All Questions
10,934 questions
6
votes
0
answers
213
views
Hölder's inequality for trace-class maps of $p$-liquid spaces and a related conjecture of Grothendieck
In Condensed Math and Complex Geometry Proposition 8.8, Clausen-Scholze describe trace-class maps between projective objects in the $p$-liquid category as sums of rank 1 operators against ${<}p$-...
- 61
1
vote
0
answers
111
views
References on the partial trace
For the Hilbert space $H^N:=L((\mathbb R^{3})^N,\mathbb C)$, consider the projection operator $D: H^N\to H^N$ as follows :
$$D(\Phi):=\left(\int_{(\mathbb R^{3})^N}\overline{\Psi(x_1,\ldots, x_N)}\Phi(...
- 1,399
0
votes
1
answer
188
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Does the second Bourgain–Delbaen space belong to C_p?
The second Bourgain–Delbaen space $Y$ is a separable $\mathcal{L}_\infty$ space such that $Y$ contains no copy of $c_0$ or $\ell_1$. The dual space is isomorphic to $\ell_1$.
An operator $T:X\to Y$ ...
2
votes
0
answers
43
views
Distributions and time-kernels
Let $U\subset\mathbb{R}^{d}$ be an open subset and set $M:=I\times U$, where $I=(a,b)\subset\mathbb{R}$ is some open subset. Lets consider a linear operator $B:C^{\infty}_{c}(M)\to C^{\infty}(M)$ that ...
- 1,429
3
votes
0
answers
135
views
Possible gaps for a function and its Fourier transform
This is another question on the possible shape of sets $A,B\subset \mathbb{R}^d,d\geq 2,$ where resp. a non-null Schwarz function $f$ and its Fourier transform can vanish.
A nice remark by Christian ...
- 1,352
-1
votes
0
answers
94
views
Why define Schwartz by supremum rather than limit?
The Schwartz space is defined as the set of all indefinitely differentiable functions such that the supremum over the free variable of any (order) derivative times any (order) power is finite.
However,...
- 107
-1
votes
0
answers
53
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convergence of convolution in Bochner space
I want to prove a well-known fact in $L^p(R^n)$ namely that, the convolution of an element in $L^p$ with an element of $L^1$ is in $L^p$
let: if $u∈L^p (R;X) , f∈L^1 (R)$ and $X$ is Separable and ...
1
vote
1
answer
90
views
Sobolev inequality with weight in the case $1<n\leq p$
Assume that $1<n\leq p$. Does there exist a (non-negative) measure $\mu$ (preferably with some positive density function with respect to the Lebesue measure $dx$) and $q>p$ so that for all $f\in ...
- 459
0
votes
1
answer
93
views
A question on finite Fourier series
Let $\mathcal F(N)$ denote the space of finite Fourier series up to frequency $N > 0$, i.e. $f\in \mathcal F(N)$ if and only if it can be written as
$$f(x) = \sum_{k=0}^N a_k\cos(kx+\theta_k)$$
for ...
- 71
3
votes
0
answers
192
views
What smoothing to use for PNT-like results?
Consider a Dirichlet series $\sum_n a_n n^{-s}$ with desirable analytic properties (e.g., analytic extension to $\Re s>0$); one example would be $a_n=\mu(n)$. Say we want to estimate $\sum_{n\leq x}...
- 20.2k
1
vote
1
answer
117
views
Lower bound for a commutator trace
I have this Hilbert space of square-integrable complex-valued functions on a square, $\mathbb{L}^2([0,1]^2)$. And let $M_x$, $M_y$, and $M_{x+y} = M_x+M_y$ be the operators of multiplication by the ...
- 121
16
votes
1
answer
969
views
Pedagogically intuitive reformulation of Zorn's Lemma for functional analysis
While teaching an applied functional analysis class, I’ve noticed that students often struggle to develop an intuitive understanding of Zorn’s lemma. It’s relatively straightforward to explain why ...
- 5,824
0
votes
1
answer
169
views
Existence of a "universal" measure-preserving transformation on the unit interval
Let $I = [0,1]$ be the unit interval equipped with the Lebesgue measure $\lambda$. Let $\mathcal{M}$ be the set of all Lebesgue measure-preserving transformations $T: I \to I$. We say a transformation ...
user545331
3
votes
0
answers
155
views
Colimits in commutative Banach algebras?
Let $K$ be a complete non-Archimedean field. It is known that the category $\mathrm{Ban}_K$ of $K$-Banach spaces with bounded linear maps does not have infinite colimits. The usual argument for $\...
- 426
1
vote
2
answers
117
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If $f\in C([0,\infty))$, does $\delta>0$ and $g\in C^1((0,\delta))\cap C([0,\delta])$ s.t. $g\geq f$ on $[0,\delta]$ and $g(0)=f(0)$ exist?
The question is the following:
Suppose $f : [0,\infty) \rightarrow \mathbb{R}$ is a continuous function. Can I find $\delta \in (0,\infty)$ and a function $g : [0,\delta] \rightarrow \mathbb{R}$ such ...
- 309
2
votes
0
answers
82
views
The support of the functions in the closed span of the Rademacher functions in $L_1(0,1)$
Given a measurable function $f:(0,1)\to \mathbb{R}$, we denote by $M(f)$ the measure of the set $\{t\in (0,1) : f(t)\neq 0\}$.
It is not difficult to prove that if $(f_n)$ is a normalized sequence in $...
- 4,461
0
votes
0
answers
55
views
reference request: conditions for pointwise and operator-norm convergence of kernel projections
At a very high level, I’m interested in the following question. Suppose $X$ is a (separable) Hilbert space, and $T_n : X \rightarrow X$ is a sequence of finite rank self-adjoint maps that converges (...
- 101
7
votes
2
answers
394
views
Tangent space to infinite dimensional manifolds
In finite dimensional geometry, there is a single invariant of a vector space - its dimension. This characterizes finite dimensional manifolds as being glued from Euclidean balls.
This situation is ...
- 593
4
votes
1
answer
227
views
Problem in Probability Theory and Functional Analysis
Let's consider the vector space V of bounded scalar functions, which includes the constant function 1. We assume that any uniform limit of a bounded monotonic sequence of functions from V also ...
0
votes
1
answer
53
views
Exponentially weighted norms are not equivalent
Let $\|u\|^2_{L^2_\eta}$ be the exponentially weighted norm of the space of functions $u(x)$ for which $u(x)\mathrm{e}^{\eta\cdot x}$ with $\eta\in \mathbb{R}$ is in $L^2(\mathbb{R})$. How can I show ...
0
votes
0
answers
146
views
On the pointwise limit of a sequence of analytic functions
I have been confused with this problem for weeks now. Suppose I have Banach spaces $E$ and $F$ and a sequence of functions $f_{n}: U \subset E \to F$, where $U$ is open and nonempty. Let $x \in U$ be ...
0
votes
0
answers
50
views
Self-adjoint operators and index of quadratic form associated to it
Let $B$ a bounded self-adjoint operator on a real Hilbert space $H$ with an associated inner product $(\cdot,\cdot).$ Take $V=\operatorname{span}\{f_1, f_2, \ldots, f_n\}$ a finite dimensional ...
- 101
1
vote
0
answers
65
views
Fractional Sobolev embedding
Let $s\in (0,1)$ and $1<p<\infty$. Let $H^{s,p}(\mathbb{R}^n)=H^{s,p}$ the Bessel potential space, defined as the image of $L^p(\mathbb{R^n})$ by the Bessel potential. It is known that these ...
3
votes
1
answer
307
views
Approximate square root of Dirac delta function on $\mathrm{SL}_2(\mathbb{R})$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\AdS{AdS}$I hope to find a sequence of complex-valued functions $\{f_i(g)\}$ on the group element $g$ of a locally compact group $\SL(2,\mathbb{R})$ so ...
- 175
1
vote
0
answers
87
views
Proof mistake of: $M_0A(G) = B(G)$ for a locally compact group
I am posting my question of mathstack exchange here. (see: My post on MSE)
Let $G$ be a locally compact group with Haar measure $\mu$, and $B(G),A(G),C_r^*(G),L(G)$ be its Fourier-Stieltjes algebra, ...
- 261
2
votes
1
answer
78
views
Is there a relative projective tensor (cross-)norm for Banach $A$-algebras?
$\newcommand\norm[1]{\lVert#1\rVert}$I am interested in a relative version of the projective tensor product and projective tensor (cross-)norm for Banach algebras. Let $A$, $B$, $C$ be commutative (...
- 7,127
0
votes
1
answer
139
views
Existence of infinite rank compact operator
Given any separable Banach space $X$, we know that always there exists a Banach space $Y$ such that there is an injective compact operator from $X$ to $Y$. Can we show that given any infinite ...
- 585
1
vote
0
answers
52
views
Sufficient conditions for boundedness of Fourier transform
This should be a well studied topic: I am looking for sufficient conditions on a function $u(x)$ on $\mathbb{R}$ ensuring that its Fourier transform is bounded. Of course one such condition is $u\in L^...
- 9,007
1
vote
0
answers
127
views
Trace type convergence of the Laplacian on the box to the Laplacian on $\mathbb R^d$
Let $-\Delta \colon H^2(\mathbb R^d) \to \mathbb R^d$ be the (negative) Laplacian on the full space and $-\Delta_L$ the Laplacian acting on $L^2([-L,L]^d)$ with some boundary conditions making it self-...
- 61
1
vote
1
answer
183
views
Metric currents on singular measures in $\mathbb R^d$
Unless I am misunderstanding a lot of works, it is my understanding that a finite and non negative measure $\mu=g\mathcal{H}^\alpha$, where $\mathcal{H}^\alpha$ is the $\alpha$-Haudorff measure, ...
- 391
2
votes
0
answers
228
views
Any rigorous construction of $\phi^4$ theories without the mass term in the Lagrangian? (revised)
There are various papers on rigorous construction of massive $\phi^4$ theories in $2$ or $3$ Euclidean dimensions.
In 2D, there are in fact more general results such as this one by Glimm, Jaffe and ...
- 3,477
2
votes
0
answers
90
views
Representation of Dirac-delta distribution in subspace of functions
Suppose I have a subspace $V\subset L^2(\Omega)$ where $\Omega\subset \mathbb{R}^d$ is a bounded and closed set. $V$ is defined by
\begin{align}
V=\text{span}(\{\varphi_i(x): i=1,2,\dots,n\})
\end{...
- 93
4
votes
1
answer
195
views
Asymptotic spectrum of a complex Sturm-Liouville differential operator
Let $\varepsilon > 0$ and consider the (complex) Sturm–Liouville differential operator on $[0,1]$ given by
$$
\mathcal{L}_\varepsilon f(x) = \varepsilon^2 f''(x) + i V(x) f(x),
$$
with Neumann ...
- 333
4
votes
2
answers
389
views
Gaussian mixtures are dense in total variation?
Let $M_{TV}(\mathbb{R}^d)$ denote the set of probability measures on $\mathbb{R}^d$ with finite total variation norm which are absolutely continuous with respect to the Lebesgue measure.
By a Gaussian ...
- 5,405
1
vote
0
answers
104
views
Commutative Banach $\mathbb{R}$-algebras without complex structure, but with path-connected group of units
For a finite-dimensional commutative (associative, unital) $\mathbb{R}$-algebra $A$, the condition $\pi_0(A^\times) = 1$ (i.e. the group of units of $A$ being path-connected) is equivalent to $A$ ...
- 7,127
0
votes
0
answers
66
views
Taking trace of a tensor product of matrix-valued smooth functions on the thin diagonal
Let $V$ be a finite dimensional real / complex vector space and consider the space $L(V,V)$ of linear operators on $V$.
Fix $n \in \mathbb{N}$ and let $\mathcal{M}$ be the real / complex vector space ...
- 3,477
1
vote
0
answers
71
views
Integral formula of quantum dilogarithm
In the paper"Level N Teichmüller TQFT and Complex Chern-Simons Theory" arXiv:1612.06986, the authors study the quantum dilogarithm function:
\begin{equation}
\mathrm{D}_{\rm b}(x,n)=\prod_{...
- 109
3
votes
0
answers
122
views
Analytic functions and Hyperfunction as TVS
I have several related questions on Analytic functions and Hyperfunction as topological vector spaces (I am mainly interested in questions 4,6,10):
For an open set $U\subset \mathbb C^n$ we can ...
- 2,649
0
votes
0
answers
105
views
Generalizing the property of linear independent set in infinite dimensional TVS
Given a infinite dimensional Hilbert space $H$, and a countable set of vectors $\{v_{i}\}_{i=1}^{\infty}$. I want to study the following property of $\{v_{i}\}_{i=1}^{\infty}$:
There exists sequences $...
- 523
3
votes
0
answers
144
views
Minimizing vertical integral of a Mellin transform
Let $\eta:[0,\infty)\to [0,\infty)$ satisfy $\eta(0)=1$ and $\int_0^\infty \eta(x) dx = 1$ (say).
Write $M\eta$ for the Mellin transform of $\eta$. Let $\epsilon>0$ be small.
What is the choice of $...
- 20.2k
9
votes
0
answers
163
views
Moore-Penrose partial isometries and hermitian elements
Let $A$ be a unital Banach algebra. An element $a \in A$ is hermitian if $\|\mathrm{exp}(ita)\|=1$ for every $t \in \mathbb{R}$. An element $a \in A$ is Moore-Penrose invertible if there exists $b \in ...
- 3,497
3
votes
1
answer
91
views
Conditional Expectation in Diffusion Process
Consider a $d$-dimensional diffusion process $\mathbf{X}=(\mathbf{X}_t)_{t\in [0,T]}=([X^1_t,...,X^d_t])_{t\in [0,T]}$ that is the unique strong solution of the following SDE:
$$\left\{\begin{matrix}
...
- 143
0
votes
0
answers
57
views
Double-periodic functions with (possible) poles
Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a ...
1
vote
1
answer
100
views
Is Nelson-Symanzik positivity compatible with fermionic statistics?
Let $\{ S_n \}_{n =0}^\infty$ be a sequence of tempered distributions where $S_n \in \mathcal{S}'(\mathbb{R}^{nd})$ where $d \in \{2,3,4\}$ is fixed. Moreover, we put three additional conditions:
$...
- 3,477
1
vote
0
answers
38
views
About Carleson measures on the Hardy space on the bidisc
I have been reading the paper "Carleson Measures in Hardy and Weighted Bergman Spaces of Polydiscs" by F. Jafari and there are a few things that going on that I am not entirely convinced of.
...
2
votes
1
answer
124
views
Choice of the eigenbasis for the Dirac operator on $S^d$
This question is a simplified version of my previous one. I think that adding a gauge potential complicates the problem too much.
Let us consider the Dirac operator $D$ on the $d$-sphere $S^d$ with ...
- 3,477
1
vote
1
answer
129
views
Is every operator range a Baire space in the relative topology?
Let $X$ be a Banach space and let $U\subseteq X$ be a (not necessarily closed) linear subspace. One says that $U$ is an operator range if there is another Banach space $E$, and a bounded linear map $...
- 483
7
votes
1
answer
415
views
Is there a “Closure-of-Range Theorem” for Banach spaces?
The classic Closed Range theorem states that for a linear bounded operator $T:X\to Y$ between Banach spaces, and its transpose $T^*:Y^*\to X^*$, the four conditions:
$T(X)$ is $s$-closed; $T(X)$ is $...
- 60.5k
1
vote
0
answers
45
views
Existence of optimal entropic weights for empirical modeling
Let $\mathcal{X} = [0,1]^n$ be the input space and $\mathcal{Y} = \{1, ..., n_c\}$ be a discrete output space. Let $D = \{(x_i, y_i)\}_{i=1}^N \subset \mathcal{X} \times \mathcal{Y}$ be a training ...
- 111
4
votes
1
answer
162
views
Topology on $O_M$, the space of slowly increasing smooth functions?
A smooth function on $\mathbb{R}^n$ is called slowly increasing if each of its derivatives is polynomially bounded. It seems that the collection of such functions is denoted as $O_M$.
Obviously, $O_M$ ...
- 3,477