All Questions
Tagged with fa.functional-analysis ca.classical-analysis-and-odes
524 questions
2
votes
1
answer
137
views
Discrete dynamical system and bound on norm
Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following:
Consider the dynamical system with $x_i \in \mathbb C^2:$
$$ x_{i} = \left(\begin{matrix} z &&...
user121558
2
votes
0
answers
149
views
Off-diagonal estimates for Poisson kernels on manifolds
Let $(M,g)$ be a complete Riemannian manifold, $\Delta$ its Laplace-Beltrami operator and $T_t = (e^{t \Delta})_{t \geq 0}$ the associated heat semigroup. We can define the subordinated Poisson ...
- 1,090
3
votes
1
answer
212
views
Eigenvalue estimates for operator perturbations
I edited the question to a general mathematical question, since I found the answer in Carlo Beenakker's reference and think that my initial question was mathematically misleading.
What was behind ...
- 536
3
votes
1
answer
791
views
Real part of eigenvalues and Laplacian
I am working on imaging and I am a bit puzzled by the behaviour of this matrix:
$$A:=\left(
\begin{array}{cccccc}
1 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 &...
user136033
5
votes
1
answer
430
views
Oscillation and Hölder continuity
Where can I find a proof of the following fact?
If
$$w(u,x_{0},r)=\sup _{B_{r}(x_{0})}u-\inf _{B_{r}(x_{0})}u$$
for some function $u(x)$ satisfies
$$ w\left(u,x_{0},{\tfrac {r}{2}}\right)\leq \lambda ...
user60665
0
votes
0
answers
63
views
Feller semigroups and fractional operators
Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
user60665
0
votes
0
answers
273
views
Local "boundary comparison principle" for harmonic functions
Let $u$ be a positive solution of the elliptic equation $\mathcal Lu = 0$ on $B^+_1 \subset \mathbb{R}^n$ such that $u$ vanishes continuously on $\{x_n = 0\}$. To fix ideas, we may take $\mathcal L = ...
user123456
1
vote
1
answer
123
views
Interpolation of a trilinear functional
Let $f,g,h\in L^2([0,1]^2)$ and let $K:\mathbb{R}^3\to \mathbb{C}$ be some smooth kernel with support containing $[0,1]^3$. Denote by $\|f\|_2$ the $L^2([0,1]^2)$ norm of $f$, and same with $g,h.$ If ...
- 203
2
votes
0
answers
126
views
On the infimium of a functional
Let $(M^n,g)$ be a closed Riemannian manifold. Define
$$\lambda(g)=\inf\{\mathcal{F}(g,f),\;0<f\in C^{\infty}(M),\; \int_Mf^2\;d\nu=1\},$$
where $$\mathcal{F}(g,f)=\int_M\left(|\nabla f|^2+ af^2\...
- 169
7
votes
1
answer
245
views
Lower estimate of the minimal eigenvalue of a Hamiltonian
Consider a linear operator $H\colon L^2(\mathbb{R}^3)\to L^2(\mathbb{R}^3)$ given by
$$H(\psi)(x):=-\Delta\psi(x)+V(x)\cdot \psi(x),$$
where $V\colon \mathbb{R}^3\to \mathbb{R}$ is a continuous (or ...
- 21.8k
1
vote
1
answer
247
views
Elliptic interface problem without conditions on the interface
Consider an open domain $U$ split in two non-overlapping subdomains: $U = U_1 \cup U_2$.
For a model case, consider a ball split in a smaller ball and an anulus.
Consider the following elliptic ...
user60665
1
vote
0
answers
97
views
Fredholm integral equation of third kind
Let us consider the following integral equation
$$a(x)u(x) + \int\limits_0^2 {K(s,x)u(s)ds} = f(x)$$
Let f in $L^p(0,1)$ for some $p \in [1,\infty]$ and let $K \in L^q((0,2) \times (0,1))$. Assume ...
- 617
0
votes
0
answers
113
views
Conditions for the embedding of the space $L^\infty(I, W^{1,2}(U))$ into $L^\infty(I \times U)$
Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$.
If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like ...
user123457
2
votes
1
answer
300
views
Optimal control theory of PDEs
This is a somewhat openly phrased question because I am not quite sure what has been done in that direction.
Imagine one has two evolution equations
$$\partial_t u = p(x,\partial_x,f)u$$
$$\...
- 536
3
votes
1
answer
243
views
Prescribing a gradient direction
Let $\Omega= \{(x,y) : \frac{1}{2} \leq x^2+y^2 \leq 1\}$ and $S = \{(x,y) : x^2+y^2 = 1\}$ the unit circle, and $X=w^{1.\infty}(\Omega;\mathbb{R})$ the space of Lipschitz valued functions. We denote ...
- 2,494
4
votes
1
answer
145
views
Power series in functions other than monomials
I would like to understand how approximations by monomials and approximations by other kinds of functions are related which I illustrate with an example.
Consider the interval $[-\pi,\pi]$ let's say.
...
- 536
1
vote
1
answer
268
views
Best constant for Hölder inequality in Lorentz spaces
It's well known (and proved by R. O'neil) that there is a version of Hölder's inequality for Lorentz spaces, namely
$$\|fg\|_{L^{p, q}} \lesssim_{p_1, p_2, q_1, q_2} \|f\|_{L^{p_1, q_1}}\|g\|_{L^{...
1
vote
0
answers
110
views
Using semigroup theory for nonautonomous semilinear equations
We have the abstract evolution equation $$U^{\prime}=A(t)U+F(U), \quad U(0) = U_0.$$ If the operator $A$ is independent of time, we can get local existence of solutions by proving that $A$ is the ...
- 11
6
votes
1
answer
134
views
Multi-parameter stationary phase asymptotic expansion
I am looking for an asymptotic expansion of the oscillatory integral of the form
$$\int_{\mathbb{R}^n}f(x)\exp(i(\lambda_1\phi_1(x)+\dots+\lambda_k\phi_k(x))dx,$$
as $\lambda_i\to \infty$ ...
- 1,692
3
votes
0
answers
223
views
Sobolev space under Mellin transform
The Mellin transform is known to be an isomorphism see wikipedia
between $M:L^2(0, \infty) \rightarrow L^2(-\infty, \infty)$
where $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...
user127612
7
votes
2
answers
682
views
Hölder continuity for operators
Let $x,y$ be positive real numbers then
$$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$
we obtain $1/...
user127612
10
votes
1
answer
586
views
Nonlinear Schrödinger equation with discrete Laplacian
In the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$" by Colliander, Keel, Staffilani, Takaoka and Tao it is argued in the beginning ...
user127411
0
votes
1
answer
242
views
Harnack inequality for fractional laplacian
Let u be a positive solution of $s\in (0, 1) $
\begin{equation}
\left\{\begin{aligned}
(-\Delta )^{s} u &= 0 \text{ in } (-2T, 2T)\\
u &=g\quad\text{in}\quad \mathbb R\setminus(-2T, 2T).
\...
- 402
11
votes
2
answers
478
views
$x f'$ bounded by $x^2f $ and $f''$?
Consider the Hilbert space of functions $f \in L^2(\mathbb R)$ such that $x^2f \in L^2(\mathbb R) $ and $ f'' \in L^2(\mathbb R).$
I am wondering whether it is true that $xf'\in L^2(\mathbb R)$ as ...
- 177
1
vote
0
answers
324
views
Conditions for Poisson summation (for discontinuous functions)
Let $G$ be an locally compact abelian group with $\Gamma$ a discrete cocompact subgroup. I'm looking for precise conditions by which Poisson summation formula holds. That is, for some function $f$ on $...
- 3,799
2
votes
1
answer
93
views
Lipschitz bound on semigroups
Let $T$ be a self-adjoint operator (possibly unbounded) and $S$ a bounded self-adjoint operator.
Then one can study the unitary groups $R_T(t):=e^{itT}$ and $R_S(t):=e^{itS}.$
Now if you think about ...
2
votes
0
answers
293
views
Average of irrational flow on the torus
Let $$F(x,y) = \frac{1}{\sqrt{2-\sin(2\pi x) - \sin(2\pi y)}}$$
defined on $\mathbb{T}^2$. Here $\mathbb{T}^2 = \mathbb{R}^2/ \mathbb{Z}^2$ is the 2-torus. How can I show that
$$ \lim_{T\...
- 375
2
votes
2
answers
270
views
An inner product on the vector space $\mathbb{R}[x_1,\cdots,x_n]_m$
For any given integers $m,n\geq1$, let $\mathbb{R}[x_1,\cdots,x_n]_m$ be the vector space of homogeneous polynomials of degree $m$ in $x_1,\cdots,x_n$ over the field of real numbers $\mathbb{R}$. ...
- 1,997
1
vote
0
answers
237
views
On the bound of the Stein-Wainger oscillatory integral
Let $\lambda\in \mathbb{R}$, $\phi\in C^\infty(\mathbb{R})$. We define the Stein-Wainger oscillatory integral by
$$I=p.v.\int_\mathbb{R} e^{i\lambda\phi(t)}\frac{dt}{t}.$$
Stein-Wainger [1] showed ...
- 11
1
vote
0
answers
146
views
Functional equation with Fourier transform
What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$:
$$f(x) - \frac{C}{x} \hat{f}(\frac{1}{x}) =x^{\alpha}$$
Where $\hat{f}$ is the Fourier transform of $f(|x|)$ and $C$ a constant....
- 1,199
3
votes
2
answers
1k
views
Limit of the logarithm of the $L^p$ norm over the logarithm of $p$ as $p$ goes to infinity
Let $\mu$ be some positive measure on $\mathbb{R}$. For technical reasons, I would like to know if the limit
$$\lim_{p\rightarrow\infty}\frac {\ln \|f\|_{L^p(\mu)}}{\ln p}$$
exists in $[0,\infty]$ for ...
- 1,429
0
votes
2
answers
387
views
Derivative of fractional Laplacian is the fractional Laplacian of the derivative
Is it true that $$\partial_x ((-\Delta)^s u(x)) = ((-\Delta)^s \partial_x
u(x))?$$
user104647
2
votes
0
answers
70
views
Can the STFT decrease arbitrarily quickly near the origin?
For $f,g \in L^2(\mathbb{R}^d)$ we can define the Short Time Fourier Transform (STFT) $V_gf \in C_0(\mathbb{R}^{2d})$ as $$V_gf(x, \omega) = \int_{\mathbb{R}^d} f \overline{g(t - x)} e^{-2 \pi i t \...
- 1,010
7
votes
1
answer
2k
views
Regularity of Fourier transforms of $L^p$ functions for $2<p\le\infty$
I was recently reading about the Mikhlin and Hörmander Multiplier Theorems, which give conditions for a measurable function $m:\mathbb R^d\to\mathbb C$ to be an $L^p$ multiplier, i.e. for there to ...
- 1,247
3
votes
1
answer
318
views
On some points of Bari's theorem
I am looking to prove some parts of Bari's theorem (1946). Let $H$ be a Hilbert space, and let $(e_n)_{n \in \mathbb{N}}$ be an orthonormal basis of $H$. Let $(x_n)_{n \in \mathbb{N}}$ be a sequence ...
- 309
2
votes
1
answer
183
views
A minimal sequence in $L^2([-\pi, \pi])$
Let $\lambda_n = n + \delta_n $ for all $n \in \mathbb{Z}$ where $\delta_n$ are a sequence of real numbers in $\ell^2(\mathbb{Z})$. How can one show that the sequence $(x_n)_{n \in \mathbb{Z}} = (e^{i ...
- 309
2
votes
0
answers
89
views
Extension of a generalized function to the plane
Let $\phi$ be a generalized function on $\mathbb{R}^2\backslash\{0\}$, and assume that its differential $d\phi$ extends to the whole plane $\mathbb{R}^2$.
Q. Does $\phi$ also extend to $\mathbb{R}^...
- 21.8k
8
votes
1
answer
380
views
Lavrentiev phenomenon between $C^1$ and Lipschitz
Does there exist a (onedimensional) integral functional of calculus of variations (with $f$ finite everywhere)
$$
F(y)=\int_a^b f(t,y(t),y'(t))\,dt
$$
such that
$$
\inf_{y\in Lip([a,b])}F(y)<\inf_{...
- 1,309
7
votes
0
answers
619
views
Lavrentiev Phenomenon
Does there exist a (onedimensional) integral functional of calculus of variations
$$
F(y)=\int_a^b f(t,y(t),y'(t))\,dt
$$
such that not only
$$
\inf_{y\in\operatorname{Lip}([a,b])}F(y)>\inf_{y\in ...
- 1,309
-1
votes
1
answer
136
views
An elementary question about integration by parts! [closed]
Let $f,g: R \rightarrow R$ be two positive increasing functions. Under what (non-trivial) conditions one can guarantee that $\int_{0}^{\infty}f'g dx\geq \int_{0}^{\infty}g'fdx$.
2
votes
0
answers
683
views
Laplace problem with Robin boundary condition on a wedge
I'm trying to understand what the essential differences between Dirichlet/Neumann and Robin boundary conditions are. Therefore, let $\omega \in \left(0, 2\pi\right)$ and let
\begin{equation*}
\Omega = ...
4
votes
0
answers
125
views
Properties of solution to Schrödinger equation
Given a Schrödinger equation with, let's say continuous, periodic potential
$$-y''(x)+V(x)y(x)=\lambda y(x)$$
where $V(x+1)=V(x)$ and $V$ is even, i.e. for $x \in (0,\frac{1}{2})$ we have $V(x+\frac{...
- 501
4
votes
2
answers
410
views
Asymptotic behavior of the solution of the high degree differential equation $(x^{2n}y^{(n)})^{(n)}-x^2y=\lambda \; y$
The following differential equation has two independent solutions, one of the two is decreasing exponentially at infinity (k-Bessel function).
$$(x^2y')'-x^2y=\lambda \;y$$
Now for a higher-degree ...
- 1,199
0
votes
1
answer
218
views
Heat semigroup dissipative
Consider the heat semigroup on $L^1(\mathbb{R}).$ I would like to know if the generator of this semigroup is dissipative in the sense of this definition.
On $L^2$ it would be completely trivial, but ...
- 167
0
votes
1
answer
186
views
Meromorphic solutions to Legendre's equation
I just saw the following question that was asked yesterday on math overflow on meromorphic solutions to ODEs
Although, I understand the answers and comments to the questions, I did not understand how ...
- 501
2
votes
2
answers
258
views
Meromorphic extension of solutions to ODEs
I encountered the following question in my studies:
Let us assume we have a real anlaytic solution to an ODE on $\mathbb{R}$ of Schr\"odinger type
$-\psi''(x)+V(x)\psi(x)=\lambda \psi(x)$
but we ...
- 167
3
votes
1
answer
148
views
Prove existence of continuous function on $(0,1)$ with special properties [closed]
Consider the interval $I=(0,1)$ and let $f,g$ be two linearly independent continuous functions on $[0,1]$.
I am asking if there is a continuous function $h$ such that
$$\int_0^1 h(s) f(s) ds=0$$
$$...
- 501
7
votes
3
answers
2k
views
Collections of examples and counterexamples in (real, complex, functional) analysis, ODEs and PDEs
What books collect examples and counterexamples (or also "solved exercises", for some suitable definition of "exercise") in
real analysis,
complex analysis,
functional analysis,
ODEs,
PDEs?
The ...
user60665
2
votes
0
answers
79
views
One-dimensional integral equation uniquely solvable?
I recently met a question similar to this one and I would like to post it here, because I basically found nothing:
We define the (possibly unbounded) integral operator $T:D(T) \subset C_0(\mathbb{R}) ...
- 329
2
votes
0
answers
350
views
Fractional iteration of a variant of the $\sin()$ function - how to fractionally iterate $ f(x)=\sum_{k=1}^\infty (-1)^k a_{2k}x^{2k}$?
I was reconsidering the fractional iteration of the sine-function and remembering that the power series for the fractional iterates have convergence radius zero I looked at the variant of the sine ...
- 5,342