All Questions
Tagged with fa.functional-analysis real-analysis
1,447 questions
5
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2
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358
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Linear transport equation with unbounded coefficients
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = f_0(x) \in \mathscr S(\mathbb R^n).$
I am wondering then if $q$ and all its ...
- 124
5
votes
4
answers
613
views
Optimizing the gradient norm on the unit sphere
Let $ \Bbb S^{d-1}=\{(x_1,\cdots ,x_d): x_1^2+ \cdots +x_d^2=1\}\subset \Bbb R^d$ be the unit
sphere. Let $\nabla u= (\partial_{x_1}u,\cdots, \partial_{x_d}u)$ be the gradient of a function $u\in C_c^\...
- 1,101
5
votes
2
answers
977
views
Symbol of the Laplace-Beltrami on $\mathbb{S}^2$
This question is about how the principal part (or symbol) is defined on a manifold?-I assume that the answer is: As in $\mathbb{R}^n$ using local coordinates, i.e.
A differential operator $P=\sum_{|\...
- 329
5
votes
1
answer
353
views
Family of functions with prescribed derivatives
Suppose $f: \mathbb C \times (-1,1) \to \mathbb C$ is a smooth function that satisfies $f(0,t)=1$ for all $t\in (-1,1)$. Assume that for any $k\in \mathbb N$, any $z \in \mathbb C$ and any $t \in (-1,...
- 4,135
5
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2
answers
2k
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Relationship between KL, chi-squared, and Hellinger
There are many well-known relationships between the KL divergence, chi-squared ($\chi^2$) divergence, and the Hellinger metric. In the paper "Assouad, Fano, and Le Cam" by Bin Yu, the author ...
- 63
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2
answers
1k
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How to prove the second Korn inequality?
$\textbf{Theorem}.1$ (The first Korn inequality) Suppose that $ \Omega $ is a bounded domain in $ \mathbb{R}^d $ with Lipschitz boundary. Then\
\begin{eqnarray}
\sqrt{2}\left\|\triangledown u\right\|_{...
- 1,219
5
votes
2
answers
699
views
Ground state for non-linear Schrödinger
When studying the blow-up for focusing non-linear Schrödinger equation (NLS) one often compares the initial-state to a stationary solution.
In the energy-critical case, this stationary solution is ...
- 536
5
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2
answers
321
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If the Hausforff dimension of the graph of a function $u$ is $N$ and $\tilde u = u$ a.e. then $\dim_H \mathrm{graph} \, \tilde u = N$ too
Let $\Omega$ be an open (non empty) set and $u:\Omega \subset \mathbb{R}^N \to \mathbb{R}^M$ be a function such that the Hausdorff dimension of its graph is $N$.
Let $\tilde u = u$ a.e. Is it true ...
- 839
5
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2
answers
840
views
Decompostition of a Lipschitz domain
We say that $\Omega$ is a strongly star shaped domain (with respect to $0$ for example) in $\mathbb R ^n$ if:
$$\Omega = \{x\in \mathbb R ^n : \left \| x \right \| < g(\frac{x}{\left \| x\right \...
- 291
5
votes
2
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359
views
Proof without distributions
I was wondering whether there is a way to show this identity
$$\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 ...
- 111
5
votes
2
answers
459
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Backward heat equation and forward perturbed heat equation well posed?
I consider the following scenario. Let $I$ be a compact interval in space and $f$ a nice function in the space $C^{\infty}(I)$. In the following we consider a self-adjoint realization of our operators ...
- 536
5
votes
1
answer
151
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Existence of operator with certain properties
I am curious to know the answer to the following question:
Does there exist a continuous linear operator on some Banach space $X$ such that $\Vert T \Vert=1$, and $\sigma(T)\supset \{1\}$ is isolated ...
user69109
5
votes
2
answers
415
views
Existence of Solution, System of Equations
Suppose $P(\lambda, i)$ is the probability that a Poisson random variable with average $\lambda$ is equal to $i$, i.e. $\frac{\lambda^i}{e^{\lambda}i!}$
I think the following system of equations ...
- 53
5
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1
answer
3k
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Equicontinuity and $L^2$ convergence imply uniform convergence
I'm currently working through an old Paper of Garsia, Rodemich and Rumsey (A Real Variable Lemma) and theres one thing i don't get. Suppose $(f_n)_{n\in\mathbb{N}}$ is a sequence of continuous real ...
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5
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1
answer
542
views
If $f$ is bounded, decays fast enough at infinity and $\int f=0$, does this imply that $f$ is in the Hardy space $\mathcal H^1(\mathbb R^n)$?
Let $\mathcal H^1(\mathbb R^n)$ be the real Hardy space (as in Stein's "Harmonic Analysis", Chapter 3). It is well known that $\mathcal H^1(\mathbb R^n)\subset L^1(\mathbb R^n)$ and its ...
- 1,075
5
votes
2
answers
233
views
Analytic approximations of smooth vector fields
Let $M$ be the set of smooth divergence-free vector fields $u$ on $\mathbb{R}^3$ with
$$|\partial_x^{\alpha} u(x)| \leq C_{\alpha K}(1+|x|)^{-K}$$
on $\mathbb{R}^3$ for any $\alpha,K$.
Further, we ...
- 749
5
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2
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2k
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Chain-rule and change of variables in BV/Sobolev
A lot of results are available for the following chain-rule problem:
(CRP1) Let $f\colon \mathbb R \to \mathbb R$ be a $C^1$/Lipschitz function and let $g \colon \mathbb R^d \to \mathbb R$ be a ...
- 608
5
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1
answer
350
views
Set of translations of a real function having a dense linear span
Let $W$ be the space of continuous functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $\lim_{x\rightarrow \pm \infty} f(x)=0$, and consider the sup-norm topology on $W$.
Problem. does there ...
- 537
5
votes
3
answers
2k
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Morrey's inequality for Sobolev spaces of fractional order
Let $H^s(\mathbb T)$, where $s\in\mathbb R$, be the space of $2\pi$-periodic functions (or distributions), $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx}$, such that
$$
\|u\|_{H^s}^2=\sum_{k\...
- 2,933
5
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2
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774
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Can we calculate the inner product of a semicontinous function with the Dirac delta function?
Dear all,
It is clear that if $f:R\mapsto R$ is a continuous function, than $< f, \delta_x >=f(x)$. Now, if $f$ is only semicontinous, can we say that $< f, \delta_x >=f(x)$? I think this ...
- 1,649
5
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1
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340
views
How to give a counterexample of this estimate related to Paley-Littlewood theorem?
I am studying Paley-Littlewood theorem in Harmonic analysis, and I met an exercise. I would like to construct a function $f$ as a counterexample to show that the inequality
\begin{equation}
\| f \|^...
- 187
5
votes
1
answer
216
views
Bounds on dimension of a subspace
Let $I=(0,1)$ and let $C>1$ be a constant. Let $L^2(I)$ and $H^1(I)$ be the standard Sobolev spaces on $I$. Suppose that $U$ is a subspace of $H^1(I)$ with the additional property that:
$$ \| u\|_{...
- 4,135
5
votes
1
answer
526
views
Boyd & Chua 1985: Is the proof of Lemma 2 correct?
$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$I'm reading this article by Boyd and Chua [1], in which they prove the approximability of arbitrary time-invariant (TI) operators ...
- 153
5
votes
2
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594
views
Taylor $k$-differentiability of a real function at a point
I am interested in the standard name for the following weak form of $k$-differentiability.
Definition. A function $f:\mathbb R\to\mathbb R$ is called Taylor $k$-differentiable at a point $x_0$ if ...
- 41.8k
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1
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189
views
Subsequences of an orthonormal basis generating a strongly embedded subspace in $L_2(0,1)$
A closed subspace $M$ of $L_2(0,1)$ is said to be strongly embedded if the norms $\|\cdot\|_2$ and $\|\cdot\|_1$ are equivalent on $M$.
Let $(f_n)_{n\in \mathbb N}$ be a orthonormal basis of $L_2(...
- 4,461
5
votes
2
answers
338
views
Approximation of analytic function by a fixed number of monomials
This question seems simple but I can't manage to disprove it. Let $N\in \mathbb{N}$. We know that by its analyticity that this precise linear combination of monomials
$
\sum_{n=0}^K \frac1{n!} x^n
$
...
5
votes
2
answers
134
views
Prove that $K \ast f \in W^{1,\infty}(\mathbb R)$ if $K \in BV(\mathbb R)$
Let $f \in L^1 \cap L^\infty(\mathbb R)$ and $K \in BV(\mathbb R)$. Do these assumptions suffice to prove that for the convolution $K \ast f$ we have that $$K \ast f \in W^{1,\infty}(\mathbb R)$$ ...
- 131
5
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1
answer
260
views
Approximate Sobolev embedding
It is well-known in $H^2(\mathbb R^3)$ embeds into $L^{\infty}(\mathbb{R}^3).$ Now consider a function $u \in \ell^{\infty}(h\mathbb Z^3)$ and a grid of points $x \in h\mathbb{Z}^3.$
We then define ...
- 124
5
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1
answer
571
views
Schrödinger operator with Coulomb potential
The free Laplacian $-\Delta$ has absolutely continuous spectrum $[0,\infty).$ The Coulomb Hamiltonian $H=-\Delta-\frac{1}{\vert x\vert}$ on $L^2(\mathbb R^3)$ has absolutely continuous spectrum $[0,\...
- 119
5
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1
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171
views
Invariant subspace in infinite dimensions
Let $A(t)$ be a family of skew self-adjoint operator defined on some Hilbert space $H$ with common domain $D(A).$
The dependence on $t$ is in the strongly continuous sense, i.e. for all $x \in D(A)$ ...
- 177
5
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1
answer
669
views
Compact operators on $\ell^1$
Let $T$ be a compact symmetric operator on $\ell^2$ and $T\vert_{\ell^1}$ be bounded on $\ell^1$. Are there any non-trivial conditions that $T\vert_{\ell^1}$ is compact as well (for example would $T$ ...
- 329
5
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1
answer
101
views
Does minimum of an analytic map restricted to analytic curves implies minimum?
Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be an analytic function such that its restriction to any arbitrary analytic curve $\gamma$ passing through the origin $0\in \mathbb{R}^n$ attains a local ...
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5
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1
answer
136
views
Reference for higher order Campanato Lemmas, e.g. `Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}'
Whence can I reference the following fact (I have seen it quoted as `standard' in respectable places, so I hope it is so)?:
Let $f : B_2(0) \to \mathbb{R}$, say $f \in L^2(B_2(0))$ . Suppose that ...
- 1,771
5
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1
answer
246
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An asymmetric quadrilinear estimate
Fix $1<p<2$ and let $a_{i}=1-\frac{\theta_{i}}{p^{\prime}}$
where $\theta_{i}\in (0,1/2)$, $i=1,2,3,4$, and $p^{\prime}$ is the conjugate exponent of $p$. Note here that $0<a_{i}=1-\theta_{i}+...
- 852
5
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1
answer
151
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On existence of a concave function
Let $a$ be a strictly positive $C^\infty$ smooth function on the unit interval. Does there exist a strictly positive $C^\infty$ smooth function $f$ on $I$ such that
$$ f’’(x) \leq 0\quad \text{and} \...
- 4,135
5
votes
1
answer
564
views
Convergence of discrete Laplacian to continuous one
I make the following observation:
Let $\Delta^{(n)}$ be the discrete Laplacian on $\mathbb{C}^n$ (ie the $n\times n $ matrix with diagonal $-2$ and upper/lower diagonal $1$.)
This one has eigenvalues ...
- 536
5
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1
answer
155
views
Which averages of products of a function give a norm?
Let $f: [0,1] \rightarrow \mathbb{R}$ be a bounded measurable function. For some real non-negative numbers $a_1, a_2, b_1, b_2$ with $a_1+b_1=a_2+b_2=1$ consider the quantity
$$N(f)=\int_{[0,1]} \int_{...
- 2,288
5
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1
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499
views
Hausdorff dimension of the graph of a BV function
Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function.
Is the Hausdorff dimension of the graph of $u$ equal to $N$? How can we prove it?
Update.
In an answer to this post, it ...
- 839
5
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1
answer
332
views
Convergence of a sequence by iteration
Let $F:\mathbb R^d\to\mathbb R$ be a convex function. Assume that $F$ has a uniformly bounded gradient, $|\sup_{x\in\mathbb R^d}\nabla F(x)|<+\infty$. Define the sequence as follows: Take an ...
- 345
5
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1
answer
211
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Pointwise convergence in functional calculus
Let $A_n$ be a family of (bounded) self-adjoint operator converging pointwise to some (unbounded) self-adjoint operator $A,$ i.e. for all $x$ in the domain of $A$
$$\left\lVert A_n x-Ax \right\rVert \...
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5
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1
answer
2k
views
Commuting with self-adjoint operator
Let $T$ be an (unbounded) self-adjoint operator. Assume that there is a bounded operator $S$ such that $TS=ST.$ For which kind of $f$ do we have that $f(T)S=Sf(T)?$
My thought was that using a ...
- 501
5
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1
answer
187
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Getting out a system of linear ODEs by knowing the Magnus expansion
Assume we are given for a transition between two time points $t_0 = 0$ and $t_1$ a matrix relationship, eventually describing the solution of a system of linear with non-constant coefficients,
$$Y(t_1)...
- 749
5
votes
1
answer
229
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 ...
- 53
5
votes
1
answer
510
views
Norm inequality for the inclusion $L^2(\partial \Omega)\hookrightarrow H^{-1/2}(\partial \Omega)$
Let $\Omega \subset \mathbb{R}^3$ be a lipschitz domain. We then have the trace operator $\tau : H^1(\Omega) \to L^2(\partial \Omega)$ and can define the space $H^{1/2}(\partial \Omega) := \tau(H^1(\...
- 75
5
votes
1
answer
805
views
Arzelà-Ascoli for $C_b(0,1)$? Or more generally, why is that continuous functions "live most naturally" on compact spaces?
I’m wondering if there is a version of Arzelà-Ascoli for continuous functions on not-necessarily compact metric/Hausdorff spaces $X$, i.e. a characterization of the compact subsets of $C_b(X)$ (under ...
- 831
5
votes
1
answer
279
views
Connecting PDE notions for functions $[0,T] \to (\Omega \to \mathbb{R})$ to related notions for functions $[0,T] \times \Omega \to \mathbb{R}$
Fix $\Omega \subseteq \mathbb{R}^N$ a bounded domain (of whatever smoothness you end up needing, let's say $C^1$ domain for definiteness) and fix some $0 <T < \infty$. In considering evolution ...
- 324
5
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1
answer
1k
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The spectrum of the discrete Laplacian
Consider a connected (we define connected components by defining the set of vertices where every vertex has one neighbour) sublattice $V$ of the square lattice $V \subset\mathbb{Z}^2.$
On this we ...
- 53
5
votes
1
answer
1k
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Trace-norm of integral operator
Let me start by saying that I do appreciate any insight on this. So also if you have a partial result, please share it as a comment or answer.
This is somewhat unrelated to what I normally do, so I ...
- 175
5
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2
answers
429
views
Does the truncated Hausdorff moment problem admit absolutely continuous solutions?
Let $\mu$ be a (Borel) probability measure on $[0,1]$ and define $m_j(\mu) = \int x^j\,\mu(dx)$. Let $k$ be a positive integer and consider the set $\mathcal C_{\mu,k}$ of probability measures $\nu$ ...
- 311
5
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1
answer
327
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Convergence in energy of bounded (semi)subharmonic functions
Consider a sequence $(f_n)$ of functions in the flat torus $T^d$ converging Lebesgue-a.e. to a limit function $f$.
Assume that:
1) $|f_n|(x)\leq 1$ for every $n,x$
2) $\Delta f_n\geq -1$ in the ...
- 660