All Questions
3,630 questions with no upvoted or accepted answers
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67
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Order of ultradistribution
I know that the order of any distribution of compact support is finite. Is this true in the case of ultra distribution of compact support ( dual of Denjoy-Carleman space)?
- 75
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84
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A Riemann Hilbert problem on the unit square
Let $p_0=0$, $p_1=1$, $p_2=1+i$ and $p_3=i$ be the four vertices of a square $Q$ on the complex plane $\mathbb C$.
Let $f \in C^{\infty}_c((0,1))$ and consider the following Riemann-Hilbert problem on ...
- 4,153
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73
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The $w^{*}$-convergent sequences and the Mackey topology on $X^{*}$
Let $X$ be a Banach space. Recall that the Mackey topology $\mu(X^{*},X)$ on $X^{*}$ is the topology of uniform convergence on weakly compact subsets of $X$. Let $(x^{*}_{n})_{n}$ be a sequence in $X^{...
- 3,343
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56
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Moduli of continuity and Wasserstein differentiability of functions between measures
Let $X=\mathbb{R}^n$; I am also interested in the general case $X$ is a metric space but for simplicity let's focus on Euclidean space. Let $\mathcal{P}(X)$ denote the space of Borel probability ...
- 11
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72
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Multivarate "RKHS" Examples
I've been reading about RKHSs and Hilbert spaces of functions these days a bit these days and I haven't yet come across an example of a hilbert space $H$ whose elements are all functions $f:\mathbb{R}^...
- 5,405
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Toeplitz operators for other measures then Lebegue
In the standard setting there is a lot known about Toeplitz operators i.e that the compression of a multiplication operator restricted to the Hardy space.
Are there any results when one has a ...
- 243
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139
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Property $(\mathcal{L}(\phi),\phi)\geq 0$ about a operator $\mathcal{L}$
Consider the operator $\mathcal{L} : H^2(\mathbb{T}_L) \subset L^2(\mathbb{T}_L) \longrightarrow L^2(\mathbb{T}_L)$ given by
$$\mathcal{L} = -\omega \partial_x^2+3\varphi^2-1,$$
that is
$$\mathcal{L}(...
- 205
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79
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Conditions on triangle inequality for integral kernel
Consider $\int_RK(x,y)f(y)dy$, where $K(x,y) \in M_+(R^2)$.
Let $L(t,s)$ be an iterated rearrangement of $K$. Let also $$
A(t,v)=\int_0^{1/v}L(1/t,s)ds,
$$
which is decreasing with $v$ and ...
- 343
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On some bounds on two constants concerning the disconnectedness of the spectra of small perturbations of operators
Let $H$ be a separable, infinite dimensional, complex Hilbert space. In the book:
Jiang, C. L.; Wang, Z. Y. (1998). Strongly Irreducible Operators on Hilbert
Space. CRC press
above the statement of ...
- 1,175
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181
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Eigenvalues of product of operators
Let $A,B$ be two Trace class operators with spectral decomposition $\sum_{j\geq 1} \lambda_j \phi_j(\cdot)\otimes \phi_j(\cdot)$ and $\sum_{j\geq 1} \gamma_j \psi_j(\cdot)\otimes \psi_j(\cdot)$ ...
- 519
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294
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Regarding subspace generated by the polynomial multiples of outer functions
Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The ...
- 31
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115
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Algorithm/iterative procedure for constructing hypercyclic vectors?
Let $B$ be a separable Banach space and let $L:B\rightarrow B$ be a hypercyclic operator; here I use the definition of hypercyclicity given implicitly by Birkhoff's Transitivity Theorem: continuous ...
- 5,405
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353
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Eigenvalues of convolution matrices
Let $h: \mathbb{R}\to \mathbb{R}$ be a smooth function. Fix $0\leq s_1\leq \cdots \leq s_m\leq 1$ and $0\leq t_1\leq \cdots \leq t_n\leq 1$. Construct $A\in \mathbb{R}^{m\times n}$ by letting $A_{i,j}:...
- 315
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147
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Using Paley-Wiener Theorem to prove the decay of $G(x-y)$
This question is related to my previous one, where I was looking for some help to prove the decay of the lattice Green function:
\begin{eqnarray}
G(x-y) = \int_{[-\pi,\pi]^{d}}\frac{d^{d}k}{(2\pi)^{d}}...
- 1,305
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74
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Empty Weyl/Fredholm spectrum of an operator on an infinite dimensional Banach space
Let $X$ be a complex infinite dimensional Banach space, and let $T \in B(X)$ be nonscalar. The Fredholm spectrum of $T$ is defined by:
$$ \sigma_{\Phi} (T) := \lbrace \lambda \in \mathbb{C} : T- \...
- 1,175
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Minimax theorems in nonconvex setting
Let $X$ be a topological space, $Z$ be a compact convex subset of $\mathbb R^m$, and let $f:X \times Z \to \mathbb R$ be a continuous function (w.r.t the product topology on $X \times Z$).
Question. ...
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126
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Does my functional satisfy the Palais Smale condition?
Consider the functional
$$ I(u)=\frac{1}{2} \int_\Omega |\nabla u|^2\ dx + \frac{1}{4} \int_\Omega (1-|u|^2)^2 \ dx - \frac{c}{2} \int_\Omega \langle i\partial_1 u , u\rangle ,$$
where $u:\mathbb{R}^2 ...
- 383
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277
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Why the name `Lipschitz-Free Banach spaces'?
There are many names for the same objects that is known as the Arens--Eells spaces, transportation cost spaces, free Banach spaces over a (pointed) metric space, and Lipschitz-free Banach spaces.
The ...
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379
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Creating an inverse system which "stratifies density"
Setting:
Let $X'$ be a dense subset of an infinite-dimensional Fréchet space $X$ and suppose that $(X_n')_{n \in \mathbb{N}}$ is a nested sequence of non-empty subsets of $X'$ satisfying
$$
\bigcup_{n ...
- 5,405
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139
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Any reflexive space has the property of Banach-Saks?
We say that a Banach space $(X,\|.\|)$ has the Banach-Saks property if every bounded sequence $(x_m)_m$ in $X$ admits a subsequence $(x_{m_n})_n$ which converges in the sense of Cesàro, that is, there ...
- 115
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66
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Outer-regular product of $\tau$-additive measures
Due to the deficiencies of the simple product measure defined on measurable rectangles, there have been many different constructions of product measures in more specialized circumstances.
Originally, ...
- 3,816
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54
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Constrain representation of tempered distribution
This is a follow-up to this question.
Let $T$ be a tempered distribution on $\mathbb{R}^d$.
Then there is a multiindex $\alpha \in \mathbb{N}_0^d$, an $n \in \mathbb{N}_0$ and a bounded continuous ...
- 651
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153
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Maximal ideal space of $\ell^\infty(A)$, $A$ a commutative unital Banach algbera
Let $A$ be a commutative unital complex Banach algebra with norm $\|\cdot\|_A$, and let $\ell^\infty(A)$ denote all bounded sequences $(a_n)_{n\in \mathbb{N}}$ with $a_n\in A$, $n\in \mathbb{N}$, with ...
- 21
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236
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Fredholmness of elliptic operator on Hölder spaces
Let $(M,g)$ be a smooth oriented closed Riemannian manifold, $E\to M$ a smooth vector bundle, and $C^{k,\alpha}(E)$ the Banach space of sections of $E$ that are $k$-times differentiable (with respect ...
- 163
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36
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Does a total preorder on lotteries that preserves countable mixtures preserve arbitrary mixtures?
Let $X$ be a countable set. A lottery on $X$ is a function $\lambda: X \to [0,1]$ such that $\sum_x \lambda(x) = 1$. Let $\Delta X$ be the set of lotteries on $X$.
A total preorder $\preceq$ on $\...
- 869
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30
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Hypercylic operators with sets of hypercyclic vectors almost covering the space
Let $\{T_i\}_{i \in I}$ be a family of hypercylic operators on a separable Banach space $X$. From the transitivity theorem, we know that $HC(T_i)$, the set of vectors $x \in X$ with $\{T_i^n(x):n \in ...
- 5,405
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55
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$ \{x\in X:h(x)\leq r\} $ is sequentially compact subset of $X$?
Let $(X,\|.\|)$ be a reflexive Banach space and $(D,\|.\|)$ be a Suslin subspace of $X$ such that $D$ is weakly closed subset of $X$.
Take $h:X\to [0,+\infty]$ such that $h(x)=\|x\|$ if $x\in D$ and ...
- 117
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106
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strict convexity of the Legendre-Fenchel transform
Let $d$ be a positive integer.
Let $L:\mathbb{R}^d\to\mathbb{R}$ be a differentiable function with continuous derivatives.
Assume that the Legendre-Fenchel transform of $L$ exists everywhere, is ...
- 11
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126
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Continuity of Helmholtz-Hodge projection in $H^1(\Omega)$
Let $\Omega \subset \mathbb{R}^d$ (for simplicity $d = 2$ or $3$) be a bounded Lipschitz domain. For any vector-valued function $\mathbf{f} \in \mathbf{L}^2(\Omega):= \left ( L^2(\Omega) \right )^d$, ...
- 23
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175
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Compute Frobenius inner product of two tensor-trains in terms of tensor contractions
Let $p\in\mathbb N$, $n\in\mathbb N^p$ and identify the Hilbert space tensor $\bigotimes_{k=1}^p\mathbb R^{n_k}$ with $\mathbb R^{n_1\times\cdots\times n_p}$ (equipped with the Euclidean inner product)...
- 167
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312
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History of microlocal analysis
Was the study of pseudo-differential operators the basis for the birth of microlocal analysis? I'm trying to find out how this branch of analysis was developed...
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231
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Dual space of mean-free Sobolev space
I am considering the space $V:=\{v \in H^1(\Omega): \int_\Omega v = 0\}$ of mean free functions. What is the dual space of this space? Is the dual space given by $D:= \{f \in (H^1(\Omega))^*: \langle ...
- 11
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Can we say that $f$ admits a $m(X,X^*)$-continuous extension to $X$?
Let $X$ be a Banach space equipped with the Mackey topology $m(X,X^*)$. We suppose that $\big(X,m(X,X^*)\big)$ is separable space. Let $H$ be a countable, $m(X,X^*)$-dense subset with $(H=-H)$.
Let $...
- 117
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183
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G-abelian systems
Let $(\mathfrak{A},\alpha,\phi)$ be a $C^*$-dynamical system made of a unital $C^*$-algebra, a $*$-automorphism and an extremal invariant (i.e. ergodic) state.
Consider the covariant GNS ...
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55
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Operational quantities characterizing upper semi-Fredholm operators
An operator $T:X\rightarrow Y$ is said to be upper semi-Fredholm if its range is closed and its kernel is finite-dimensional. M. Schechter (1972) introduced a quantity $$\nu(T):=\sup_{\operatorname{...
- 3,343
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Show that a tensor-train is contained in a recursive sequence of subspaces
Let
$p\in\mathbb N$;
$n_k\in\mathbb N$ and $\left(e^{(k)}_1,\ldots,e^{(k)}_{n_k}\right)$ denote the standard basis of $\mathbb R^{n_k}$ for $k\in\{1,\ldots,p\}$;
$u\in\bigotimes_{k=1}^p\mathbb R^{n_k}...
- 167
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Determining the behavior of a contraction mapping with undefined points
Label $X$ as the real interval $[0, a]$ where $a \in \mathbb{R}^+$, so that $\text{int}(X) = (0, a)$ labels the interior of $X$ and $\partial X$ labels the boundary of $X$. I have a function $f:\text{...
- 1,087
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75
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"Constructive" proof that compact sets $K\subseteq L_1(\mu\times\nu)$ are contained in products $S\widehat{\otimes} T$
A.Defant and K.Floret in chapter 7 of their Tensor Norms and Operator Ideals prove the equality
$$
L_1(\mu\times\nu)\cong L_1(\mu)\widehat{\otimes}L_1(\nu)
$$
for measures $\mu$ and $\nu$. At the same ...
- 7,426
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148
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Spectrum of Laplacian-like operator
Let $\kappa_1, \kappa_2>0$ be fixed.
Consider the unbounded operator $A: D(A) \rightarrow L^2(-1,1)\times\mathbb{R}$ defined by
$$
A\begin{bmatrix} y \\ h \end{bmatrix} = \begin{bmatrix} \...
- 309
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203
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Construction of weight function to satisfy condition on given functional
Consider the following function :
$$F(z) = \omega(z){\sin^2\left(\frac{c\Gamma(z)}{z}\right)}$$
Here, $\omega(z)$ is a weight we are going to consider
The following two conditions should meet for $\...
- 375
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39
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$C^2$-control using orthonormal frame on a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold. Let $E=(E_1,\dots,E_n)$ be an orthonormal frame for $M$.
So for $M$ itself we have a natural $C^k$-norm $\|f\|_{C^k_g(M)}:=\max\limits_{1\le m\le k}\sup\limits_{...
- 938
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109
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Is this a positive definite kernel?
Under which conditions on the function :
\begin{array}{l|rcl}
K : & \mathbb R^+ & \longrightarrow & (0, 1)\\
&t & \longmapsto & K(t) \end{array}
is the symmetric ...
1
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0
answers
67
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Comparison of inductive limit topology with very-rapidly decaying non-convex $L^{!/2}$-space topology
This is related to these posts and here.
Let $L^1([n,n+1])$ denote the subspace of $L^p$-functions on $[0,\infty)$ essentially supported on $[-n,n]$. Denote the accelerated $\ell^1$-direct sum ...
- 5,405
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0
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64
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Bound on number of linearly independent eigenvectors of adjoint of composition operator
Fix $N>1$. Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ be such that the composition operator via
$$
\begin{aligned}
C_f:C^{\infty}(\mathbb{R},\mathbb{R}) &\rightarrow C^{\infty}(\mathbb{R},\...
- 5,405
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0
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114
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Spins in classical statistical mechanics
I'm reading Kupiainen's notes on the renormalization group and also caught my attention. Actually, this is something that often causes my some confusion. On page 43, in the section about Ginzburg-...
- 3,535
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102
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Given a finite set of points, does there exist a linear function pass through a point and strictly below the other points for all the points?
I guess my question is a follow up question of this one: usul, Existence of a strictly convex function interpolating given gradients and values, version: 2019-04-13.
In usul's question, the answer ...
- 29
1
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0
answers
104
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"Global" topologies between compact convergence and uniform convergence
Let $X$ and $Y$ be locally compact (but not compact), second countable, Hausdorff spaces with $Y$ metric. It is easy to see that the topology of compact convergence is weaker than the topology of ...
- 5,405
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0
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102
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Commutator estimates for $-(-\Delta)^s$, with $s \in (1,2)$
I'm currently trying to work with the non-local operator given by
$$
(-\Delta)^{\frac{s}{2}}f(x)= c_s\text{P.V} \int_{-\infty}^\infty \frac{-f(x+y)-f(x-y)+2f(x)}{|y|^{1+s}} dy,
$$
where $f :\mathbb ...
- 446
1
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0
answers
179
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Polar decomposition of the Volterra integral operator
Repost of this Math.SE question due to a lack of answers (No one was able to help me find the closed form of $U_T$ and $|T|$ after two bounties). I also searched extensively online but couldn't find ...
- 67
1
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0
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158
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Regularity theory for parabolic PDEs in fractional Sobolev spaces
I am trying studying on my own parabolic PDEs throughout the book "Linear and Quasi-linear Equations of Parabolic Type" by Vsevolod A. Solonnikov, Nina Uraltseva, Olga Ladyzhenskaya and the ...
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