Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
0
votes
0answers
4 views
Epi-convergence to indicator function
Let the indicator function be defined as $$I(x) \triangleq \begin{cases} 1, & \quad x \geq 0 \\ 0, & \quad x < 0 \end{cases}$$ and $I_{\nu}(x) \in [0,1]$ be a continuous approximation of ...
0
votes
0answers
52 views
Comparison between operators
I have found the following two concepts:
$\bullet$ Let $L$ be a linear operator in a Hilbert space $H$. The
operator $B$ is said to be $L$-compact if $D(L)\subset D(B)$
and for any ...
0
votes
0answers
45 views
A parabolic PDE with Lipschitz nonlinearity, how to obtain well-posedness?
Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$ (or more generally a compact manifold). I'm interested in well-posedness (existence most importantly) of equations of the form
$$u_t(t) - ...
3
votes
1answer
151 views
The norm of a separable Banach space can be determined by countable continuous linear functionals?
Recently I'm reading Stochastic Equations in Infinite Dimensions, a result is used many times. It is
If $E$ is a separable Banach spaces, then there is a sequence $\{ \phi_n \}$ in its dual ...
-4
votes
0answers
55 views
Lim inf and lim sup for a specific sequence of sets [on hold]
Let the indicator function be defined as $$I(x) \triangleq \begin{cases} 1, & \quad x \geq 0 \\ 0, & \quad x < 0 \end{cases}$$ and $I_{\nu}(x) \in [0,1]$ be an approximation of the $I(x)$ ...
-4
votes
0answers
62 views
Sequence of connected sets converging to a disconnected set [on hold]
Let us have the disconnected set $\mathcal{X} = \{0\} \cup [\underline{x},\overline{x}]$, with $0 > \underline{x} > \overline{x}$.
Is it possible to define a sequence of connected sets ...
0
votes
1answer
87 views
Can we expect $\left\||f|^{2}f-|g|^{2}g\right\|\leq C ||f-g||$ in the Banch algebra $A(\mathbb T)$ ?
Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space,
$$A(\mathbb T):= \{f\in ...
0
votes
0answers
26 views
Sequence of solutions of approximated problem converging to stationary solution of the original problem
Let $I(x)$ be the indicator function defined as $$I(x) \triangleq \begin{cases} 1, & x \geq 0 \\ 0, & x < 0 \end{cases}$$ and let $L(x,\alpha) \in [0,1]$ be a smoothing function that ...
0
votes
0answers
17 views
Parabolic partial differential equation, initial conditions
Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$.
Given the parabolic PDE $$\partial_tg+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...
0
votes
1answer
32 views
uniqueness for Poisson equation in R^d with mildly regular data
I'm interested in Poisson's equation $-\Delta u=f$ set in the whole space $R^d$ (let's say $d\geq 3$ for simplicity) when $f$ has very little integrability, specifically $f\in L^{1+\varepsilon}$ for ...
4
votes
1answer
111 views
Simultaneous Orthogonal basis for $L^2(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$
Given a smooth bounded set $U\subset \mathbb{R}^n$, there is a simultaneous orthogonal basis for $L^2(U)$ and $H^1_0(U)$ by the existence of eigenvectors to the Laplacian in a bounded domain, which ...
0
votes
1answer
178 views
A variation of the Banach fixed-point theorem
Let $(X, d)$ be complete metric space, $q \in [0, 1)$ be a real number, and $f$ be a map that satisfies $$d(f(x), f(y)) \leq q \cdot d(x, y)$$ for all $x, y \in X$. Then, Banach fixed-point theorem ...
1
vote
2answers
115 views
Smooth but non-analytic kernel functions
Does there exist a (stationary) covariance kernel function which is $C^\infty$-smooth but not real analytic? If so, could you please provide an example?
0
votes
0answers
50 views
Finding the Fractional Derivative of This Function [duplicate]
I've been trying to find an answer to this question, and it seems as though the question has gone unanswered. The question regards the derivative of $f(x)=1+n^{-x}$ where $n$ is a natural number. Is ...
12
votes
3answers
574 views
Sobolev spaces and geometry
This is a very naive question, is there a way to geometrically understand Sobolev spaces without going through analysis and PDE's? To my knowledge, Sobolev spaces where created precisely to study ...
2
votes
2answers
135 views
Lower bounds for norms of commutators
For various reasons I became interested in bounds on the norm of commutators of operators. For instance, if $B(H)$ is the algebra of bounded operators on a Hilbert space, one may ask for a lower bound ...
0
votes
0answers
72 views
How to use, $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in a Banach space $C([0,T]; M^{p,1})$?
(May be this is very basic question for MO)
(For details or this question you may see the paper page no. 9, MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; ...
0
votes
0answers
52 views
Weak derivatives and dual of Hölder functions
Let $0<\alpha<1$ and $f \in C^{\alpha}$ be a Hölder function (either with compact support on $\mathbb R^n$ or on a closed Riemaniann manifold).
From what I understand, the derivative of $f$ in ...
0
votes
0answers
40 views
Can inverse Lipschitz function approximate continuous function?
Let $(X,d)$ be a compact metric space and $C(X, R^n)$ the space of continuous funcions from $X$ to $R^n$. Given $f\in C(X, R^n)$ and $r>0$. My question is:
Can we find a inverse Lipschitz function ...
9
votes
0answers
235 views
Does anybody know if the Fourier algebra of SL(3,Z) has an approximate identity?
(Note to those who like to tidy LaTeX, or ${\rm \LaTeX}$: I kindly request that you don't put any LaTeX in the title of this question, nor change the bolds below to blackboard ...
1
vote
0answers
79 views
Derivative of Set Functions
For any closed set $I\subseteq (0,1]$, I have a set function in the form of
$$\theta(I)=\int_{q(I)}YdF(Y)$$
Where $q(I)$ if the image of I under the mapping $u \mapsto q(u)$. And $F(\cdot)$ is the ...
-1
votes
0answers
316 views
Given an even function how to obtain the most close odd function and vise versa?
Given an even function $f(x)$, how to obtain the most close to it continuous odd function $g(x)$?
By most close I mean that $\int_0^\infty |f(x)-g(x)| dx$ be the minimum possible and the difference ...
0
votes
0answers
51 views
Fourier Analysis in Kahane and Zelasko's characterization of maximal ideals in commutative Banach algebra
I've been told that Kahane and Zelasko (in A characterization of maximal ideals in commutative Banach algebra's Studia Math 29 1968) use a "non-trivial result" from Fourier analysis in their proof of ...
4
votes
0answers
83 views
Solving a Fredholm equation with a piecewise kernel : Karhunen-Loeve of a stopped Brownian motion
Is there a way to solve analytically the Fredholm integral equation of the second kind
$$
\int_0^{100} K(s, t) f(s) ds = \lambda f(t)
$$
where the kernel has the piecewise 'linear' form
\begin{align}
...
4
votes
1answer
127 views
Besicovitch Almost Periodic Functions a subspace of what?
The common example of a nonseparable Hilbert space comes from the collection of Besicovitch almost periodic function spaces. Starting with $L^p_{\text{loc}}(\mathbb{R})$ we look at those elements ...
2
votes
3answers
197 views
Weak convergence in the space of Lipschitz Functions
Consider the Banach space $X$ of Lipschitz functions $g:[0,1] \to \mathbb{R}$ such that $g(0)=0$, with the norm of $g$ given by its Lipschitz constant, i.e. $||g||_L = \sup_{x \neq y } ...
3
votes
1answer
107 views
General additive function of probability
Let $H$ be a function of finite sequences of probabilities (non-negative numbers summing up to 1) into real numbers, such that:
$H$ is continuous,
$H$ is symmetric w.r.t. the order of its arguments,
...
1
vote
2answers
172 views
Characterisation of compact operators
It is known (and easy) to prove that if $T: H\longrightarrow H $ is compact, where $H$ is a Hilbert space, then for any orthonormal basis $ e_n $ we have $||Te_n||\longrightarrow 0$.
My question is ...
1
vote
1answer
167 views
Are smooth functions dense in the space $\{u \in H^1(Q) \text{ with } \Delta_\Gamma u \in L^2(Q)\}$?
Define $$Q = \bigcup_{t \in (0,T)}\Gamma \times \{t\}$$ where $\Gamma$ is a compact (without boundary) hypersurface. Assume whatever smoothness is required.
Define $L^2(Q) := L^2(0,T;L^2(\Gamma))$ ...
0
votes
0answers
70 views
$2$-normed Spaces
Someone suggested today that $2$-normed spaces are actually equivalent to normed spaces. Can anyone who's familiar with the topic provide a counterexample? (I can't access Gähler's original paper ...
0
votes
0answers
85 views
Extending Continuous Basis [migrated]
It is given $(k-d)$ continuous vector-valued functions $K_1,\dots,K_{k-d}:\mathbb{R}\mapsto\mathbb{R}^k$, with $d\leq k$. Suppose that for all $x\in\mathbb{R}^k$, the set ${\cal ...
0
votes
0answers
130 views
Positive definite matrix in Banach algebra
Let $V$ be a Banach algebra lattice, endowed with an ordering $\leq$. And $T:V\rightarrow V$ be a positive operator. For $v_1,v_2,...,v_n\in V$, I want to show that (following matrix is positive ...
3
votes
2answers
330 views
When is it $C(X)$?
Suppose that $\tilde{X}$ is a compact space. If $C(\tilde{X})$ is isometrically isomorphic to the second dual of a Banach space, does there exist a locally compact space $X$ such that ...
0
votes
0answers
111 views
On ultrafilters
Let $\mathcal{U}$ be a free ultrafilter of $\mathbb{N}$. Is true or false what $(C([0,1],\ell_p)^*)_{\mathcal{U}} = L_1(\mu, \ell_q)$, where $\mu$ is a measure, $C([0,1],\ell_p)^*$ is the dual of ...
0
votes
0answers
80 views
On the existence of embeddings of $\ell_r$ into $L_1([0,1], \ell_p)$ for $r<p$
If $2<r<p$, is it true or false that $\ell_r \not \!\hookrightarrow L_1([0,1], \ell_p)$ ? In other words, if $r< p$, is it true or false that $L_1([0,1], \ell_p)$ contains a copy of $\ell_r$ ...
0
votes
0answers
60 views
Fractional Derivative of A specific function
I've tried and tried to do this problem myself, but I've hit some snags on the way.
I'm trying to take the fractional derivative of:
$f(x)=1+n^{-x}$ where n is an integer and $n\geq2$ and $x>1$.
...
2
votes
0answers
53 views
Are there pathological examples of log-concave measures that admit no shifts?
Does there exist a random vector $X$ in, say, the space $\mathbb{R}^\infty$ of sequences that has the following properties?
The distribution of $X$ is log-concave, i.e. for every $n$ the joint ...
1
vote
1answer
141 views
Tietze's extension theorem for compact subspaces
The topological question:
Are there Hausdorff topological spaces $X$ which are compactly generated (=Kelly spaces = $k$-spaces, that is, a subset is closed if its intersection with every compact set ...
2
votes
0answers
69 views
Birkhoff orthogonal of a Banach space in its bidual
Let $X$ be a Banach space embedded in $X^{**}$ in the usual way.
We consider the set
$$
O_X := \{ x^{**}\in X^{**} : \|x^{**}-x\|\geq \|x\| \textrm{ for all }x\in X\}.
$$
I think this is the ...
4
votes
1answer
122 views
Lieb: Stability of matter, problem with variational method
I am trying to recalculate 'Stability of Matter' Paper from Lieb.
I have a problem at the last step of the proof at page 3, where Lieb calculates the minimizer. I will explain my ideas and my problem ...
1
vote
1answer
70 views
L logL space and compactness
I think that if a sequence of L^1 functions have the integral
$$
\int f_n \log (f_n)dx
$$
uniformly bounded, then there is a subsequence that converges strongly in $L^1$.
The questions are:
1) Is ...
1
vote
1answer
119 views
On the relation between the set of extreme points of the unit ball of $M(X)$ and $M(X)^{**}$
Suppose that $X$ is a locally compact topological space. Let $M(X)$ denote the Banach space of regular Borel measures on $X$. It is known that the bidual of $C_0(X)$ is a commutative $C^*-$algebra. ...
5
votes
3answers
125 views
Method to compute fundamental solutions which are distributions
The Malgrange-Ehrenpreis theorem tells us that there is a fundamental solution for any linear differential operator of constants coefficients. The original proof was not constructive (it was based on ...
1
vote
1answer
91 views
Characterisation of adjoint operators
Let $X$ be a Banach space and $X^*$ denote his dual space. Then, it is well-known that if $T$ is a bounded linear operator on $X$, then $T^*$ is a bounded linear operator on $X^*$. My question is the ...
0
votes
0answers
89 views
Fractional Derivatives Of Sums
I have a question regarding the definition of a fractional derivative. I've searched, but I can't find a definition of fractional derivatives that explain the concept in terms of an operator on some ...
1
vote
0answers
67 views
Regularity of weak solutions for a quasilinear problem
Theorem 6.2.7 in the book Nonlinear Analisis of Gasisnski and Papageorgiou states: If $u\in W^{1,p}(\Omega)\cap L^\infty(\Omega)$ and $\Delta_pu \in L^\infty(\Omega)$ then we have $u\in ...
4
votes
0answers
69 views
Groups of operators between local unitaries and full unitaries
Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...
1
vote
1answer
95 views
Is this operator trace class?
Let $T:H\to H$ be a compact operator on a complex Hilbert space.
Assume that
$$
\sup_{(e_j)}\sum_j\left|\langle Te_j,e_j\rangle\right|<\infty,
$$
where the supremum extends over all orthonormal ...
2
votes
1answer
74 views
Reproducing kernels and equivalent inner products
Suppose $H$ is a reproducing kernel Hilbert space and $K_{1}\left(x,\cdot\right)$ and $K_{2}\left(x,\cdot\right)$ two reproducing kernels with respect to two equivalent inner products on this space. ...
8
votes
0answers
216 views
Reference for invariance of essential spectrum under relatively compact perturbations
I'm looking for a proof of the following statement:
Let $X$ be a Banach space and $T$ be a closed map on $X$. For any relatively compact map $A$ the essential spectrum of $T$ and $T+A$ are the same.
...
