All Questions
951 questions
2
votes
1
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61
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$K *g_n$ converges in the topology of smooth functions, $K$ approximates $\delta(x)$ and $g_n$ is a.e convergent to $g$, then regularity of $g$?
This question is continuation from If $K *g_n$ converges in the Fréchet topology of smooth functions and $K$ approximates $\delta(x)$, is $g_n$ itself convergent? - revised.
As before, let us ...
- 3,477
2
votes
0
answers
194
views
Extension of universal approximation theorem
Let $I_d:=[0,1]^d$ with $d\ge 2$. Define $C(I_d):=\{F: I_d\to\mathbb R \mbox{ is continuous}\}$ and
$$N(I_d):=\{F\in C(I_d): F(x)=\sum_{k=1}^n f_k(v_k\cdot x), \mbox{ where } n\ge 1 \mbox{ and } f_1,\...
user128095
2
votes
1
answer
166
views
Cocompact lattices in $\mathrm{Sp}(n, 1)$
This is a continuation from my previous question. I am reading the following paper of Cowling-Haagerup, and I was wondering whether there are uniform lattices in $\mathrm{Sp}(n, 1)$. Is there some way ...
- 131
2
votes
1
answer
453
views
Weak convergence of probability measures on weak versus strong dual
The space of temperate distributions $S'(\mathbb{R}^d)$ is often equipped with the weak-$\ast$ or with the strong topology. When defining the notion of a probability measure on $S'(\mathbb{R}^d)$, ...
2
votes
0
answers
279
views
Relationship between $p$-capacity and Riesz $s$-capacity of a set
What is the relationship between the definitions of $s$-capacity (page 13 here) and $p$-capacity (here) of a set?
Are they equivalent? If not, what inequalities hold? What is the difference (in terms ...
- 839
2
votes
0
answers
136
views
Linear independence of functions
Let $x_1,x_2,\ldots,x_n\in\mathbb{R}^d$ be points so that no one point is in the positive span of another. That is, there is no pair of points $x_i,x_j$ such that $x_i=\alpha x_j$ for a positive ...
- 859
2
votes
2
answers
223
views
Relating function value to $L^2$ norm in Holder space
Suppose that $f \colon [0, 1] \to \mathbb{R}$ is $k$ times continuously differentiable and Holder in the sense that for some
$t = k + \beta$, where $\beta \in (0, 1]$ and $k$ is a nonnegative integer ...
- 420
2
votes
2
answers
457
views
A Fixed point Theorem that does not need the convexity of set valued map?
I am looking for a fixed point theorem for set valued maps that does not assume the set valued map should be convex valued.
Something like contractiblity or other properties can be replaced with ...
- 383
2
votes
1
answer
228
views
Given an eigenvalue equation (elliptic PDE) in a ball $B_R$, prove the convergence of the first nonzero $\lambda_R$ and its eigenfunction $\phi_R$
Let $H: \mathbf{R}^n \rightarrow \mathbf{R}$ be a bounded continuous function. Set
$$\tag{1}
\int_{\mathbf{R}^n}\left\{|\nabla \xi|^2+H(x) \xi^2\right\} \mathrm{d} x \geqslant 0, \quad \forall \xi \in ...
- 809
2
votes
1
answer
265
views
characterization of normality by selection theorem
The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...
- 121
2
votes
1
answer
238
views
Hilbert-irreducible Banach space
A Banach space $X$ is called Hilbert-irreducible if it satisfies the following condition:
If a subspace $Y\subset X$ satisfies the parallelogram equality, then $Y$ is necessarilly a one ...
- 356
2
votes
0
answers
193
views
If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dotsb+ B_n'$ also satisfies the same inequality
Related: On a deceptively tricky calculus problem.
The way that Leonard Gross proves the log Sobolev inequality is in the following stages:
He proves that for any operator $B$ that satisfies the log ...
- 90
2
votes
1
answer
336
views
Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$?
Let
$\Omega$ be a metric space,
$C_b(\Omega)$ the space of all real-valued bounded continuous functions on $\Omega$, and
$\mathcal{M}(\Omega)$ the space of all finite signed Borel measures on $\Omega$...
- 657
2
votes
3
answers
3k
views
dual space of a subspace of the space of bounded measures
Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Equipped with the weak convergence, the dual space of $\mathcal{M}$ is $\mathcal{C}_b(\mathbb{R})$ consisting of continuous ...
- 1,835
2
votes
1
answer
296
views
An abstract characterisation of weak* topologies
Is there a way of endowing the unit ball $B_X$ of a Banach space $X$ (we may assume that $X$ is an AL space, if that helps) with a topology $\tau$, so that $\tau=\sigma(Y^*,Y)$ (the weak* topology) if ...
- 153
2
votes
2
answers
528
views
Characterizations of amenable groups which use the space $\ell_1(G)$ and convolution
Let $G$ be a discrete group.
Do you know characterizations of amenable groups which use the space $\ell_1(G)$ and convolution?
I only know Johnson's theorem:
A group is amenable if and only if the ...
2
votes
0
answers
138
views
Smooth derivations of a Banach space
Let $E$ be a real (or complex) Banach space. By $C^\infty(E) $ we mean the space of all functions $f:E\to \mathbb{R}(f:E\to \mathbb{C})$ which are smooth in the sense of Frechet diffetentiability. A ...
- 356
2
votes
1
answer
315
views
Parabolic PDE Long Time Asymptotics and Elliptic Operator Spectrum
How does one show directly that the solution following parabolic partial differential equation (PDE) of $p(t,v)$ approaches its stationary solution which is a solution of an elliptic partial ...
- 2,239
2
votes
1
answer
142
views
Estimating a solution to an Euler-type ODE
Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer and $a$ be a real number.
Let $u(r)$ be a function on $[1,\infty)$ ...
- 969
2
votes
2
answers
602
views
Image of the trace operator on W^{1,1}
Let $\Omega \subset R^n$ be a bounded region with Lipschitz boundary. Is the trace operator $T: W^{1,1}(\Omega)\rightarrow L^1(\partial \Omega)$ surjective? If not, what is the image?
- 21
2
votes
1
answer
155
views
Tempered distributions at non-coinciding points and density of Schwartz functions
In the previous question, I find that situation is much less favorable than expected…. So I add more details to focus on the specific case I have in mind.
Let us consider the Schwartz space $\mathcal{...
- 3,477
2
votes
4
answers
3k
views
Splitting a space into positive and negative parts
Let $V$ be a vector space over $\mathbb R$. A symmetric bilinear pairing on $V$ is a linear map $a: V\otimes V \to \mathbb R$. Because $\mathbb R$ is characteristic not-two, I will freely confuse ...
- 54.6k
2
votes
2
answers
867
views
Decomposition of an abelian von Neumann algebra
Hi, I came across the statement below and I couldn't figure out why it is true. I was hoping someone could explain it or give me a good reference. Thank you in advance.
"Let $\pi$ be a non-degenerate ...
2
votes
0
answers
71
views
Gluing together mixed normed vector spaces with mixed topologies
This is a variant of this question.
Definitions/Facts
$Ball_1(X)$ denotes the unit ball (about $0$) in a normed vector space $X$.
MixTop of triples of pairs $(X,\tau)$ of normed vector spaces $X$ ...
- 5,405
2
votes
1
answer
702
views
Correction term in the relation between the Itō and Stratonovich integrals in Hilbert spaces
I'm reading the paper On the relation between the Itō and Stratonovich integrals in Hilbert spaces and there is something I don't understand.
In the notation of the paper, let
$H,H_1$ be separable $\...
- 167
2
votes
0
answers
83
views
Reference request for (co-)free constructions
Following a comment of user131781, posted to an answer of this question on MO, I am looking for references to the construction of (co)-free functors from categories into the category of Banach spaces ...
- 5,405
2
votes
1
answer
471
views
Takesaki II proposition 3.15 proof about modular automorphism groups: mistake in book?
Consider the following fragment from Takesaki's second volume "Theory of operator algebras" in chapter VIII "Modular automorphism groups" p122-123.
Why is it possible to choose an ...
- 175
2
votes
2
answers
257
views
Reference request on Min-Max theorem
Consider the following min-max problem
$$\inf_{x\in M} \sup_{y\in N} F(x,y),$$
where $F: M\times N\to\mathbb R$ is Lipschitz and $y\mapsto F(x,y)$ is concave for all $x\in M$. Could we derive $\...
user128095
2
votes
1
answer
378
views
Does the norm on a sequence space have to be monotone?
Let $\rho:[0,+\infty)^{\mathbb{N}}\to[0,+\infty] $ satisfy the following properties:
$\rho(\lambda u)=\lambda\rho( u)$, for every $u\in [0,+\infty)^{\mathbb{N}}$ and $\lambda\ge 0$;
$\rho(u+v)\le \...
- 5,529
2
votes
1
answer
228
views
Combination of simple tensors
I aksed this question on Math Stack Exchange 6 days ago, with no answer: https://math.stackexchange.com/q/4875445/1297919
Let $X$ and $Y$ two Banach spaces and let $X\otimes Y$ their tensor product. ...
2
votes
1
answer
775
views
Tietze's extension theorem for compact subspaces
The topological question:
Are there Hausdorff topological spaces $X$ which are compactly generated (=Kelly spaces = $k$-spaces, that is, a subset is closed if its intersection with every compact set ...
- 16.4k
2
votes
1
answer
320
views
Fourier series but different waveform
Given a nondegenerate smooth simple closed convex curve $f: [0,2\pi]\to \mathbb C \setminus \{0\}$ with winding number (around origin) $1$, and $f$ have zero mean. Let $f_n: [0,2\pi]\to \mathbb C \...
- 807
2
votes
0
answers
131
views
Can we conclude $\sup_g\int f_1g\le\sup_g\int f_2g$ from $\int f_1\le\int f_2$ in this situation?
Disclaimer: Please bear with me, the question isn't as complicated as it looks like, but I wasn't able to find any simplification for which no counterexample comes to my find.
Let $(E,\mathcal E,\...
- 167
2
votes
0
answers
119
views
Reverse Sobolev inequality for holomorphic functions
Problem. Let $U \subset \mathbb{C}$ be open and $[0,1] \subset U$. Assume $f(z)$ is holomorphic on $U$. Is it possible to find a constant $C$ (that depends on $f$) such that, for all $0 \leq a < b \...
- 981
2
votes
1
answer
284
views
Eigenfunctions of an infinite summation operator
I would like to find ALL eigenfunctions to the operator, for $f$ a real function on R+*:
$f \rightarrow \sum_{1}^{\infty} f(nx)$
So to find $f$ such that: $\sum_{1}^{\infty} f(nx) = \lambda f(x)$
...
- 1,199
2
votes
0
answers
240
views
3 questions around the Stone space of the free $\sigma$-algebra on $\omega_1$ free generators
During my studies, I came across several different Stone spaces, e.g.:
(i) The Cantor cube $X=\{0,1\}^{\omega_1}$, which is the Stone space of the free Boolean algebra on $\omega_1$ free generators;
...
- 207
2
votes
0
answers
77
views
Homomorphism of composition to additive structure
Consider the following topological groups
$\operatorname{Homeo}(\mathbb{R}^d)$ be the topological group of all homeomorphism from $\mathbb{R}^d$ onto itself; equipped with the compact-open topology (...
- 5,405
2
votes
0
answers
71
views
Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is an $s$-dimensional Hausdorff measure restricted to the Koch curve?
Motivated by my previous question Alberti rank-one theorem and irregular jump discontinuities, I'd like to ask the following:
Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is an ...
- 839
2
votes
1
answer
1k
views
Is a polynomial decay sufficient for a smooth function to be in $\mathcal{F}(L^1)$?
Background: I have a function $g(\omega)\in C^{\infty}(\mathbb{R})$, which vanishes like $O(|\omega|^{-\beta})$ at infinity for some $\beta>0$.
This answer states that functions that decays "too ...
- 243
2
votes
2
answers
494
views
Polynomial approximation (Weierstrass theorem) with bounds
Consider the closed interval $[0,1]$ and let $f \in C[0,1]$. Let $g$ be a real valued function on $[0,1]$ such that $g \leq f$.
Suppose $g = f$ at atmost finitely many points. Does there exist a ...
- 431
2
votes
1
answer
959
views
Do kernels provide a basis for a RKHS?
Let $H$ be a Reproducing Kernel Hilbert Space with elements $f:X\rightarrow \mathbb{C}$, with kernel $K(x, y)$. My question is whether, for some choice of $x_i\in X$, it is the case that $u_i:=K(x_i, \...
- 461
2
votes
1
answer
713
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Reference request: numerical analysis of PDEs and integro partial differential equations
I'm very new to the field of numerical analysis of PDE and integro partial differential equations.
My advisor (who is not a specialist in this area) highly recommended to read
Randall J. LeVeque'...
user102027
2
votes
1
answer
416
views
Questions about Maharam's classification theorem
I am studying von Neumann algebras. In the wiki article abelian von Neumann algebras, it mentions that every abelian von Neumann algebras acting on a separable Hilbert space is *-isomorphic to $L^{\...
- 523
2
votes
1
answer
3k
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A comparison principle for parabolic equation
(Crossposted from https://math.stackexchange.com/questions/757672/how-to-prove-comparison-principle-for-parabolic-pde-nonlinear)
Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for ...
- 266
2
votes
1
answer
1k
views
Existence of a projection operator onto subspace of Hilbert space
Let $V \subset H$ be Hilbert spaces with a continuous, compact and dense imbedding. Let $\{w_j\}_j \subset V$ be a basis of $V$ and of $H$ (so finite linear combinitions are dense) which is not ...
- 35
2
votes
1
answer
107
views
Lower bounds on translates of a function over a compact set
Let $f\in L^p(\mathbb{R})$ and define $f_\theta(x)=f(x-\theta)$. Let $K\subset\mathbb{R}$ be a compact set. I would like to compute (or at least lower bound) the following:
$$
\inf_{\theta\ne\theta'\...
- 45
2
votes
2
answers
1k
views
Approximation of smooth compactly supported functions on $\mathbb{R}^2$ using sums of products of one variable functions
Let $f \in C^{\infty}(\mathbb{R}^2)$ be smooth and compactly supported. Can we approximate $f(x,y)$ by sums of the form $\sum_{i=1}^m g_i(x) h_i (y)$ where $g_i, h_i \in C^{\infty}(\mathbb{R})$ are ...
- 33
2
votes
1
answer
366
views
Weak sequential continuity of certain bilinear forms on Banach algebras
Let $A$ be a Banach algebra and $Bil(A)$ denote the space of bounded bilinear forms on $A$.
$Bil(A)$ is a Banach $A$-bimodule with the module operations
\begin{eqnarray*}
\beta a(x,y) &:=& \...
- 2,605
2
votes
1
answer
1k
views
Packing number of Lipschitz functions
For some $L>0$ say ${\cal L}$ is the space of all $L-$Lipschitz functions mapping $(X,\rho) \rightarrow [0,1]$ where $(X,\rho)$ is a metric space.
For any $\alpha >0$ do we know of a ...
- 2,246
2
votes
1
answer
520
views
Fréchet derivative of evaluation-like functional (multivariate)
I'm fairly new to functional calculus but and posting here since the question seems more appropriate than for MSE. When coming across this post I could not help but wonder the following.
Let $H$ be ...
- 5,405