All Questions
10,447 questions
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211
views
Gauss transformation in fractional Sobolev space
Let $g_{\mu}(x) = \mu^{d/2}\exp(-\pi\mu|x|^2)$ for every $\mu > 0$. Prove that
$$
\int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} u\right|^{2} \geq \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}...
- 101
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votes
0
answers
88
views
Intersection of Sobolev Spaces
Suppose $\Omega$ is a bounded open subset of $\mathbb{R}^n$ with a "nice" boundary. We have the Sobolev spaces $W^{k,2}(\Omega)$, which are all contained within each other: $W^{m,2}(\Omega)\...
- 807
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votes
0
answers
63
views
Existence of a measurable maximizer
Let $F$ be a continuous cdf with full support on $[0,1].$ Let $A$ be a compact subset of $\mathbb{R}$ and $\mathcal{M}$ be the set of measurable functions $\alpha:[0,1]\rightarrow A.$ Let $\bar \alpha ...
- 1
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0
answers
126
views
A question about associated operator on continuous functions space equiped with L2 norm
For M a connected compact manifold, $T$ is in $C^{1+\nu}(M,M)$ with $\nu\in(0,1)$, i.e., $DT$ is some Hölder continuous function with Hölder exponent $\nu$. Denote by $m$ the Lebesgue measure on $M$ ...
- 39
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votes
0
answers
35
views
What is the impact of individual estimate on each other in matrix inversion?
I am looking to understand the impact of each estimate on each other in matrix inversion.
Lets say I have a vector $A = \left[a_1, a_2 \right]^T$ of size $2 \times 1$ and $a_1$ and $a_2$ are related ...
- 1
0
votes
0
answers
157
views
Dependence of functional integral on the function space
In physics, the following functional integral is considered
\begin{gather}
Z[J]= \int Df \exp(-\int d^dx( f\Box f+\lambda f^4 +Jf ))
\end{gather}
It is usually said that the integration is performed ...
- 593
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0
answers
59
views
Series representation of functions
Let $H$ be a Hilbert space, consisting of functions $f:\mathbb{R} \to \mathbb{R}$. Let
$$
V = \left\{ f_J \in H: f_J= \sum_{j=1}^J c_j^{(J)} g_j, c_j^{(J)}\in\mathbb R, J\in \mathbb N \right\}
$$
...
- 93
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votes
0
answers
163
views
Generalization of polynomial coefficients
I'm dealing with a hard combinatorial problem where for every positive integer value of a variable $n$ I have to calculate a list of numbers, specifically $n^2$, that depend on $n$ and its list index ...
- 47
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0
answers
81
views
Is a bounded measurable convex function above its interior lower semi-continuous convex envelope?
Let $E$ be a locally convex topological vector space, let $C$ be a convex set which matches the closure of its relative interior $\mathring C=\{ x\in C : \forall y\in C,\exists z\in C,~x\in\mathopen]y,...
- 204
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votes
0
answers
37
views
Finding an element of Gelfand triple with a designated time derivative
Let $V$ be a real separable Banach space and $H$ be a real separable Hilbert space such that
\begin{equation}
V \subset H \subset V'
\end{equation}
where $V'$ is the dual of $V$ and the inclusions are ...
- 3,477
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votes
0
answers
94
views
The asymptotic behaviour of the Fourier transform of a certain class of radially symmetric functions
Fix $\theta\in (-\pi/2,\pi/2)$ and let $a>0$. Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is analytic in $S:=\{z\in \mathbb{C}: |\arg{z}|<\pi/2\}$ and
$$|f(z)|\sim |z|^{-a},\qquad |z|\to \...
- 852
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votes
0
answers
51
views
Rescaling of cosine families
First of all, the best wishes for 2024. Recently, I got aware of cosine operator families (in the framework of evolution equations). It is well-known, that operator semigroups can be rescaled (see for ...
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answers
192
views
Reference request: an introduction to nuclear spaces
I am looking for a short introduction to nuclear spaces and nuclear operators. I am interested in these spaces as they often arise in mathematically rigorous quantum field theories. I have read the ...
- 721
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answers
56
views
Why is von Neumann inequality important for equivalence of $\forall_j \ T_j^n\rightarrow 0$ in A-topology and abs continuity of $(T_1,\ldots, T_N)$?
The whole theorem goes as follows:
Let $(T_1, \ldots, T_N)$ be a tuple of commuting operators in Hilbert space $H$ satisfying:
$$\exists_{M > 0} \ : \ \forall_{p \in \mathbb{C}[z_1, \ldots, z_N]} \ ...
- 63
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votes
0
answers
98
views
Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?
(Cross posted from Math StackExchange: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?)
Assume $(\Omega, \mu)$ is a probability space. Consider a ...
- 2,830
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votes
0
answers
50
views
Kirszbraun-like extension of periodic functions
Let $\Lambda \subset \Lambda' \subset \mathbb{R}^n$ be lattices. Let $f : \Lambda' \rightarrow \mathcal{H}$ be a $a$-Lipschitz function, where $\mathcal{H}$ is a finite-dimensional Hilbert Space. ...
- 1
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answers
272
views
How to prove that the uniform limit of $C^k$ functions is $C^{k-1,1}$?
Already asked in SE but no response, I think it also reasonably belongs here.
https://math.stackexchange.com/questions/4829428/uniform-convergence-of-ck-functions
Basically what the title says, plus ...
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0
answers
252
views
Self-adjoint operator with pure point spectrum
Suppose that A is a self-adjoint (possible unbounded) operator from a separable Hilbert space H to itself. I would like to know if the following statement is true:
A has pure point spectrum (i.e., the ...
- 43
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answers
49
views
Reference needed for powers of semi-group generators
Let $\mathcal{L}$ be the infinitesimal generator of a Markov semi-group. I am looking for references that study powers of $\mathcal{L}$; i.e. $\mathcal{L}^n$, for $n\in\mathbb{N}$.
For example, if the ...
- 90
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votes
0
answers
42
views
Measurability of the weak completion of an orthogonal representation
Let $G$ be a locally compact group and let $\pi$ be a strongly continuous orthogonal representation of $G$ in a real Hilbert space $H$. Denote by $E$ the real Hausdorff locally convex space obtained ...
- 407
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votes
0
answers
67
views
Constants in the entropy number of the Sobolev space
For a Sobolev space with $W^s(\Omega)$, where $\Omega\subset R^d$ is a compact space with smooth boundary, we know that the entropy number satisfies $e(\delta, W^s(\Omega, 1),\|\cdot\|_{L_\infty})\leq ...
- 39
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votes
0
answers
48
views
Generator of contraction semigroup
Assume $A$ is a symmetric operator in a Hilbert space, which generates a contraction semigroup (a priori it is not known, whether this semigroup is self-adjoint). Is A then self-adjoint?
- 41
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0
answers
63
views
Chapter 2, Section 5 of Chavel's book “Eigenvalue In Riemann Geometry" is about the zero-point distribution of the derivatives of eigenfunctions
In Chapter 2, Section 5 of Chavel's book, regarding the Neumann eigenvalues of the Laplacian in space forms, how did Chavel determine that $T'_{l,j}$ has ($j-1$) zeros? I have consulted books on the ...
- 11
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votes
0
answers
97
views
Amenability of $\textrm{w}_0(A)$ for a $C^*$-algebra $A$
Let $A$ be a $C^*$-algebra with only finite dimensional irreducible representations. As in a previous question, let $\textrm{w}_0(A)$ denote the subspace of $\ell^{\infty}(A)$ consisting of all weakly ...
- 2,605
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votes
0
answers
107
views
Evaluating a matrix Pick function via its integral representation
In the proof of Theorem 3.1 of the paper Inequalities for M-matrices, Ando evaluates a matrix function (see equation boxed in orange below) via an integral representation of a Pick function (see ...
- 1,719
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votes
0
answers
146
views
Non-degenerate representation of a Banach algebra
Let $\mathcal{A}$ be a non-reflexive Banach algebra. For the definition of Arens product, please refer to this link. Here we let $\square$ denote the first Arens product and $\diamond$ denote the ...
- 307
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votes
0
answers
94
views
When can an affine functional on the dual be represented as an element of a Banach space?
In Measures Which Agree on Balls by Hoffmann-Jørgenson, we are given a functional $\varphi: T(x_0)\to (-\infty, \infty]$, which is a lower semicontinuous, affine, Baire function on a subspace $T(x_0)$ ...
- 297
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answers
145
views
$L_\infty([0,1], \mathbb{C})$ is it isomorphic to $\ell_\infty(\mathbb{N}, \mathbb{C})$?
By a result of Pełczyński, $L_\infty([0,1], \mathbb{R})$ is isomorphic to $\ell_\infty(\mathbb{N}, \mathbb{R})$. That is the case of real valued functions and sequences.
A natural question then is: ...
- 141
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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
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votes
0
answers
115
views
Existence of Green functions and some properties
Let $\Omega$ be a smooth domain in $\mathbb{R}^N$， $N\geq 3$, $p\in \Omega$ is a fixed point, $\lambda$ is a parameter (can be 0,>0,<0), if there exisits a Green function $G_{\lambda}(x,p)$ ...
- 471
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votes
0
answers
119
views
About definition of stable solution. $Q_u(\phi) \ge 0$ for all $\phi \in C_c^1(\Omega)$ replaced by "for all $\phi \in W_0^{1,2}(\Omega)$"
I want to ask about a remark about the stable solution of elliptic PDE Remark 1.1.1.
We say $u$ is stable solution of $-\Delta u=f(u) \ \text { in } \Omega$ and $u=0$ on $\partial \Omega$ if it ...
- 809
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
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votes
0
answers
138
views
Question about a step in the proof of the min-max principle
I honestly do not think this is a hard question, maybe it is even obvious but I tried MSE and had no success so far, so I am reproducing the question Question about the proof of the min-max principle ...
- 1,305
0
votes
0
answers
80
views
Continuity of linear map on tensor product spaces with different norm properties
I originally asked this question on StackExchange, but I think that it may be more suitable to here.
Let $V$ and $U$ be Banach spaces. I'm considering a linear map $\phi: V \rightarrow U$, and ...
- 177
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votes
0
answers
51
views
Reparameterizing a function to be linearly bounded
Trying to find a reparameterization of a function from $f(y, z, \ldots)(x)$ to $f(y(a_1), z(a_2), \ldots)(x)$ so that for all $x \in [r, t]$ we have
$$
|f(y(a_1), z(a_2), \dots)(x) - f(y(b_1), z(b_2), ...
- 101
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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
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0
answers
75
views
Goldstine theorem in quasi-Banach spaces
A classical theorem of Goldstine is the following: Let $X$ be a Banach space and $J \colon X \to X''$ the natural inclusion. Then $J(B_X)$ is $\sigma(X'', X')$-dense in $B_{X''}$, where $B_Y$ is the ...
0
votes
0
answers
56
views
Zero flux along lines
I am considering the $L^1$ ball in $\mathbb{R}^d$, and a conservative vector field $V$ on it, which arises as the gradient of a bounded, almost-everywhere Lipchitz-function. Denoting by $e_i$ as the i’...
- 232
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votes
0
answers
47
views
Is the embedding $i: (L^p_\text{loc} (Y), \| \cdot \|_{L^p_\text{loc}}) \to (L^0(Y), \hat \rho)$ continuous or Borel measurable?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...
- 657
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votes
0
answers
113
views
The set of measurable functions together with convergence in measure is a completely metrizable abelian topological group
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple ...
- 657
0
votes
0
answers
96
views
Closed formula for Laguerre
Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$.
Assume $0<\beta<1$. Is there a closed formula for this sum
$$\sum^\infty_{j=0} \frac{1}{(b+j)^{1-\beta}}L^{m}_j(x)$$
where $b>0$ and $...
- 521
0
votes
0
answers
48
views
A direct proof for non-zero limit points of weighted backward shifts
Fix a sequence $(w_1,w_2,\ldots)$ of positive reals such that the linear operator $T: \ell_2\to \ell_2$ given by
$$
T(x_1,x_2,x_3,....)=(w_2x_2,w_3x_3,\ldots) \text{ for all sequences in } \ell_2
$$
...
- 1,511
0
votes
2
answers
125
views
Is there a modification of $f$ on a null set such that $F: [0, T] \to L^p ({\mathbb R}^d), t \mapsto f(t,\cdot)$ is Bochner measurable?
Let $T>0$ and $p \in [1, \infty)$. Let $f \in L^p ([0, T] \times {\mathbb R}^d)$. By a theorem in this thread, there is a Lebesgue null subset $N$ of $[0, T]$ such that $f(t, \cdot)$ is Lebesgue ...
- 825
0
votes
0
answers
145
views
Why is this function in $L^1$?
I had a question about a claim made in the paper "Group Invariant Scattering" and why it is true. Consider the function $h_j(x) = 2^{nj}\psi(2^jx)$, where $\psi$ is a function such that $\...
- 101
0
votes
0
answers
149
views
Validity of Hölder inequality for the homogeneous Besov spaces $\dot{B}^0_{1,2}(\mathbb{R}^n)$ and $\dot{B}^0_{2,2}(\mathbb{R}^n)=L^2(\mathbb{R}^n)$
I am looking at Corollary 1. in p.244-245 of the book
"Sobolev Spaces of Fractional Order,
Nemytskij Operators,
and Nonlinear
Partial Differential Equations" (1996) by Thomas Runst
Winfried ...
- 3,477
0
votes
0
answers
77
views
Property (H) in the dual norm
Consider the Hilbert space $l_2$ with an equivalent norm
$$\Vert x \Vert = \max \{2 \Vert x \Vert_1, \Vert x \Vert_2 \},$$
where $\Vert x \Vert_1 =( \sum_{n=2}^\infty x_n^2 )^{\frac{1}{2}}$ and $\Vert ...
- 85
0
votes
0
answers
99
views
Efficient method to determine minimum eigenvalue of $2 \times 2$ block diagonal matrix
Suppose $H$ is a $2 \times 2$ block-diagonal symmetric matrix in $\mathbb{R}^{2^N \times 2^N} $. That is
$$ H = \begin{pmatrix} A_1 & 0 & \cdots & 0\\ 0 & A_2 & \cdots & 0 \\
...
- 131
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
0
votes
0
answers
138
views
Under what conditions is $\lim_{x\to a}\left|\varphi\circ f(x)-\tau \circ g(x)\right|=0$ true?
This question is inspired from another much easier problem I was trying to solve which I tried to generalize. The question is essentially as follows (assuming all the limits exist)
If $a\in \mathbb R\...
- 791
0
votes
0
answers
355
views
On a Duality between Riemann-weil explicit formula and Abel- Plana summation of trigonometric prime counting function:
Consider the analytic function $g(x)$
Now define
$f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$
Such that
$|f(x+it)|=o(e^{2πt})$
uniformly for every $x$...
- 790