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

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8
votes
3answers
360 views

Thin large subspaces of $\ell^N_1$

Consider a sequence $V_N$ of subspaces of $\ell^N_1$ so that $\dim V_N = N- n$ and $n$ is $\mathsf{o}(N)$. Is it true that these spaces are "thick" (unofficial terminology), i.e. are there constants ...
2
votes
0answers
90 views

A modification of Minty's trick?

I have the following result: $$0 \leq \int_0^T (a(t)- |w(t)|)(b(t) - g^{-1}(|w(t)|))\quad\forall w \in L^2(0,T)$$ where $a$ and $b$ are both non-negative. Does it follow that $b(t) = g^{-1}(a(t))$? ...
9
votes
2answers
319 views

Completeness of nonharmonic Fourier Series

I have the following question: The Exponential System $(\exp(2\pi i n \cdot ))_{n\in \mathbb{Z}}$ constitutes an orthonormal basis of $L^2([-1/2,1/2])$. Thus, certainly the oversampled system ...
4
votes
0answers
95 views

Smooth perturbation of a positive self-adjoint operator with compact resolvent

Consider a one-parameter family $A_t$ of unbounded positive self-adjoint operators with discrete spectrum (for example, one can consider a one-parameter family of Laplacians on a compact Riemannian ...
0
votes
0answers
48 views

Left introversion operators associated to function spaces on semigroups

I am stuck on the following question for quite sometime now. Please help, any comment is welcome. Let $S$ be a topological semigroup and $\mathcal{F}$ be a translation invariant, conjugate closed ...
3
votes
2answers
149 views

Lower bound for Euler's function

Euler function is defined, for $|x|\le 1$, as follows: $$\phi(x)=\prod_{i=1}^\infty(1-x^i)$$ Upper bounds for $\phi$ can be simply derived from ending the product early, e.g. ...
4
votes
0answers
335 views

Properties of the solution of the heat equation

Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention. Note 2: This is my research problem, but the original problem ...
0
votes
0answers
32 views

How the diffusion in the unit ball induce the boundary process on the boundary directly?

This is Example 1.2.3 from Fukushima, Masatoshi, Oshima, Yoichi and Takeda, Masayoshi's book "Dirichlet Forms and Symmetric Markov Processes". In this example, we define a Dirichlet form in the unit ...
3
votes
0answers
134 views

Proof without distributions

I was wondering whether there is a way to show this identity $$\pi \int_{\mathbb{R}^3} \frac{f(x)}{|x|} dx = \int_{\mathbb{R}^3} \frac{\widehat{f(x)}}{|x|^2} dx $$ without using distributions for $f ...
5
votes
1answer
98 views

Is the Feichtinger's algebra $(S_0(\mathbb{R^d}),||\cdot||_{S_0})$ reflexive?

The Feichtinger's algebra $S_0(\mathbb{R^d})=M^{1,1}(\mathbb{R^d}):=\{f\in L^2(\mathbb{R^d}):V_g(f)\in L^1(\mathbb{R^{2d}})\}$, where $V_g(f)(x,\omega)$ is the short-time Fourier transform of $f$ with ...
9
votes
1answer
247 views

The intuition behind the Hilbert projective metric and the Perron Frobenius Theorem

Recently I have read a proof of the Perron Frobenius Theorem for positive aperiodic matrices. In this proof, the trick is to put a metric in the "positive quadrant" of $\mathbb{R}^n$, ...
6
votes
0answers
198 views

Commutation preserving operators

Let $A$ and $B$ be unital $C$*-algebras and let $T\colon A\to B$ be a bounded linear bijection that preserves commuting elements, i.e., $ab=ba$ implies $TaTb=TbTa$. Does $T^{**}$ then also preserve ...
3
votes
0answers
104 views

Eigenvalues and eigenvectors of the q-Bernstein operator

The Bernstein operator maps $f\in C[0,1]$ to its Bernstein polynomial $B_n f.$ The eigenvalues and eigenfunctions of the Bernstein operator on $C[0,1]$ have been described in [1]. Similar description ...
1
vote
0answers
79 views

A version of the Weak Regularity Lemma

Definitions: Given a graph $G$ and $S$, $T \subseteq V(G)$, let $e_G(S, T)$ denote the number of edges of $G$ with one endpoint in $S$ and the other in $T$ and let $$d_G(S, T) := \frac{e_G(S, ...
4
votes
1answer
127 views

Invariant subspaces are reducing subspaces in $L^2(\mu)$; where $\mu$ is a singular measure w.r.t Lebesgue measure

I have already posted this question on math.stackexchange but didn't get any answer. I hope this is the right place to ask this question. Recently I was reading a book "Operator Function and system" ...
0
votes
1answer
144 views

Subspaces of $H^{\infty}(\mathbb{D})$ which contains a nontrivial weak* closed subalgebra

Let $H^{\infty}(\mathbb{D})$ denotes the Banach space of bounded holomorphic functions in the unit disc. Consider the weak* topology on $L^{\infty}(\mathbb{T})$ that it inherits as the dual of ...
1
vote
0answers
71 views

One-parameter group of unitary operators and Core

Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...
0
votes
1answer
113 views

Nonstable $K$-theory question

Let $Y$ be a compact, Hausdorff topological space, and $X$ be a locally compact, contractible, Hausdorff space which is homeomorphic to a dense subset of $Y$. Question A: Is ...
4
votes
1answer
121 views

Is Hessian operator self-adjoint on infinite dimensional environment?

As we know the Hessian matrix is symmetric in a finite-dimensional environment. What about the Hessian operator $D^2F$ for a functional $F:H\rightarrow \mathbb{R}$, where $H$ is a Hilbert space and ...
0
votes
1answer
122 views

Averaging measurable functions over amenable group actions

Let $G$ be an amenable group acting on a space $X$. Amenability means there is a $G$-invariant mean on $L^\infty(G,{\mathbf R})$. Given a bounded function $f\colon X\to {\mathbf R}$ one can use the ...
0
votes
1answer
87 views

The monotone operator in $BV$ space

I am considering the following minimizing problem: $$ \min_{u\in BV(\Omega)}\{\frac12\|u-u_0\|_{L^2}^2 + |u|_{TV(\Omega)}\} $$ where $u_0\in BV(\Omega)$, $\Omega\subset \mathbb R^2$ is open bounded, ...
1
vote
0answers
45 views

Mathematical difference between broad and narrow band Spectral estimation [closed]

Is there different mathematical formulation behind spectral estimation of narrow band and wide band? By spectral estimation I mean estimating the frequencies in a given signal. Fourier transform is ...
5
votes
0answers
107 views

Differential operators acting on the Schwartz space

I asked a similar question on math stack exchange but didn't get an answer so I will try to ask it here. Any help/suggestion is most than welcome! Let $D$ be a linear differential operator with ...
2
votes
0answers
34 views

Trace space of $\{ t^su \in L^2(0,\infty;X) \mid t^su_t \in L^2(0,\infty;Y)\}$ for $s \in (-\frac 12, \frac 12)$

Let $s \in (-\frac 12,\frac 12)$ and let $X=D(\Lambda)$ be a Hilbert space with $\Lambda$ the infinitesimal generator of a bounded semigroup of class $C^0$ in $Y$ (which is another Hilbert space), and ...
0
votes
0answers
44 views

The properties of the solution pf minimizing problem with different parameters

I asked an similar problem before but received no respond. Here I modified the problem, add in more informations and assumptions, and with an extra question... Let $\Omega\subset \mathbb R^2$ be open ...
1
vote
1answer
196 views

How are the assumptions used in the proof of Bourbaki-Alaoglu Theorem?

I've posted this question in MSE for a while without getting any answer. In the proof of the following theorem, where are the assumptions "Hausdoff" and "locally convex" used?
2
votes
0answers
115 views

Is $C^{\gamma}(\Sigma)$ dense in $C(\Sigma)$?

Consider $\{0,1\}$ with the discrete topology and $\Sigma=\{0,1\}^{\mathbb{N}}$ with the product topology. We know that this product topology is generated by the metric $$ ...
0
votes
0answers
131 views

Amenable hypergroups

I needed some information about the intersection of the kernels of invariant means on hypergroups. So I read the discussion made for the question " The kernel of all invariant means " which answer my ...
3
votes
3answers
136 views

Characterizing (up to permutations) finite sequences of real numbers

Let $S=\{x_1,\cdots, x_N\}$ be a finite sequence of real numbers. I am interested in characterizing the family of functions $F$ such that for any $f\in F$ the function $$ c(\lambda) ...
1
vote
1answer
100 views

Comparison between spectra

Let $G$ be a normal operator with compact resolvent on a Hilbert space $H$ such that ${\rm ker}(G) \neq {0}$. Further let $P$ be the orthogonal projection onto ${\rm ker}(G)$, and let $G_{0}:=G+P$. ...
5
votes
0answers
174 views

Constructive approximation of Lipschitz functions

There are a number of theorems in classical functional analysis about approximation of Lipschitz functions by smooth functions. I was wondering if there are any similar constructive and explicit ...
3
votes
1answer
295 views

On the embedding of a function space $X$ into $L^2\cap L^4$

It is well-known that if $\Omega\in \mathbb{R}^n$ is a bounded domain, then we have the embedding $$ L^4({\Omega})\subset L^2({\Omega}) $$ since $||f||_{L^2(\Omega)}\leq C(\Omega) ||f||_{L^4(\Omega)}$ ...
16
votes
2answers
410 views

Hahn-Banach theorem with convex majorant

At least 99% of books on functional analysis state and prove the Hahn-Banach theorem in the following form: Let $p:X\to \mathbb R$ be sublinear on a real vector space, $L$ a subspace of $X$, and ...
8
votes
2answers
291 views

Constructing a function over a metric space through given points

Suppose there is a compact metric space $(X,\rho)$ and a Euclidean space $\mathbb{R}^n$. There is a sequence of unequal points $\{x_1,...x_N\}$ in $X$ such that all metrics $\rho(x_i,x_j)$ are known ...
1
vote
0answers
52 views

Existence and uniqueness of Abel integral equation

I consider the following Abel's integral equation: $$ \int_0^t \frac{k(t,s)f(s)}{\sqrt{t-s}}=g(s) $$ where $g(s)\in C^{\infty}[0,T]$ and $k(t,s)=C+\sqrt{t-s}$. To the best of my knowledge, there ...
9
votes
1answer
165 views

Optimal exponent in the Lojasiewicz-Simon gradient inequality

Lojasiewicz's theorem asserts that if $F: \mathbb{R}^n\to \mathbb{R}$ is a real-analytic function in a neighborhood of its critical point $0$, then there exist constants $\theta\in (0,1/2]$, ...
2
votes
0answers
52 views

Factorization of linear bounded operators in Banach spaces

I am looking for a reference, if any, to the following statement: "Let $X$ and $Y$ be Banach spaces, $A\in\mathcal{B}(X,X)$ be a linear bounded operator, and $B\in\mathcal{B}(X,Y)$ be surjective. ...
4
votes
0answers
166 views

Discrete groups G whose full C*-algebra C*(G) is not MF?

This is a cheap rip-off of this question, but I am genuinely interested in an answer. Is there a known example of a countable discrete group G whose full group C*-algebra C*(G) is not MF? Let us ...
2
votes
1answer
99 views

On the dependence on $\epsilon$ in Dvoretzky's theorem

A $k$-dimensional section of a convex body $K \subset {\mathbb R}^n$ is just the intersection of $K$ with a $k$-dimensional hyperplane $h$. Such a section is said to be $(1+\epsilon )$-almost ...
4
votes
0answers
161 views

Convergence of spectrum

Consider a one-parameter real analytic family of metrics $g_t$ on a compact manifold $M$ converging to a metric $g$ in $C^k$-norm, for some $k$. It is known that the Laplace spectrum of $g_t$ will ...
13
votes
2answers
391 views

$\frac{d}{dt} (A+t B)^p\,\text{ for } p\geq 1$

Given two positive self-adjoint operators $A,B$ on a Hilbert space. Let $p\geq 1$. I would like to calculate $$\frac{d}{dt}|_{t=0} (A+tB)^p,$$ where the power is defined through the spectral ...
3
votes
2answers
217 views

Do Hausdorff locally convex inductive limits always exist?

The following is from Schaefer, "Topological Vector Spaces", 1999, p. 56/57: Let $(E_\alpha)_{\alpha \in A}$ be a family of locally convex spaces with $\alpha$ in a directed poset $A$ and $h_{\beta ...
4
votes
2answers
197 views

An extremal type problem on segments

I am interested in the following extremal-type problem. Let us define $\Psi$ as $$\Psi(x)=\max_{f\in L^2[0,x] \,\,with\,\,\|f\|_2=1}\Bigg|\int_0^x\int_0^xf(t)f(s)\ln|t-s|dsdt\Bigg|$$ on ...
7
votes
2answers
476 views

$H^{*}$ algebras as a generalization of $C^{*}$ algebras

Let $H$ be the quaternions algebra. An $H^{*}$ algebra is a normed ring $A$ which is simultaneously a unital left $H$ module and has an involution $*$ with the following properties: $\forall \lambda ...
3
votes
0answers
106 views

Compact embedding of ${\rm L}^1_{loc}$ space

I was reading one preprint and stumbled upon a part in the proof where one particular embedding was used. Namely: Let $\Omega\subset{\bf R}^2$ be open and bounded and let $p\in\langle 1,2\rangle$. ...
2
votes
1answer
87 views

Why is the ker-hull-topology on $Irr(A)$ is the discrete topology?

Let $A$ be a C$^*$-algebra. Let $Irr(A)=\{[\pi]: \pi$ is an irreducible representation of A}, here is $\rho\in [\pi]$ if there is an unitary operator $V:H_{\pi}\to H_{\rho}$ such that ...
4
votes
1answer
87 views

Elliptic regularity for two dimensional domains

Suppose $ \Omega$ is a smooth bounded domain in $ R^2$. I am interested in the regularity of solutions to $$-\Delta u(x) = f(x) \mbox{ in } \Omega$$ with $ u=0$ on $ \partial \Omega$. If $ f \in ...
3
votes
1answer
100 views

Markov-semigroup sobolev inequality

I have a question about the following definition: A probability measure $\mu$, such that the Markov semi-group $e^{Lt} \in L(L^2)$ exists and is symmetric, satisfies the Sobolev inequality iff for ...
2
votes
0answers
132 views

Can one integrate around a branch-cut?

How meaningful is it to try to integrate around the branch-cut of a function? For example lets say I have the function $\log(z^2+a^2)$ for $a>0$ and I choose my branch-cuts to be starting at $\pm ...
1
vote
1answer
71 views

degree theory for elliptic equations; special solutions

I am interested in using degree theory to examine some semilinear problems. But instead of just looking for solutions lets assume i am looking for a certain class of solutions; for instance lets ...