Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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1answer
91 views

Prove a function is polynomial?

How would one prove that a function is a polynomial? I can't seem to find anything about this on the internet. I would like to know if there are any unique properties that only polynomials can ...
0
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0answers
23 views

Reproducing Kernel of a RKHS of continuous functions may not be continuous in two variables together

Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$. That is $\mathcal{K}$ is RKHS, ...
3
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0answers
41 views

Kullback Leibler “variance”: does that divergence have a name?

If you consider two probability distributions $p$ and $q$, one way to measure the distance between the two is the Kullback-Leibler divergence: $$KL(p,q)=\int p \log (p/q) = E_p(\log p/q)$$ and this ...
1
vote
0answers
53 views

Does strong convergence in $W_p^1$ imply strong convergence of derivatives of absolute values in $L_p$?

Let $G$ be a Lipshitz domain and $u_n \to u$ in $W_p^1(G)$. Is it correct, that $\frac{\partial |u_n|}{\partial x_m} \to \frac{\partial |u|}{\partial x_m}$ in $L_p(G)$? I know, that $\frac{\partial ...
0
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0answers
27 views

solutions of elliptic linear pde depending analytically on a parameter

Fix $ \Omega$ a bounded smooth domain in $ R^N$ and suppose $0<w(x)$ is a smooth solution of $ -\Delta w(x)=w(x)^2$ in $ \Omega$ with $ w=0$ on $ \partial \Omega$ (were are assuming $2< ...
4
votes
1answer
84 views

Equivalent Norms on Sobolev Spaces

When $k$ is a positive integer and $1<p<\infty$, we know that there is some $C>0$ such that for all $u\in W^{k,p}\left(\mathbb{R}^{N}\right) :$ $$ \left\Vert \left( -I+\Delta\right) ...
5
votes
3answers
170 views

Can phase significantly concentrate a function's spectrum?

Let $F$ denote the Fourier transform over some group. What is known about the following quantity? $$\gamma:=\inf_{x\neq 0}\frac{\|Fx\|_1}{\|F|x|\|_1}$$ Here, $|x|$ denotes the pointwise absolute ...
0
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0answers
21 views

Writing eigen functions of one Stochastic Process in terms of the eigen functions of another

Let us consider a centred square integrable stochastic process $\{X_t:t\in [0,2]\}$. Also let the eigen values and the eigen function of the kernel of the covariance operator of $X_t$ are ...
0
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0answers
25 views

Questions about the definition of ``stabilization entropy" for dynamical systems

Let $\phi (t,x,u)$ be the solution to the differential equation, $\dot{x}(t) = f(x(t),u(t))$ where $x(t) \in \mathbb{R}^d$, $u : [0,\infty) \rightarrow \mathbb{R}^m$ and $f: \mathbb{R}^d \times ...
0
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0answers
80 views

Topic in functional analysis [on hold]

i'm a graduate student and i like an analysis. What are current research topics in the functional analysis especially in geometry of Banach spaces? I would like to read about them.
0
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0answers
50 views

Differentiating and integrating an infinite series arising from a PDE

Let us work on a bounded domain $\Omega$ with Neumann BCs, with $(\varphi_k, \lambda_k)$ being the orthonormalised eigenvectors and eigenvalues of the Neumann Laplacian. Given $u \in H^{\frac ...
3
votes
1answer
126 views

Discretizing probability measures

Consider a probability distribution on $\mathbb{R}^k$, say $\mu$. Then there is a sequence of probability measures $\mu_n$ that converge weakly to $\mu$ so that each of them is discrete (takes ...
2
votes
2answers
133 views

Extremal functions for Gagliardo-Nirenberg inequality

Recently I read about the Gagliardo-Nirenberg inequality. And I would like to ask about the attainability and the maximizers of the GN inequality: $(∫|u|^{r}dx)^{\frac{1}{r}} \leq ...
2
votes
1answer
67 views

Is this series involving hyperbolic functions uniformly convergent?

Suppose that $\mu_k$ is an increasing sequence of numbers such that $0 < \mu_1 \leq \mu_2 \leq ..$ with $\mu_k \to \infty$ as $k \to \infty$ $\sum_{k=1}^\infty |u_k|^2 < \infty$ and ...
2
votes
1answer
149 views

An elementary functional inequality

Let $g$ be a $C^1$ function with $g(0)=0$ and $g(t)>0$ for all $t>0$. I am surprised that for all such $g$ the following seems to hold $\frac{\int_0^t(g'(s))^2ds}{g^2(t)}\geq \frac{1}{t}$ for ...
1
vote
1answer
61 views

Conformally covariant distributions

In Conformal Field Theory (in $D$ dimensions) one considers (in particular) correlation functions of the form $$ \langle O(x)O(y)\rangle, $$ where $O$ is a scalar primary field. Scale covariance ...
1
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1answer
86 views

'Test Functions' to Lower Bound the Norm of Elements of Dual Quantum Group

There may well be an answer to this question in a simpler category than that of finite dimensional quantum groups and in that case this question is more suitable to math.stack and I apologise in ...
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0answers
73 views

Is this question complete? [closed]

Let $(X, \|\cdot\|_X)$ is Banach space and $(Y,\|\cdot\|_Y)$ is any normed space. Let $T:X \to Y$ be a linear, bounded operator. Prove that if $int(cl(T(B_1(0))))$ is non-empty then $int(T(B_1(0)))$ ...
0
votes
1answer
46 views

Sum of two parts of a continuous stochastic process

Let $X$ be a centered continuous stochastic process which is square integrable on $[0,2]\times \Omega$ and the basis of $L^2(0,2)$ is $\{e_i\}$. By using Karhunen-Leove Theorem one can write for all ...
0
votes
1answer
78 views

How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?

Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary. My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that ...
2
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0answers
50 views

An estimate for the maximal C* norm in the group algebra of a free group

Let F--->G be an epimorphism of groups, F being finitely generated and free. Let H be its kernel. Consider a lifting i: G--->F of the epimorphism. Every element of C[G] is of the form a=sum a(g) i(g) ...
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votes
0answers
41 views

Any theory for functional equations with shift operator [closed]

I'm looking for some ideas to solve this functional equation. $$\psi(x)T^t\psi(x)=1$$ where $T^t\psi(x)=\psi(x+t)$ and $\psi(x) \in C^\infty$. The solution should be like $$t=F(\psi,x)$$ Hope ...
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votes
0answers
33 views

Prove in any normed space not Hilbert space [closed]

Let (X,||) be any normed space and suppose x(n) is a sequence in X. If x(n)--->x weakly then show that ||x||<= lim inf||x(n)||
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votes
0answers
57 views

If Y is Banach then L(X,Y) is Banach where L(X,Y) is space of linear bounded operators [closed]

Let (X,||) be a normed space. If x(n) is Cauchy in X then it has a subsequence y(k) such that Sum||y(k+1)-y(k)|| is finite. K from 1 to infinity. By using this how can we ...
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0answers
52 views

When does analytic in the operator norm imply analytic in the trace class norm?

This is a crosspost from MSE. It's been up there for a few weeks now. A 200 rep bounty yielded no results (or even comments). I'm hoping someone here has some helpful ideas. See this post for the ...
2
votes
1answer
63 views

Domain of fractional powers of operators

Let $A$ and $B$ be non-negative ($(A x, x) \geq 0$ for all $x \in \mathcal{D}(A)$, similarly for $B$) densely defined self-adjoint operators on a Hilbert space $H$. Then the spectral theorem defines ...
0
votes
1answer
84 views

Uniform convergence of Fourier (orthonormal) expansion of series

Let $u \in L^2(M)$ on some closed Riemannian manifold. We can write $$u = \sum_{k \geq 0}(u,\varphi_k)\varphi_k$$ if $\varphi_k$ is some o.n basis of $L^2$ with is orthogonal in $H^1$ (eg. ...
1
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0answers
25 views

Condition for maximizer of convex combination to be expansion mapping

I have $\Pi_n:\mathbb R^{n+1}\rightarrow \mathbb R$ and $F_n:\mathbb R^2\rightarrow \mathbb R$ with $$F_n(x,a)=\Pi_n(x,...,x,a)$$ $$f_n(x)=\operatorname{ArgMax}_{a\in\mathbb R}\{F_n(x,a)\} $$ such ...
1
vote
1answer
151 views

Spherical harmonics and ellipticity of the Laplacian

Let us consider the sphere $S^n$ and the Laplacian $-\Delta$ on it. Let $L^2(S^n) = \bigoplus_k V_k$, where $V_k$ represents the eigenspace of the Laplacian with eigenvalue $k(k + n - 1)$. We know ...
1
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0answers
32 views

Parametric Sard-Smale theorem - when is the generic set open?

I am learning Morse/Floer Theory and in my work on my master's thesis I want to apply the parametric Sard-Smale theorem. I.e. I consider a Banach bundle $\mathcal{L} \to \mathcal{M}\times \mathcal{G}$ ...
1
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0answers
21 views

Equivalence of fractional power of second-order positive differential operator as pseudodifferential operator and a fractional definition

Let $A$ be a second-order differential operator on a closed manifold $M$ satisfying $$(Au,u) \geq 0$$ with $A=-\Delta$ the Laplace-Beltrami the model case. One can define for $s \in (0,1)$ the ...
-1
votes
1answer
55 views

Infinitesimal generator is bounded [closed]

Consider a strongly continuous semigroup of bounded linear operators $S(t):X\to X$. The infinitesimal generator of $S(t)$ is the linear operator $A:D(A)\subseteq X \to X$ defined by ...
3
votes
1answer
78 views

Generalization of maximum principle to other norms

Consider the Laplace equation $\Delta u=0$ in $\Omega \subset \mathbb{R}^d$ with Dirichlet boundary conditions, i.e. $u=g$ on $\delta \Omega$. By the maximum principle we know that the solution $u$ ...
2
votes
1answer
106 views

questions about the proof of the theorem of completely positive order zero maps

I hope my question is ok for mathoverflow. I first asked on math.stackexchange but received no answer and then delated it. I want to understand the proof of the theorem (which you can find in the ...
2
votes
0answers
111 views

Complex sum of squares of vector fields (hypoelliptic operators)

Consider a compact $M$ of dimension $n$. Consider real smooth vector fields $X_0, X_1, X_2,..., X_n$ on $M$ and consider the differential operator $\mathcal{L}_1 = \sum_{i = 1}^n X_i^2 + X_0.$ Now, by ...
2
votes
1answer
69 views

Besov and Triebel-Lizorkin spaces

Let me start with a couple of notational reminders. For $\xi\in \mathbb R^n$, $$ 1=\varphi_{0}(\xi)+\sum_{\nu \ge 1}\varphi_{\nu}(\xi),\quad \varphi_{0}\in C^\infty_c(\mathbb R^{n}),\quad ...
3
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0answers
61 views

Chain rule for weakly differentiable functions

Given are $f\in L^1(\mathbb R^n)$, $f>0$, such that $\log f\in L^1_{\mathrm{loc}}(\mathbb R^n)$ and $\nabla \log f = g$ in the sense of distributions, with $g\in L^1_{\mathrm{loc}}(\mathbb R^n)\cap ...
1
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0answers
35 views

pettis integral

Maybe this is rather a refernce question on Pettis integrals. Some naive questions arise: 1) Assume that $F$ is Pettis-integrable on $\Omega$ and that $\omega \subset \Omega$ is measurable. Is $f$ ...
5
votes
1answer
256 views

Is the unitary group of a pre Hilbert space contractible?

I already posted my question on mathstackexchange For a separable Hilbert space $H$ it is known that the unitary group $U(H)$ is contractible, both for the norm topology (Kuiper's theorem) and for ...
3
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0answers
48 views

Dilation of positive operators into martingales

In Rota's paper (An Alternierende Verfahren for General Positive Operators), Theorem 2 says that: Let $P$ be a doubly stochastic operator which is selfadjoint in $L^2 (S, \Sigma, \mu)$. Then there is ...
9
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2answers
327 views

Do locally convex topological vector spaces embed into diffeological spaces?

The nLab casually remarks that locally convex tvs embed into diffeological spaces by (discussion around) a corollary in Kriegl and Michor, namely 3.14, but this deals with Boman's theorem and results ...
15
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1answer
937 views

The list of problems for Grothendieck's thesis

Is the list of open problems which were given by Dieudonne and Schwartz to Grothendieck for his thesis published somewhere? I know a quotation of Dieudonne that the problems concerned duality theory ...
0
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1answer
133 views

Sobolev type embedding

Consider a compact manifold $M$ and a point $q \in M$. Let us say that that the following inequality holds: $$ \Vert \varphi u\Vert_{L^p} \leq C\Vert \varphi u\Vert_{H^1},$$ where $\varphi \in ...
8
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0answers
131 views

Subspaces and quotients in Banach space theory

In Banach space theory (closed) subspaces and quotient seem to play a symmetric role. However, since the behavior of subspaces is more intuitive, subspaces appear more frequently. E.g., the theory of ...
2
votes
1answer
146 views

Elliptic regularity of second order pseudos

Let $M$ be a compact manifold. Choose a point $q \in M$. Let $P$ be a second order positive self-adjoint pseudodifferential operator such that $\text{Spec}(P) \subset (0, \infty)$. Also, we know that ...
0
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0answers
13 views

Unboundedness of Laplacian [migrated]

I am currently considering the following operator ("modified Laplacian"): $T \colon \left( W^{2,2}(\mathbb{R}), \| \cdot \|_{L^2} \right) \longrightarrow \left( L^2(\mathbb{R}), \| \cdot \|_{L^2} ...
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0answers
37 views

Is the following definition of a functional derivative natural?

if $\delta S = \int \sqrt g F[\phi] \delta \phi$ Then is it natural to define the functional derivative as follows, $\frac{\delta S}{\delta \phi} = F[\phi]$. In particular does this definition ...
4
votes
2answers
176 views

Is the space of signed finite measures on a compact set $M([0,1])$ a sequential space?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$ (with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all ...
8
votes
1answer
181 views

Is the Jordan decomposition of a self-adjoint functional constructive?

Let $A$ be an abstract C*-algebra, and let $\varphi\colon A \rightarrow \mathbb C$ be a bounded linear function. Assuming the axiom of choice, there exist unique positive bounded linear functions ...
0
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0answers
40 views

Maximizing Expected Utility

I am currently trying to solve a maximization problem given by $\max_{f(x)} \int_0^1 \int_\mathbb{R} (c-y\cdot f(x)-d\cdot (x+f(x)-b)^2) \ h(x) \ dx \ dy$. Or in other words, I have a utility ...