev, Kashaev ...
- 1,391
0
votes
0
answers
73
views
Computationally efficient solution for the measure of central tendency minimizing Lp loss for p > 1
We know that the measure of central tendency that minimizes the Lp loss is $\min_c \sum_{i=1}^n |x_i - c|^p$
For $p=1$ (L1 loss), this is the median. For $p=2$ (L2 loss), this is the mean. Both of ...
- 109
18
votes
6
answers
4k
views
What is the best place to learn about the mathematical foundations of quantum mechanics?
I'm looking for good references to learn about the mathematical foundations of quantum mechanics. By mathematical foundations, I do not mean rigorous quantum mechanics in general but the axioms behind ...
- 1,305
-1
votes
2
answers
408
views
Conditional expectation: commuting integration and supremum
Let $X$ and $A$ be compact Polish spaces endowed with Borel $\sigma$-algebras. Let $\mathcal{A} = X\times \mathcal{B}(A)$ be the $\sigma$-algebra consisting of cylinders whose projections on $A$ are ...
- 109
0
votes
1
answer
91
views
Construct next polynomial from predecessor and resulting GCD
I have a sequence of polynomials built from an interpolation derived in a combinatorial problem. For each integer value of a parameter $n$ there is a different polynomial.
After trying to find a way ...
- 47
5
votes
2
answers
330
views
Functional equation of bounded analytic functions
Let $\mathbb{D}:=\{z \in \mathbb{C}|~|z|<1\}$. Consider $f,~g \in H^{\infty}(\mathbb{D}):=\{f\in\text{ Hol}(\mathbb{D}): \|f\|_{\infty}:= \sup_{z \in \mathbb{D}} |f(z)| < \infty\}$ such that $f^...
- 149
9
votes
2
answers
628
views
How was Claim 5 in "A non-linear generalisation of the Loomis–Whitney inequality and applications" thought up?
In Bennett, Carbery and Wright's paper A non-linear generalisation of the Loomis–Whitney inequality and applications, Claim 5 was used to generalise the case from characteristic functions to simple ...
- 275
2
votes
0
answers
329
views
Conditions for an existence of smooth solution to a parabolic PDE
I'm interested to know the conditions of when the parabolic PDE ($U \subset \mathbb{R}^n$ is some bounded open subset):
\begin{equation*}
u_t - \sum_{i,j=1}^n(a^{ij}(x,t)u_{x_i})_{x_j} + \sum_{i=1}^nb^...
- 425
2
votes
1
answer
210
views
What is a subset of $\mathbb{Z}^3$ making $\Bigl( \sin(n \cdot x),\cos(n \cdot x) \Bigr)_{n \in \mathbb{Z}^3}$ linearly independent?
This question was originally posted in ME: https://math.stackexchange.com/questions/4725157/what-is-an-explicit-subset-of-mathbbz3-that-makes-bigl-sinn-cdot-x
but more and more I think about it, this ...
- 3,477
2
votes
1
answer
145
views
Orthonormal bases in RKHSs via interpolating sequences
Definitions and setting
Let $\mathcal{H}$ be a separable, infinite-dimensional, reproducing kernel Hilbert space on a nonempty set $X$. As usual, denote the reproducing kernel on $\mathcal{H}$ by $K$ ...
- 5,405
3
votes
1
answer
219
views
Is there a real/functional analytic proof of Cramér–Lévy theorem?
In the book Gaussian Measures in Finite and Infinite Dimensions by Stroock, there is a theorem with a comment
The following remarkable theorem was discovered by Cramér and Lévy. So far as I know, ...
- 657
7
votes
1
answer
737
views
Converse of closed graph theorem
Suppose $X$ is a normed linear space. If for every Banach space $Y$ and for every linear operator $T:X\to Y$, graph of $T$ is closed implies $T$ is continuous, then can we prove that $X$ is a Banach ...
- 585
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
$$a \ \ = \ \frac{1}{\sqrt{2}} \left(\ \ \ \!\frac{d}{dq} + q\...
0
votes
0
answers
155
views
Reducing subspaces of unitary operators
Let $U$ be a unitary operator on a Hilbert space $\mathcal{H}$. We can assume $\mathcal{H}$ is an $L^2$ space and $U$ acts as multiplication by a function $u$ with $|u(x)| = 1$ a.e (by the spectral ...
- 23
2
votes
0
answers
188
views
Self-adjointness of fractional laplacian
Lets consider the following functional analytic definition of the fractional Laplacian: Consider a (complete, connected, oriented) Riemannian manifold $(M,g)$ with corresponding Laplacian $\Delta_{g}$....
- 1,171
2
votes
0
answers
62
views
Localized estimate for divergence free vector field
Suppose $\Omega \subset \mathbb{R}^3$ is a simply connected Lipchitz domain. For a divergence free field $w\in [L^2(\Omega)]^d$, it is well known that there exists a vector field $v\in [W^{1,2}(\Omega)...
- 31
59
votes
9
answers
10k
views
Motivation for and history of pseudo-differential operators
Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds). Which prominent example problems lead you to work with pseudo-differential ...
- 5,327
2
votes
1
answer
174
views
Maximizing the first Neumann eigenvalue on disks
Let $D^2$ be a smooth disk and for any Riemannian metric in $D$, let $\mu_1(g)$ be the first positive Neumann eigenvalue of the Laplacian on $(D, g)$. Li and Yau proved that
$$\mu_1(g) \operatorname{...
- 2,095
4
votes
1
answer
202
views
A problem on Demailly's proof of finiteness theorem of elliptic differential operator
I am reading Demailly's notes on pseudodifferential operators on manifolds. And I cannot understand a statement he had made when he tried to prove that the image of an elliptic differential operator ...
- 206
1
vote
1
answer
208
views
Weak-star convergence implies trace-norm convergence
By definition, if bounded operators $a_i$ converge to $0$ in the weak*-star topology, then $\operatorname{tr} a_it \to 0$ for any trace-class $t$.
Does this also hold for the trace-norm instead of the ...
- 896
0
votes
0
answers
43
views
When does the Hermite series converge pointwise and when is it uniformly bounded?
Let $\gamma$ denote the standard Gaussian measure on the real line, and consider $f \in L^2(\gamma)$. Since the Hermite polynomials $\{H_n\}_{n \geq 0}$ are a complete orthonormal system, we may ...
- 380
3
votes
2
answers
147
views
Lumer-Phillips-type theorem for non-autonomous evolutions
The classical Lumer-Phillips theorem characterizes the generators of contraction semigroups. I am looking for a similar characterization or at least a sufficient condition for a family of unbounded, ...
- 300
3
votes
1
answer
131
views
Spectrum of continuous functions as a semigroup
Let $X$ be a countable group (with the discrete topology) and let $C_b(X)$ be the ring of continuous bounded functions $X \to \mathbb{R}$. It is known that the maximal spectrum of $C_b(X)$, namely the ...
2
votes
1
answer
254
views
A question about equivalence of weighted Sobolev space norm in S. Benzoni-Gavage and D. Serre's book
This question may not be at the research level, but it has really bothered me for a long time. The following space is used for handling initial boundary value problem for first order hyperbolic ...
- 187
6
votes
1
answer
230
views
Integration along fibres of continuous map on compact Hausdorff spaces
Let $p:Z\to X$ be a continuous surjective map between compact Hausdorff spaces.
Does there exist a family $m=(m_x)_{x\in X}$ of Radon probability measures on $Z$, such that
the support of $m_x$ is ...
user473423
2
votes
2
answers
228
views
Hardy space inclusion in the right-half plane
I'm looking for an example of a function $u \in H_2$ such that $u \notin H_\infty$, where $H_p$ is the Hardy space on the right-half plane. Since this notation is perhaps not standard, here is a ...
- 427
4
votes
1
answer
161
views
For centered Gaussian measures, is $E[\lVert X\rVert^2] \lesssim E[\lVert X\rVert ]^2$ true in infinite dimensions as well?
In the proof of Corollary 5.7 in the following link:
https://arxiv.org/pdf/1610.05200.pdf
the author claims that $E[\lVert X\rVert^2] \lesssim E[\lVert X\rVert ]^2$ for the standard normal ...
- 3,477
1
vote
1
answer
207
views
Differential operators in $\Bbb R^n$
Put $P_j=\frac{\partial}{\partial \xi_j}$ et $Q_j=2 i \xi_j$ with$\xi=\left(\xi_1, \ldots, \xi_n\right)$ et $x=\left(x_1, \ldots, x_n\right)$. How to prove :
$\exp \left(\sum_{j=1}^n x_j P_j\right)(...
- 531
13
votes
2
answers
915
views
Topological vector spaces (reference request)
In his book Topological Function Spaces Arhangel'skii says that "it is well known that every nontrivial locally convex linear topological space $X$ is homeomorphic to a space of the form $Y \...
- 674
2
votes
0
answers
98
views
Geometric interpretation of uniform convexity condition
I first want to recall the moduli of uniform smoothness (US), uniform convexity (UC), asymptotic uniform smoothness (AUS), and asymptotic uniform convexity (AUC). Throughout, let $X$ be an infinite ...
- 41
7
votes
2
answers
464
views
Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace
Let $(X, \mu)$ be your favourite measure space (finite or $\sigma$-finite if you like), let $g \in L^2$ (say, the scalar field of $L^2$ is $\mathbb{R}$, though this probably doesn't matter). Let $\...
- 12.6k
1
vote
1
answer
263
views
Does global boundedness ruin Stone-Weierstrass denseness?
Let $X$ be any topological space and denote by $\tau_X$ the topology on $C_b(X;\mathbb{R})$ that is induced by the family of seminorms $(\|\cdot\|_\psi\mid\psi\in B_0(X))$ with $\|f\|_\psi:=\sup_{x\in ...
- 463
71
votes
4
answers
11k
views
What are fixed points of the Fourier Transform
The obvious ones are 0 and $e^{-x^2}$ (with annoying factors), and someone I know suggested hyperbolic secant. What other fixed points (or even eigenfunctions) of the Fourier transform are there?
- 1,491
1
vote
1
answer
410
views
Takesaki lemma: existence Gelfand-Pettis integral
Consider the following fragment from Takesaki's second volume of "Theory of operator algebras" (lemma 2.4, chapter VI "Left Hilbert algebras").
In another post, it was explained ...
- 175
1
vote
0
answers
83
views
For any $\beta>0$, there is a constant $c>0$ such that $\left\|(1-\Delta)^{\frac{\beta}{2}} f\right\|_{\infty} \leq c\|f\|_{C_b^\beta}$
For any $n \in \mathbb{Z}^{+}$, let $C_b^n\left(\mathbb{R}^d\right)$ be the class of real functions $f$ on $\mathbb{R}^d$ with continuous derivatives $\left\{\nabla^i f\right\}_{0 \leq i \leq n}$ such ...
- 825
2
votes
1
answer
137
views
Disturbance of self-adjoint operator
Assume that $ A $ is self-adjoint operator and $ B $ is a bounded self-adjoint operator. The definite domain of $ A,B $, denoted by $ D(A) $ and $ D(B) $ satisfies $ D(A)\subset D(B) $. Show that
\...
- 1,219
6
votes
3
answers
267
views
Vanishing periodizations $\sum_{k \in \mathbb Z} f(t+ak)$ of a function $f$ for different values of $a$ implies $f=0$?
Consider a continuous function $f : \mathbb R \to \mathbb C$ with rapid decay (e.g. $|f(t)| < e^{-t^2}$). For a constant $a>0$ let
$$
F_a(t) = \sum_{k \in \mathbb Z} f(t+ak)
$$
be the ...
- 127
8
votes
5
answers
685
views
Distributions of distance between two random points in Hilbert space
Let $\mu$ be a probability distribution on a separable infinite-dimensional Hilbert space. Let $D$ be the distance between two independent random samples from $\mu$.
So $D$ has some probability ...
- 236
0
votes
0
answers
137
views
Convexity of an equivalent norm
Let $X=l_2$ with usual norm $\|\cdot\|_2$. We define a subspace of $X$ as $D=conv (B_{l_2} \cup B),$ where $B = \{ (x_n) \in l_2 : \sum_{n=1}^\infty \frac{n}{2} x_n^2 \leq 1\}$, conv is the convex ...
- 85
2
votes
1
answer
143
views
Decomposition of weak* convergent nets into positive weak* convergent nets
Let $F$ be an order unit Banach space with order unit $e$ and topological dual space $F^*$ ordered by the dual cone. Let $E\subset F^*$ be a closed subspace that separates points of $F$ and such that ...
- 71
3
votes
0
answers
245
views
Norm on the space of real analytic functions
The space $C^{\omega}(\Omega)$ of real-valued real analytic functions on the open bounded set $\Omega\subset \mathbb R^n$ does not have any obvious or natural metric which would make it a Fréchet ...
- 203
3
votes
0
answers
84
views
Convergence of the Gaussian integral on $\mathcal{E}'$ for a mapping supported on $L^2$
Let $F : L^2(S^1) \to L^2(S^1)$ be a (nonlinear) mapping such that
\begin{equation}
\lVert F(f) \rVert \leq \lVert f \rVert
\end{equation}
for all $f \in L^2(S^1)$. For the space of smooth periodic ...
- 3,477
9
votes
0
answers
240
views
What is known about when $vN(G)$ is a factor, for a locally compact group $G$?
When $G$ is a discrete group, it is an elementary result in the theory of von Neumann algebras that the group von Neumann algebra $vN(G)$ is a factor if and only if $G$ is an ICC group.
What is known ...
- 146
2
votes
0
answers
153
views
Riesz’s representation theorem in a weak form
Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^N$ $(N\geq 3)$, $\phi\in H_0^1(\Omega)$ is a solution of $$
\begin{cases}\Delta \phi+ \phi=h & \text { in } \Omega, \\ \phi=0 &...
- 471