**0**

votes

**0**answers

13 views

### Parseval frame, convergence of $\sum_{k=0}^\infty \left\|g_k\right\|$

Let $\mu$ be a Borel probability measure on $[0, 1)$, and $\{g_k\}_{k=0}^\infty$ be a Parseval frame for $L^2(\mu)$. Does
$$\sum_{k=0}^\infty \left\|g_k\right\|$$
converges?

**0**

votes

**0**answers

26 views

### inclusion maps interpolation spaces

I apologize in advance if the question below is a nonsense or a trivial one but I have no intuition as for the interpolation theory.
Let $\overline{A}:=(A_0,A_1)$ and $\overline{B}:=(B_0,B_1)$ are ...

**3**

votes

**2**answers

187 views

### Example of measure of non-compactness

I can't understand the following example of measure of non-compactness, which was given in this article.
Definition: A non-negative function $\phi$ defined on the bounded subsets of $X$ will be ...

**1**

vote

**1**answer

112 views

### Is the endpoint map smooth

Given $a,b \in \mathfrak{su}(n)$ and (with $U_0 = I$ taken) the following ODE:
$\frac{d U_t}{dt} = (a + w(t)b)U_t$
consider the "fixed time" endpoint map $V_T: L^2([0,T]) \rightarrow SU(n)$ for ...

**3**

votes

**1**answer

44 views

### Restriction of a semigroup to a form domain

Say, we have a Hilbert space $H$ with a semibounded self-adjoint
operator $A:D(A)\to H$ generating a strongly continuous semigroup
$T(t):H\to H$. Is it possible to restrict $T(t)$ to a form domain of ...

**4**

votes

**0**answers

56 views

### Are Sobolev trace spaces equal from both sides of the boundary?

Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure.
Assume $\partial\Omega=\partial\Omega'$.
Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...

**5**

votes

**1**answer

199 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 ...

**1**

vote

**1**answer

195 views

### Rank of a sequence of covariance matrices

Let $X_i$ ($i=1, \dots$) be an orthonormal basis for $L^2(\Omega, \mathbb P)$. In particular, it holds that
$$\mathbb E[X_iX_j] = \delta_{ij}.$$
Now take $Z\in L^2(\Omega, \mathbb P)$ and define ...

**-4**

votes

**0**answers

38 views

### Linear Independence for functions defined by integration [on hold]

Given that the set of functions $$f_i(x,y), \quad i=1,\dots,n$$
are linearly independent for $(x,y) \in [0,1]^2$.
Is the set of functions, $g_1,\ldots,g_n$, defined by
$$
g_i(x) = \int_{y\in [0,1] } ...

**2**

votes

**1**answer

328 views

### Weak topology on a pre-Hilbert Space

Since there was essentially no answers on my previous question, I will ask a partial case of it, which is very easy to state.
Let $\left(X,\left<\cdot,\cdot\right>\right)$ be an inner product ...

**0**

votes

**0**answers

116 views

### Radius of convergence of Taylor expansion of $z \mapsto (1 - z \cdot a)^{-1}$ [migrated]

Let $A$ be a Banach $\mathbb{C}$-algebra with norm $\text{N}(-)$ and let $a \in A$. Where can I find a reference to/can somebody supply a proof of the following posited equality?$$\max_{z \in ...

**3**

votes

**0**answers

126 views

### A question on $p$-approximation property

We say that a subset $K$ of a Banach space $X$ is relatively $p$-compact ($1\leq p<\infty$) if there exists a $p$-summable sequence $(x_n)_{n=1}^{\infty}$ in $X$ such that
$$ K\subseteq ...

**2**

votes

**1**answer

135 views

### Fractional Sobolev spaces and interpolation in unbounded Lipschitz domains

I am not really familiar with the topic, thus I am looking for some references about the following problem.
Let $s>0$ be a positive real and let $p\in(1,+\infty)$. We define the Bessel Potential ...

**2**

votes

**1**answer

88 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 ...

**11**

votes

**2**answers

281 views

### Quantum Hamiltonian for an Inverse Cube Force Law

If you have a nonrelativistic quantum particle in $\mathbb{R}^3$ in an attractive inverse cube force, its Hamiltonian is
$$ H = -\nabla^2 - \frac{c}{r^2} $$
where I'm keeping things simple by ...

**4**

votes

**0**answers

81 views

### Compensated compactness for system of conservation laws?

As far as I knew, the method of compensated compactness can be used only for one-dimensional scalar and $2\times 2$ systems of conservation laws, i.e. $u_t+f(u)_x=0$. But if I understood correctly ...

**5**

votes

**0**answers

131 views

### Tensorization of Orlicz norm?

Associated with a convex function $\phi:[0,\infty)\mapsto[0,\infty)$ satisfying $\lim_{x\to 0} \frac{\phi(x)}{x} = 0, \lim_{x\to\infty}\frac{\phi(x)}{x} = \infty,$ the Orlicz norm of a random variable ...

**4**

votes

**1**answer

122 views

### Kernel of the natural map between group $C^*$-algebras

Let $\Gamma$ be a discrete group. We can form two $C^*$-algebras: the universal (or full) and reduced, to be denoted by $C^*_u(\Gamma)$ and $C^*_r(\Gamma)$ (respectively). Both of them are completions ...

**2**

votes

**1**answer

124 views

### Hardy space, Lebesgue space for $p<1$,

We denote $\mathcal D'(\mathbb R^n)$ the space of distributions, and $\mathcal D(\mathbb R^n)$ the space of smooth, compactly supported functions.
Let $\rho\in \mathcal D'(\mathbb R^n)$ such that ...

**1**

vote

**0**answers

62 views

### Comparing Dirichlet energy and area of a Surface-immersion

Let $(F,g)$ be a closed Surface, $(M,h)$ a Riemannian 3-Manifold and $f: F \to M$ a smooth immersion. Denote by $f^*(h)$ the pullback metric on $TF$ induced by $f$ and let $dV_g$ and $dV_{f^*(h)}$ be ...

**0**

votes

**0**answers

57 views

### Singular integral equation

Investigating a control problem for heat equation I stacked on solution of this integral equation which seems to be singular:
$$
\int_0^1\mathcal{K}(\tau)u(\tau)d\tau=\mathcal{M},
$$
in which ...

**4**

votes

**2**answers

278 views

### The derivative of a non-tempered distribution can be tempered?

Suppose we have a non- tempered distribution $u\in \mathcal D'(\mathbb R^d)\backslash \mathcal S'(\mathbb R^d)$. Is it possible to have $\partial_{x_1}...\partial_{x_d}u \in \mathcal S'(\mathbb R^d)$ ...

**25**

votes

**6**answers

6k views

### Learning roadmap for harmonic analysis

In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas ...

**1**

vote

**0**answers

171 views

### Density of subspace with nonlocal/Wentzell boundary condition

Given the space $F$ defined by:
$$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$
I want to prove that the subspace $E$ of $F$ defined by ...

**0**

votes

**0**answers

41 views

### An eigenvalue convergence question [closed]

The following question is from Eberhard Zeidler's book in Nonlinear Functional Analysis volume 1, question 1.5b page 40.
Let $(X,(\cdot | \cdot))$ be a Hilbert space, take the operator: ...

**0**

votes

**1**answer

116 views

### On the Riesz representation theorem II

I have a follow-up question to On the Riesz representation theorem .
Let $V$ be a subspace of a Hilbert space, and let $V^\times$ be the space of all antilinear functionals on $V$, equipped with the ...

**4**

votes

**0**answers

44 views

### Determinant of quotient of unbounded operators

I have been trying to prove this for a while but failed so far.
Let $A$ and $B$ are two positive, self-adjoint operators with compact resolvent on a Hilbert space $H$ defined on the same dense ...

**4**

votes

**0**answers

157 views

### An equivalent relation on the space of polynomials in one complex variable

Let $P(z)$ be a polynomial with complex variable $z$. We consider the following distribution for the roots of $P(z)=0$: the distribution is a triple $(n_{1},n_{2},n_{3})$
where these integers are ...

**0**

votes

**0**answers

67 views

### A question in compact set

Definitions:
${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$
${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...

**0**

votes

**0**answers

74 views

### When do block sequences yield disjoint subspaces?

Let $X$ be a Banach space having a (unconditional, normalized) Schauder basis $(e_n)_n$. Suppose that $Y$ and $Z$ are (closed) block subspaces of $X$ having normalized block bases (with respect to ...

**1**

vote

**0**answers

78 views

### Fredholm subvector spaces of $B(\mathcal{H})$

Let $\mathcal{H}$ be a separable Hilbert space with orthonormal base $\{e_i\}\;\;i\in \mathbb{N}$.
Definition: We say a subvector space $W\subset B(\mathcal{H})$ is a Fredholm subspace if ...

**4**

votes

**1**answer

279 views

### On the Riesz representation theorem

Let $V$ be a vector space with inner product $(\phi,\psi)$ antilinear in the second argument - not necessarily a Hilbert space. Let $\Phi$ be an antilinear functional on $V$.
What are the precise ...

**2**

votes

**1**answer

125 views

### Isometry Group of real Hilbert space?

Does the isometry group of a real separable infinite-dimensional Hilbert space have two connected components? Or, conversely, is the there even a Kuiper's theorem in the real case?
How does the ...

**10**

votes

**5**answers

449 views

### Identities and inequalities in analysis and probability

Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of ...

**6**

votes

**2**answers

746 views

### Survey papers on the role played by PDE in mathematics

There are already several questions on MathOverflow that inquire about the many diverse relationships between PDE and several other 'areas' of mathematics (e.g., algebraic and differential geometry ...

**1**

vote

**0**answers

69 views

### Construct a PDE solution from a net of approximations

Consider $P$ a linear partial differential operator in $\Bbb R ^n$. Consider some boundary condition given in the generic form $C(u) = 0$, that guarantees a unique solution (if any) of $Pu = 0$.
Let ...

**4**

votes

**2**answers

251 views

### Vector Fields in a Riemannian Manifold

Suppose $(M,g)$ is a Riemannian manifold.
Is there a way to classify manifolds where there exists a vector field that commutes with the laplace beltrami operator?
Thanks

**0**

votes

**1**answer

101 views

### solution uniqueness of non-linear Fredholm equations

the equation is
$F(x)=G(\int k(x,y)f(y)dy)$ $(*)$
where $f(x)=dF(x)/dx$ is the unknown and it's required to be non-negative. With integral by parts we'll have the form of a non-linear Fredholm ...

**0**

votes

**0**answers

24 views

### Absolute continuity and the Luzin N-Property for functions of two variables

It is a well known fact that absolutely continuous functions of one real variable have the so-called Luzin N-property. That is, if $E\subset\operatorname{Domain}(f)$ has zero measure, then $f(E)$ has ...

**3**

votes

**1**answer

98 views

### Associative convolution on p-adic distribution

Let $\mathcal{D}(\mathbb{Q}_p)$ be the space of the locally constant functions with compact support and let $\mathcal{D}'(\mathbb{Q}_p)$ be the space of distributions: linear functionals on ...

**1**

vote

**0**answers

59 views

### Weighted Sobolev spaces over open/closed intervals

I am escalating this question from SE, as I have been unable to obtain any guidence in relation to a possible solution.
Some context, I am working with weighted Sobolev spaces of the form ...

**4**

votes

**0**answers

196 views

### Area defined with $\pm$ closedness

Denote $B_n\subset\Bbb R^n$ to be unit ball at origin.
Denote $S\subset B_n$ to region of type $\mathsf I$ if it satisfies
$$s\in S\iff\forall t\in S, s+t\in S\mbox{ or }s-t\in S$$
I am convinced ...

**-2**

votes

**1**answer

161 views

### No Hilbert space can have countable Hamel basis without using Baire's Category theorem [closed]

I want to prove that no Hilbert space can have countable Hamel basis just using the fact that any finite dimensional subspace is closed (more specifically without using Baire's theorem). I saw a paper ...

**0**

votes

**1**answer

82 views

### Is there relation between vector valued RKHS and interpolation space?

Vector valued RKHS which is covered extensively in the book "Pick Interpolation and Hilbert function spaces" . In a different context interpolation space is defined in the wikipedia link: ...

**2**

votes

**1**answer

109 views

### Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that I could not find it any where.
Is the completion of $C_0^{\infty}(\mathbb{R}^N)$ with the respect to norm
$$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, ...

**0**

votes

**0**answers

48 views

### Existence of a solution to $xf(x) = \int_0^1 k(x,y) yf(y) dy$

Crosspost - I asked very similar question on math.sx.
Let $X = (0,1)\times (0,1)$ and $k\colon X \to \mathbb{R}$ be a Lebesgue measurable non-negative function such that
$$ \int_0^1 k(x,y) dy = ...

**0**

votes

**0**answers

26 views

### Wiener amalgam space $W(\mathcal{F}L^{2}, L^{1}) \subset L^{1}$?

(I have asked this question on SE but could not get any answer and hope this is o.k for MO)
Let $X=\mathcal{F}L^{p}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{p}(\mathbb R)\},$ and $\|f\|_{X}= ...

**3**

votes

**1**answer

47 views

### Under what hypothesis on the domain is the X-ray transform/John transform operator bounded?

I asked this question on math stackexchange, without any reply yet.
Link:http://math.stackexchange.com/questions/1401580/under-what-hypothesis-is-the-x-ray-transform-john-transform-operator-bounded
...

**1**

vote

**3**answers

387 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 ...

**0**

votes

**0**answers

51 views

### $L^2$-boundedness of integral operator

Let $a:{\bf R}^d\to M^{d\times d}$ semi-definite matrix consisted of smooth functions i.e.
$$
\langle a(x) \xi,\xi \rangle=\sum\limits_{k,j=1}^d a_{kj}(x)\xi_j \xi_k \geq 0, \ \ x\in {\bf R}^d, \ \ ...