All Questions
Tagged with fa.functional-analysis ca.classical-analysis-and-odes
524 questions
24
votes
3
answers
3k
views
Can Hölder's Inequality be strengthened for smooth functions?
Is there an $\epsilon>0$ so that for every nonnegative integrable function $f$ on the reals,
$$\frac{\| f \ast f \|_\infty \| f \ast f \|_1}{\|f \ast f \|_2^2} > 1+\epsilon?$$
Of course, we ...
- 13.4k
2
votes
1
answer
268
views
Hardy-type inequality for point boundary
Let $f$ be in $W^{2,p}(\mathbb{R}^n)$ for $n\geq 3$ and $p>n/2$, with $f=0$ at the origin. I want to show that the integral $$\int_{B(0,r)} (f |x|^{-2})^p dV <\infty$$ for some small $r>0$. A ...
- 2,834
1
vote
0
answers
136
views
A linear operator equation (PDE) with non-monotone term
I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple.
There is a linear operator $L:{D}(L) ...
- 11
2
votes
0
answers
171
views
Operator theory of initial-value ODE problems
The theory of elliptic boundary value problems is usually treated from the perspective of functional analysis, and the theory of operators between Hilbert spaces.
In contrast to that, the theory of ...
- 5,327
5
votes
2
answers
320
views
Uniqueness of solutions to an ODE system
For each $i$ (up to infinity), let $u_i \in C^1(0,T)$ satisfy
$$\frac{d}{dt}u_i(t) + \sum_{j=1}^\infty b(t;w_j,w_i)u_j(t) = 0$$
$$u_i(0) = u_i(T)$$
where $b(t;\cdot,\cdot)$ is an inner product on some ...
CommunityBot
- 1
0
votes
2
answers
425
views
A book about almost periodic functions [closed]
Can anyone give me suggestions for new books about Besicovitch's almost periodic functions? Thanks a lot.
- 25.8k
0
votes
0
answers
103
views
The trivility of Besov space for large parameter
For all $s>0$, $1\le p<+\infty$ and $u\in L^p(\mathrm{d}x)$, we define
$$D_{s,p}(u)=\sup_{r>0}\frac{1}{r^{sp+n}}\int_{\mathbb{R}^n}\int_{B(x,r)}|u(y)-u(x)|^p\mathrm{d}y\mathrm{d}x$$
and
$$W^{...
- 369
1
vote
0
answers
182
views
Laplacian mapping on various function spaces
I have a question related to a certain elliptic operator on $R^N$ but I think i can clarify my confusion if I just consider the Laplacian $\Delta$ on the unit ball in $R^N$.
If $ 1 <p< \infty$...
- 539
3
votes
0
answers
146
views
Variational Principle for a System of Differential Equations
I am studying a differential operator of the form $$ L\left(\begin{array}{c} u \\ v \end{array}\right) = -\Delta \left(\begin{array}{c} u \\ v \end{array}\right) + V(x)\left(\begin{array}{c} u \\ v \...
- 516
19
votes
1
answer
5k
views
Intuition for the Hardy space $H^1$ on $R^n$
the standard intuition for Lebesgue spaces $L^p(\mathbb R^n)$ for $p \in [1,\infty]$ are measurable functions with certain decay properties at infinity or at the singularities.
In particular, a ...
- 347
2
votes
0
answers
142
views
Holder continuity of Poisson equation with divergence free drift
I am interested in the following PDE.
Suppose $u_m$ is a smooth solution of a elliptic equation of the form
$$ -\Delta u_m(x) + a_m(x) \cdot \nabla u_m(x) = f_m(x) \qquad B_1 $$ with $ u_m=0 $ on $\...
- 539
5
votes
3
answers
490
views
Continuity with values in L^2
Hi,
let $T>0$, $\Omega\subset\mathrm{R}^n$ be a bounded smooth domain and suppose
$$u\in L^2(0,T;W^{1,2}(\Omega))\cap L^\infty((0,T)\times\Omega))\ \text{and } \partial_tu\in L^2(0,T;W^{-1,2}(\...
- 3,388
3
votes
1
answer
916
views
Can I approximate Schwartz functions which integrate to zero by $C_0^\infty$ functions which integrate to zero?
Let $X$ be the closed subspace of Schwartz space $\mathcal{S}(\mathbb{R}^N)$ defined by
\begin{equation*}
X=\left\{f\in\mathcal{S}(\mathbb{R}^N):\quad \int f\; dx=0\right\}.
\end{equation*}
My ...
- 135
1
vote
2
answers
923
views
Spectrum of Mathieu equation
I have the differential equation $-f''(x)-q \cos(x) f(x) = \lambda f(x)$ and I want to find all the eigenvalues of this equation analytically on $[0,2\pi]$ that satisfy the boundary condition $f(0) = ...
- 9,853
4
votes
2
answers
658
views
Abstract ODE; PDE; uniqueness of solution
I have a somewhat vague question regarding an abstract ODE in a Banach space.
Suppose $A:D(A) \subset X \rightarrow X$ is some linear operator (let's assume it's closed) and maybe add some other ...
- 4,729
12
votes
1
answer
353
views
smooth Luzin theorem
For measurable functions $f(x)$, $g(x)$ on $[0,1]$ define the distance $\rho(f,g)$ as a Lebesgue measure of the set $\{x:f(x)\ne g(x)\}$. Then Luzin's famous theorem states that $C[0,1]$ is dense with ...
- 24.2k
27
votes
2
answers
5k
views
What can be said about the Fourier transforms of characteristic functions?
What can be said about the Fourier transform of the characteristic function $1_A$, where $A\subset \mathbb{R}^n$ is of finite Lebesgue measure? In particular,
What properties are common to ...
- 2,719
-1
votes
1
answer
237
views
Theorem with an example [closed]
i have this theorem
in the paper they gives an example:
but here $H_1$ is not satisfied !
How to correct it please?
- 256
3
votes
2
answers
717
views
Existence for ODE in Banach space (accretive operators and Crandall-Liggett)
There is a theory of mild solutions $u \in C^0(0,T;X)$ where $X$ is a Banach space for equations of the form
$$\frac{du}{dt} + Au = f$$
where $A$ is an accretive nonlinear operator under some ...
- 3,388
3
votes
2
answers
1k
views
A sufficient condition for a probability measure to have compact support
Consider a probability measure $\mu$ on, let's say, $\mathbb R$.
Is there a necessary and sufficient condition so that $\mu$ has compact support $Supp(\mu)$ ?
I agree this question is too vague, ...
- 334
3
votes
1
answer
255
views
A differentiable version of the Michael selection theorem
Assume that $X$ and $Y$ are Banach spaces and $T:X\to Y$ is a bounded surjective linear map.
Is there a Gateaux differentiable function $g:Y\to X$ such that $T\circ g=Id_{Y}$?
- 356
3
votes
1
answer
432
views
A (non trivial) continuous map on a Banach space which is nowhere Frechet differentiable
Assume that $X$ is a Banach space. Is there a continuous map $f:X\to X$ such that $f$ is nowhere Frechet differentiable, but its restriction to every finite dimensional subspace is every where Frechet ...
- 3,584
1
vote
3
answers
565
views
What are the basis functions for a product space?
Let $X=L^1\left([0,1]^3\right)$,
for numerical purpose, what are the possible basis function for $X$?
In finite element method, the basis functions are tooth functions, or polynomial functions.
Is ...
- 13
1
vote
0
answers
1k
views
Inverse Transpose of Jacobian Matrix
Let $f:\mathbb{R}^n\mapsto\mathbb{R}^n$ be a bijective function. Fixed $a\in\mathbb{R}^n$. For any $x$ closes to $a$, using Taylor's series we can approximate $f(x)$ by
\begin{equation}
f(x)\approx f(...
- 133
1
vote
1
answer
154
views
Is the speed of a curve in $ \ell^\infty $ zero a.e. if the derivative of each component is zero a.e.?
Let $ A $ be an $ \mathcal{H}^1$-measurable subset of $ \mathbb{R} $ and $ \gamma \colon A \subseteq \mathbb{R} \to \ell^\infty $ be a Lipschitz mapping with the Lipschitz constant $ L $. Also, assume ...
- 42.8k
2
votes
0
answers
118
views
A two dimensional integral equation
I have the following integral equation:
$\phi(x, y) = \frac{a}{x-y} \int_y^x \phi(s, y) ds + \frac{b}{x-y} \int_y^x \phi(x, s) ds$
where $a > 1$ and $b> 1$ are constants, and $x \geq y$. The ...
- 21
6
votes
0
answers
411
views
Birth-Death Process associated with Orthogonal Polynomials
I have read in various places the following objects are related:
orthogonal polynomials
birth-death processes
Lattice paths
continued fractions
After a lot of searching online, I found sketches ...
- 12.3k
0
votes
0
answers
537
views
matrix Khintchine inequality
The usual Khintchine inequality says that if $\{\epsilon_n\}_{n = 1}^N$ are i.i.d. random variables with $\mathbb{P}(\epsilon_n = \pm 1) = \frac{1}{2}$ for each $n$ then
\begin{equation*}
\left( \...
28
votes
2
answers
2k
views
Dynamical properties of injective continuous functions on $\mathbb{R}^d$
Let $\varphi:\mathbb{R}^d\to\mathbb{R}^d$ be an injective continuous function.
Denote by $\varphi_n$ the $n$-th iterate of $\varphi$, i.e.
$\varphi_n(x)=\varphi_{n-1}(\varphi(x))$ for all $x\in\...
- 6,548
6
votes
2
answers
2k
views
What is the translation in Fourier transform for a function to have exp. decay at $x\to -\infty$
It is known that smooth functions with exponential decay at $\pm\infty$ are functions whose Fourier transform have analytic continuation in some suited complex strip. I was wondering what happens if ...
- 91.8k
3
votes
6
answers
1k
views
Reference for complex analysis jargon
I am not a (complex) analyst but it seems that some of the questions I am working on are related to the following concepts:
logarithmic capacity
transfinite diameter
Green's function of a compact ...
- 25
1
vote
2
answers
292
views
specific improper integral involving erf
I have encountered an integral, and kindly ask for help with a solution. It is beyond my own capabilities, and neither Maple nor Mathematica were of any help:
$$
\int_{1}^{\infty} \left[\mathrm{erf}\...
- 10.3k
1
vote
0
answers
135
views
growth bound for solution of an ordinary integro-differential equation
I am considering the following ordinary integro-differential equation
$$
A g = \sigma^2(y) + \int_\mathbb{R} \nu(y,dz) z^2
$$
where
$$
A = b(y) \partial + \frac{1}{2} \sigma^2(y) \partial^2
+ \int_\...
- 6,210
2
votes
0
answers
190
views
A contradiction to do with continuity? (involves chain rule)
Suppose for each $t$, $S(t) \subset \mathbb{R}^n$ is a domain (hypersurface). We have a diffeomorphism $D^0_t:S(0) \to S(t)$ for each $t$ such that it solves the ODE
$$\frac{d}{dt}D^0_t(\cdot) = V(D^...
CommunityBot
- 1
0
votes
1
answer
905
views
Hölder continuity of uniform limit of piecewise constant functions
Consider a piecewise constant function $v: [a,b] \rightarrow \mathbb{R}$ defined by a finite partition $a=t_0 < t_1 < t_2 < ... < t_s=b$ of the interval $[a,b]$, and constants $m_1,m_2,...,...
- 60.5k
1
vote
0
answers
121
views
showing convergence of a function recursion relation
I have obtained (formally) a perturbative solution
$$
H(y) = \sum_{n=0}^\infty \delta^n H_n(y)
$$
to the following integro-differential equation ($\delta$ is a small constant, $\nu$ is a L\'evy ...
- 19.3k
18
votes
1
answer
3k
views
Let a function f have all moments zero. What conditions force f to be identically zero?
Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple ...
- 19.3k
3
votes
0
answers
860
views
decreasing rearrangements: why the asymmetry of measure-preserving maps?
Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...
CommunityBot
- 1
1
vote
0
answers
100
views
Conditions on a measure to satisfy certain relation on moments.
Suppose we have a measure $\mu$ on $\mathbb R_+$ such that $\forall s>-1$ $t^s\in L^1(\mathrm d\mu(t))$.
I'd like to impose some conditions on $\mu$ so the function
$$f:p\to \frac{\int_0^\infty t^...
- 173
7
votes
2
answers
1k
views
For what spaces is the Hardy-Littlewood maximal operator of strong type $(p,p)$ if and only if $p > p_0 > 1$?
(This is essentially a continuation of my previous question, here.)
Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. Further assume (though you ...
CommunityBot
- 1
5
votes
2
answers
1k
views
Bruhat-Schwartz functions and derivatives in p-adic numbers
First of all, I am not an expert in neither classical, nor $p$-adic functional analysis, but anyway, I stumbled over the following lately:
Let $\varphi:\mathbb{Q}_p\rightarrow\mathbb{C}$.
...
- 2,054
5
votes
0
answers
585
views
Constructing a Sobolev space containing the differential k-forms of a Riemannian manifold
I am currently writing a paper about the Hodge theorem for an algebraic topology course. The specific formulation I am proving can be stated thus. Let $M$ be a compact, orientable n-dimensional ...
CommunityBot
- 1
0
votes
1
answer
440
views
Variation on Fatou's lemma for Sobolev norms
Recall that Fatou's Lemma says that for every sequence $f_n$ of non-negative measurable functions
$$\int \liminf_{n\to \infty} f_n \ d\mu\leq \liminf_{n\to \infty} \int f_n\ d\mu \ .$$
If I am not ...
- 843
20
votes
2
answers
4k
views
Ideals of the ring of smooth functions
The ring $C^\infty(M)$ of smooth functions on a smooth manifold $M$ is a topological ring with respect to the Whitney topology and the usual ring operations. Is it possible to describe, maybe under ...
- 101
1
vote
1
answer
367
views
weak*closure of {f:||f||=1} in dual.
What is the weak* closure of {f:||f||=1}? I am sure this set is not closed in weak* topology.
So what is the weak* closure of this set. Thanks.
1
vote
1
answer
223
views
Weierstrass Approximation Theorem with an additional condition
Good afternoon:
Let $f$ be a continuous function defined on an closed interval $[a, b]\subset\mathbb R$.
By Weierstrass Approximation Theorem, for any $\epsilon>0$, there is a polynomial $p$ such ...
- 60.5k
4
votes
1
answer
910
views
Riemann-Lebesgue lemma for measures
Riemann Lebesgue Lemma states that Fourier transform of an $L^1$ function, $\hat{f}(\lambda)$ is continuous and goes to zero as $|\lambda|\to \infty$. If $\mu$ is a finite nonatomic measure then is it ...
- 13k
6
votes
1
answer
591
views
For which metric measure spaces is the Hardy-Littlewood maximal operator not of weak type (1,1)?
Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. I'll denote the Hardy-Littlewood maximal operator - either centred or uncentred, I don't mind ...
- 13k
22
votes
2
answers
2k
views
Examples of loss of regularity by "creation of topology"
I would like to have a list as general as possible of examples of situations where the density of smooth objects into some "natural class" (the meaning of "natural" depending on the problem considered)...
- 5,670
9
votes
2
answers
1k
views
Hilbert transforms of measures
Given a finite measure $\mu$ on the real line $\mathbb R$, one definition of its Hilbert transform is $(H\mu)(y) =\frac{1}{\pi}(PV)\int \frac{d\mu(x)}{x-y}$ which is known to exist almost everywhere ...
CommunityBot
- 1