**8**

votes

**3**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**3**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**2**answers

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

**2**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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 ...