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

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-4
votes
0answers
40 views

Is this set compact? [on hold]

Let A denote the set of $m\times n$ matrices whose entries bounded by $1$ in absolute one. Is the set $A$ a compact set in the smooth function space $C^1(R^n,R^m)$ with $C^1$ topology?
1
vote
0answers
64 views

Smallest degree of approximating polynomial

Let $\{0,1\}^n=S_0\cup S_1$ withh $S_0\cap S_1=\emptyset$. Let $\epsilon\in[\frac{1}2,1)$. Let $f:\Bbb R^n\rightarrow\Bbb R$ be a polynomial such that $$f(S_0)=\epsilon,\mbox{ ...
-1
votes
1answer
71 views

Question about measure lemma?

"Let (u_j) be a bounded sequence from $W^{1,p}(\Omega)$ how to prove that there exists a subsequence such that $u_j\rightharpoonup u$ in $W^{1,p}_0(\Omega)$ and $|\nabla u_j|\rightharpoonup d\mu,$ ...
-5
votes
0answers
55 views

limit exercise which requires ingenuity1 [on hold]

Hello im an eleventh grader in the best mathematics high school in my country. I wanted to see if any of you guys can help me solve a limit without integrals and L'Hospital, as i havent learnt them ...
0
votes
0answers
48 views

Proof for existence of isoperimetric minimizer by compactness theorem

Let $X$ be an $n$-dim closed Riemannian manifold, then given a number $0 <v < vol(X)$, there exists a Borel subset $A\subset X$ attaining $I(v)$, $I$ is the isoperimetric profile. The existence ...
3
votes
0answers
92 views

SubGROUPs of Banach spaces, when are they dense in a vector subspace?

It’s relatively easy to show that if $J$ is a closed subgroup of a finite-dimensional real Banach space, $B$, then it is a vector subspace iff for all bounded linear functionals $\sigma$ of $B$, ...
0
votes
0answers
23 views

Gaussian gabor frame

It is widely known that $\phi(x)=e^{-\frac{x^2}{2}}$ does not define a Gabor frame if we consider translations by units of $1$ and multiplication by $e^{2 \pi inx}$for $n \in \mathbb{N}.$ A way to ...
0
votes
0answers
64 views

the inverse of Trace theorem when $p = 2$

I can see there is an answer the inverse for the trace theorem and Image of the trace operator My question is that given $f\in H^{1/2}(\partial\Omega)$, is it possible to extended it into $\Omega$ ...
0
votes
0answers
71 views

Is the following set of infinite absolutely convex combinations closed? [on hold]

Let X be an infinite dimensional Banach space and let $(x_n)$ be a weakly-null sequence in X. Let A:=$\{\sum_{n=1}^{\infty} a_nx_n: (a_n) \in B_{l_1}\}$, where $B_{l_1}$ is the closed unit ball of ...
1
vote
0answers
95 views

How generic are Cayley graphs of non-Abelian groups with logarithmic girth?

Given a non-Abelian group $G$ I want to choose a symmetric generating set $S \subset G$ such that $Cay(G,S)$ has girth logarithmic in the size of the set. I want to know, For which $G$ can the ...
3
votes
0answers
61 views

Recognizing Schwartz regular distributions

Are there characterizations of Schwartz regular distributions other than being locally integrable (which does not lend itself to easy manipulations)? To be more detailed: if I want to show that some ...
0
votes
0answers
50 views

About equivalence of two fractional Sobolev/Hilbert spaces

Let $(\varphi_k, \lambda_k)$ be the eigenelements of the Neumann Laplacian. It's possible to define a space $$H(\Omega) = \{ u \in L^2(\Omega) \mid \sum_{k \geq 1}\lambda_k^{\frac ...
4
votes
1answer
154 views

Multivariable function analysis

Sorry if the terms I'm going to use is not professional enough:) This is about the complexity analysis of an algorithm. Let $\alpha$ be the greatest real root of the polynomial ...
4
votes
0answers
94 views

Rellich's theorem from compact resolvent

On a compact Riemannian manifold, we know that the Laplacian $\Delta$ has compact resolvent. In proving this, one typical way is to use Rellich's theorem about the compact embedding of $H^1(M)$ into ...
-2
votes
2answers
136 views

On Bohr-MollerupTheorem [closed]

In http://mathworld.wolfram.com/Bohr-MollerupTheorem.html, Bohr-Mollerup Theorem is given where it is stated that $\Gamma$ function is the unique log convex function that satisfies ...
1
vote
2answers
196 views

A question involving Mazur's Lemma

Consider the Mazur's Lemma (H. Brezis - "Functional analysis, ..."): "Assume $(x_n)$ converges weakly to $x$. Then there exists a sequence $(y_n)$ made up of convex combinations of the $x_n$'s that ...
1
vote
1answer
135 views

compact inclusion of domains of unbounded operators

Let $L$ be a positive self-adjoint operator defined densely on $L^2(M)$ where $M$ is a compact manifold. Also, let $\mathcal{D}(L) \subset H^1(M)$. It is known that $\mathcal{D}(L) \subset ...
3
votes
0answers
53 views

Pointwise (a.e) evaluation of $\sum_{n \geq 0}(u,w_n)_{L^2}w_n$ and equalities in $L^2$

Let $w_n$ be a orthonormal basis of $L^2(\Omega)$. Given $u \in L^2$ we can write $$u=\sum_{n \geq 0}(u,w_n)_{L^2}w_n.$$ Suppose $w_n$ are the eigenfunctions of the Neumann Laplacian. We can write ...
2
votes
1answer
133 views

How to show that there's a continuous function separating convex sets of Radon measures?

First, the setup: $X$ is a compact set. By Riesz's representation theorem $C(X)^*=${all Radon measures on $X$}. $K$ is a convex, closed set of probability measures. $m$ is a probability measure out of ...
-2
votes
0answers
26 views

Definition of equi-absolute continuity [closed]

Could someone provide (or point me to) a definition of equi-absolute continuity for functions defined on an open bounded subset $\Omega \subseteq\mathbb{R}^n$? I only managed to find a definition for ...
3
votes
2answers
232 views

Topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$

Are there complete TVS topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$ This question is strongly linked to is the space of all borel measures on ...
0
votes
0answers
18 views

Weak continuous convergence of operators [migrated]

Let $T_n$ and $T$ be linear maps from Banach space $X$ to a Banach space $Y$. Suppose $T_n$ satisfies $T_nx \to Tx$ (convergence in the $Y$ norm) for all $x \in X$. Let $x_n \rightharpoonup x$ in ...
11
votes
2answers
246 views

A measure on the space of probability measures

This question was originaly posted in the stackexchange https://math.stackexchange.com/questions/1226701/a-measure-on-the-space-of-probability-measures but since it only got a comment I decided to ...
3
votes
1answer
104 views

Characterizations of Wiener algebra

The Wiener algebra $\mathcal W$ is defined as $\text{Fourier}(L^1(\mathbb R))$, i.e. the image by the Fourier transform of $L^1(\mathbb R)$. Riemann-Lebesgue's lemma ensures that $$ \mathcal W\subset ...
4
votes
1answer
67 views

Does this infinite sum arising from separation of variables converge?

This problem came up in a PDE where I used separation of variables to formally get a solution. Now I need to know whether that formal solution is sensible. Let $a_k >0$ be an increasing sequence ...
1
vote
0answers
35 views

Reference request: Topological space of polygonal chains and its properties [migrated]

I'm interested in approximations of $C^1([a,b])$-functions by polygonal chains: image File:NURBstatic.svg by User:WulfTheSaxon licensed under GNU Free Documentation License A polygonal chain can be ...
1
vote
2answers
96 views

Sobolev trace map: is the fractional seminorm bounded by just the gradient?

Let $M$ be a compact Riemann manifold. Consider the trace map $T:H^1(M) \to H^{\frac 12}(\partial M)$. Is it always the case that $$|Tu|_{H^{\frac 12}(\partial M)} \leq C\lVert \nabla u ...
3
votes
1answer
119 views

Existence of state on a C*-algebra satisfying $|\tau(ab)|=\|ab\|$

(This is a repost of a question from math.SE, http://math.stackexchange.com/questions/1240966/existence-of-state-on-a-c-algebra-satisfying-tauab-ab) Let $a,b$ be elements of a unital C*-algebra $A$ ...
4
votes
2answers
146 views

Existence of a measure-preserving bijection

Let $f, g \, \colon \mathbb{R}^n \rightarrow \mathbb{R}$ be two Borel-measurable functions such that $f$ is non negative and g is radially symmetric, the function $ (0, \infty )\ni t \mapsto g ...
0
votes
0answers
50 views

Zero divisors and boundary elements of $A^{-1}$ [closed]

I need to understand basic properties of (a) zero divisors in Banach algebras or rings, and (b) spectral properties of elements of the boundary of the set of invertibles in a complex unital Banach ...
1
vote
0answers
51 views

boundedness of a sequence $ \in L^{\infty}(I,H^1(M))\cap Lip(I,L^2(M))$ implies that its temporal derivative is bounded as well [closed]

Hi I have the next claim which I would like to find a proof of it. I have a sequence of functions $u_\epsilon(t,x) \in H^1(M)$ where $M$ is a compact manifold, and $u_\epsilon \in ...
2
votes
1answer
117 views

Contraction semigroup

Let $(X, \mu)$ be a finite measure space and let $A$ be a non-negative self-adjoint operator which generates a contraction semigroup $e^{tA}$ on $L^2(X, \mu)$. If additionally, we have that $e^{tA}$ ...
0
votes
0answers
20 views

density function time series

I have a time series of density functions, say A1-A5. Each density function is defined as $f(x)=\Sigma_{i=1}^{N} \beta(x-a_i)$, where $\beta$ is a smoothing function (e.g., gaussian or delta), and N ...
2
votes
1answer
78 views

Renorming into contraction

In Pazy's book on semigroups he mentions (page 18) that when you have a commuting family of operators $B(t)$, such that $$ \sup \| B(t_1) .. B(t_n) \| \le M $$ for all finite choices $t_1, .. t_n$ ...
0
votes
0answers
63 views

Sobolev trace of $H^1(\mathcal{M} \times I)$ functions

Let $\mathcal{M}$ be a compact Riemannian manifold and let $I=(0,1)$. I seek a trace theorem saying that functions $u \in H^1(\mathcal{M} \times I)$ have a well-defined trace at $\mathcal{M} \times ...
1
vote
0answers
69 views

Equicontinuity of $\{f_{2n}\circ f_{2n-1}\}$

Let $(X,D)$ be a compact metric space and $\{f_n\}_{n\in\mathbb{N}}$ be a sequence of homeomorphisms of $(X,d)$. It is easy to see that if $\{f_n\}$ is uniformly convergent then $\{g_n\}$ defined by ...
0
votes
0answers
41 views

Densely-defined operator with closed range: conditions for operator closed

Suppose we have Banach spaces (or Hilbert spaces) $X$ and $Y$, and a densely-defined linear operator $A : \operatorname{dom}(A) \subset X \rightarrow Y$ that is densely-defined and with closed range. ...
3
votes
1answer
270 views

Moser estimates?

Consider $u$, an $L^2$ solution to the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on a ball $B_1$ of radius 1 centered at $(t_0, x_0)$, say, where $t$ can be treated as a "time" variable. I ...
1
vote
2answers
119 views

Sum of two unbounded self-adjoint operators

Let $A$ and $B$ be two unbounded self-adjoint operators. From this mathoverflow post, for instance, we know that $A + B$ is self-adjoint on $\mathcal{D}(A) \cap \mathcal{D}(B)$ if $A$ and $B$ are ...
1
vote
0answers
84 views

Size of the eigenfunction of Laplacian (reference request)

It is a classical Sobolev inequality that if $\phi$ is an eigenfunction of the Laplace-Beltrami operator on a $n$-dim compact Riemannian manifold $M$ with eigenvalue $\lambda$ then ...
0
votes
0answers
91 views

Lower order perturbations of 2nd order differential operators [migrated]

Consider the well-known Hormander's sum of squares $P = \sum_{j = 1}^m X_j^2$, where $X_j$ are vector fields on a compact manifold $M$ of dimension $n$. Also assume, as is usual to this theory, $m ...
3
votes
0answers
65 views

A Generalized Version of Maximal Correlation and Hypercontractivity of Conditional Expectation Operator

Given a pair of random variables $(X,Y)$ over a product space $\mathcal{X}\times \mathcal{Y}$, the maximal correlation coefficient is defined as ...
1
vote
0answers
49 views

Equivalence of two definitions of weak solution (subtlety with null sets)

Consider $$y_t - \Delta y = f$$ $$y(0) = y_0$$ with zero boundary condition. Let $a(t,.,.)$ be the bilinear form associated to $-\Delta$. We have two definitions of weak solutions: We have $y \in ...
1
vote
2answers
137 views

Holomorphic functional calculus and idempotents

One of the applications of the holomorphic functional calculus is with regard to idempotents. For instance, if an element $a$ in a unital Banach algebra $A$ has spectrum contained in two balls, each ...
2
votes
1answer
61 views

Does Schatten-p (quasi-)norm satisfy the norm inequality for 0<p<1?

I'm reading the paper by ANGELIKA ROHDE AND ALEXANDRE B. TSYBAKOV, ESTIMATION OF HIGH-DIMENSIONAL LOW-RANK MATRICES. And in the paper, they provide an inequation of the Schatten-p (quasi-)norm, ...
0
votes
0answers
42 views

Self adjoint operator and vertex conditions in quantum graphs

Let $\Gamma$ be a metric graph with finitely many edges. Consider the operator H acting as $\frac{-d^2}{dx_e^2}$ on each edge $e$, with the domain consisting of functions that belong to $H^2(e)$ on ...
4
votes
1answer
114 views

Diffusion semigroup generated by Laplacian

Let $M$ be a complete Riemannian manifold and $\Delta$ denote the Laplacian on it. Also assume that the spectrum of $-\Delta$ lies inside $[a, \infty)$. Let $P_t, t > 0$ denote the diffusion ...
1
vote
0answers
100 views

$L^2$ bound on solution of PDE in terms of $L^2$ norm of initial value

Let $u \in H^1((0,T)\times S)$ be the unique solution of $$u_{tt} + \Delta u =0$$ $$u|_{t=0}= u_0$$ $$u|_{t=T}=0$$ where $u_0 \in H^{\frac 12}(S)$ and $S$ is some Euclidean hypersurface without ...
14
votes
1answer
811 views

How many values determine a norm?

It is well known that for a bilinear form over an n-dimensional vector space, $n^2$ values (on all pairs of basis-vectors) determine it uniquely. How many values do we need to specify in order to ...
4
votes
1answer
243 views

Is there a name for this space?

I'm just asking if there is a name for the space of functions on $\mathbb R^n$ whose norm is defined by $$ \|f\|=\|\hat f\|_{L^p} $$ for $p\in [1,\infty]$. I find it handy to give it a name when ...