All Questions
Tagged with fa.functional-analysis ca.classical-analysis-and-odes
161 questions with no upvoted or accepted answers
8
votes
0
answers
349
views
Finding a dimension-free bound for a certain multiplier on Euclidean space
The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...
- 141
8
votes
0
answers
605
views
convergence rate in Wiener's approximation theorem
Wiener has the following fantastic results about approximations using translation families:
Given a function $h: \mathbb{R} \to \mathbb{R}$, the set $\{\sum a_i h(\cdot - x_i): a_i, x_i \in \mathbb{...
- 1,839
7
votes
0
answers
2k
views
Algebraizing topology and analysis via condensed mathematics
I asked this question on Mathematics Stackexchange, but one of the users suggested that I ask this question at MathOverflow.
I've just come across a Twitter thread by Laurent Fargues explaining a work ...
- 79
7
votes
2
answers
824
views
Fourier series of smooth functions in infinitely many variables
Let $J$ be a set (usually countable). Let $t_j$, $j\in J$, be variables in ${\mathbb R}/2\pi i{\mathbb Z}.$ Put $u_j=\exp(it_j),$ $j\in J.$ Introduce the following semi-norms on the space of Fourier ...
- 401
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
$$
\inf_{y\in\operatorname{Lip}([a,b])}F(y)>\inf_{y\in ...
- 1,309
6
votes
0
answers
220
views
Energy of harmonic maps from $\mathbb R^2$ to $S^2$ is quantized
Assume that $U:\mathbb R^2\to S^2=\{y\in\mathbb R^3:|y|=1\}$ is a smooth solution of the equation $\Delta U+|\nabla U|^2U=0$ in $\mathbb R^2$ with $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
- 517
6
votes
0
answers
208
views
Interpolation between (or: simultaneous Whitney extension for) $C^\alpha$ and $C^{1,\gamma}$ on a Lipschitz domain
I would like to know whether for a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$ (in the weak Lipschitz, so a "Lipschitz manifold", sense, not necessarily a Lipschitz graph domain), ...
- 2,670
6
votes
0
answers
411
views
Birth-Death Process associated with Orthogonal Polynomials
I have read in various places the following objects are related:
orthogonal polynomials
birth-death processes
Lattice paths
continued fractions
After a lot of searching online, I found sketches ...
- 22.8k
5
votes
0
answers
168
views
Sobolev extension from a discrete set of points
Let $1 > \alpha > 0$ and fix some $C > 0$. Consider $\Omega \subset \mathbb{R}^n$ a bounded domain and $Y \subset \Omega$ a discrete (finite) set of points. For $f: Y \to \mathbb{R}$ define
$$...
- 501
5
votes
0
answers
91
views
Reference request: sufficiently smooth functions on the plane belong to the projective tensor square of $L^2$ of the line
Let $\newcommand{\ptp}{\widehat{\otimes}}\ptp$ denote the projective tensor product of Banach spaces. Some back of the envelope calcuations, using the Fourier transform and Plancherel/Parseval, ...
- 25.8k
5
votes
0
answers
373
views
A question about Carleman linearization
Carleman linearization is a technique used to embed a finite dimensional system of analytic ordinary differential equations into an infinite system of linear differential equations:¹⁻²
Let $f$ be ...
- 1,590
5
votes
0
answers
141
views
Rate of convergence of Riemann sum of quasi-regular functions
The following result is well-known (I consider the 3-dimensional case only):
Theorem: if $f \in H^s(\mathbb{R}^3)$ with $ s > 3/2$ is compactly supported, then
$$
\left| \int_{\mathbb{R}^3} f - \...
- 53
5
votes
0
answers
913
views
Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?
Preliminary Definitions
Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
- 103
5
votes
0
answers
584
views
Constructing a Sobolev space containing the differential k-forms of a Riemannian manifold
I am currently writing a paper about the Hodge theorem for an algebraic topology course. The specific formulation I am proving can be stated thus. Let $M$ be a compact, orientable n-dimensional ...
user34072
4
votes
0
answers
127
views
What can be possible conditions for the solution of an autonomous ODE to be conservative with respect to the initial data?
Let $F : \mathbb{R}^n \to \mathbb{R}^n$ be a smooth mapping and consider the following autonomous ODE
\begin{equation}
y'(t)=F(y(t))
\end{equation}
with the initial data $y(0)=x \in \mathbb{R}^n$.
...
- 3,477
4
votes
0
answers
174
views
Techniques for showing non-degeneracy results (PDE)
Motivation:
Consider the equation,
$$-\Delta u = u^p$$
in $\mathbb{R}^n$ for $n\geq 3$ and $p=2^*-1.$ Then we know that this equation has unique positive solutions given by functions of the form $U_{a,...
- 537
4
votes
0
answers
126
views
Relationship between three different definitions of solutions for ODE with irregular coefficient
What is the difference between the notions of
Regular Lagrangian flow
Filippov solution
Caratheodory solution
of an ODE $\dot \Phi(t,x) = b(t,\Phi(t,x))$, with initial condition $\Phi(0,x) = x$, ...
- 839
4
votes
0
answers
170
views
Pointwise convergence of the eigenfunctions expansion of $f(x)=\frac{1}{|x|}$
Let $\Omega\subset \mathbb{R}^n$ a bounded domain with smooth boundary, $0<\lambda_1\leq \lambda_2 \leq \dots \leq \lambda_k\leq \dots$ the Dirichlet eigenvalues and $\{w_k\}_{k=1}^{+\infty}$ an $L^...
user39481
4
votes
0
answers
99
views
Commuting flows problem for non-Lipschitz vector fields
Let $X$ be a continuous vector field on a (say compact) manifold $M$, if $X$ has ODE uniqueness then we can define its associated flow $\mathcal F_X:\mathbb R\times M\to M$ uniquely given by $\mathcal ...
- 938
4
votes
0
answers
125
views
Properties of solution to Schrödinger equation
Given a Schrödinger equation with, let's say continuous, periodic potential
$$-y''(x)+V(x)y(x)=\lambda y(x)$$
where $V(x+1)=V(x)$ and $V$ is even, i.e. for $x \in (0,\frac{1}{2})$ we have $V(x+\frac{...
- 501
4
votes
0
answers
188
views
Evaluate a multiple integral
I want to compute this integral and I would appreciate any help: $N\geq 1$ is fixed.
$$I_N=\int_{0\le r_n\le r_{n-1}\le\cdots\le r_1} e^{-(r_1^2+\cdots+r_n^2)} \prod_{i<j} \sinh(r_i-r_j) dr_1\...
- 41
4
votes
0
answers
820
views
Calderón's complex interpolation: what is the corresponding classical theorem?
This question is closely related to my answer to Dan's question, which contains the definitions of some terms I use here. In addition, the notion of exact interpolation functor of exponent $\theta$ is ...
- 1,202
4
votes
0
answers
487
views
Convolutions and Toeplitz Operators
Let be $d>0$ an integer number and consider the Cartesian product $\mathbb Z^d$ as metric space, with the distance between $x,y\in\mathbb Z^d$ given by $\|x-y\|_1=\sum_{j=0}^d|x_j-y_j|$.
Let be $...
- 2,044
3
votes
0
answers
124
views
Estimating a solution to Euler-type ODE #2
This is a similar question to this but with a different ODE.
Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer, $R&...
- 969
3
votes
0
answers
111
views
Complex interpolation of Fourier-Lebesgue spaces and Lebesgue spaces
The complex interpolation of Lebesgue spaces $L^{p_0}$ and $L^{p_1}$ are well-known, that is,
$[L^{p_0}, L^{p_1}]_\theta = L^p$, where $(1-\theta)/p_0+1/p_1 = 1/p$ for some $\theta \in [0,1]$.
Now I ...
- 193
3
votes
0
answers
110
views
Tauberian theorem with flatness condition
Suppose $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is a series with $a_n\in \mathbb{R}$ and radius of convergence $1$ and such that $f$ restricted to $[0,1[$ admits a smooth extension to $[0,1]$ with $f^{(n)}(1)...
- 278
3
votes
0
answers
84
views
A weighted $W^{2,p}$ estimates
Let $\Omega$ be a bounded smooth domain and $u\in W^{2,p}(\Omega)\cap H^1_0(\Omega)$. By the classical $L^p$ theory of second order elliptic equation, we have
$$
\|\nabla^2u\|_{L^p(\Omega)}\leq C\|\...
- 379
3
votes
0
answers
60
views
Existence, Uniqueness, and "ODE Characterization" of Minimizers for Variational Functionals from Large Deviations
A [classical result][1] of E. Lieb is that the functional
$$\mathcal E(\phi):=\int_{\mathbb R^3}|\nabla\phi(x)|^2~dx-\int_{(\mathbb R^3)^2}\frac{|\phi(x)|^2|\phi(y)|^2}{|x-y|}~dx~dy$$
for $\phi\in W^1(...
- 891
3
votes
0
answers
151
views
Completeness of discrete shifts in $\mathbb{R}^n$
Consider the space $L^2(\mathbb{R})$. Let $(x_n)_n \subset \mathbb{R}$ be a sequence and $f \in L^2(\mathbb{R})$ a functions. It is well understood under which assumptions the span of the set
$$
S = \{...
- 173
3
votes
1
answer
490
views
Space derivative of flow of ODE with monotone source
Consider the ODE
$$
\begin{cases}
\partial_t\Phi(t,x) = f(t,\Phi(t,x)), &\ t>0, \ x \in \mathbb R \\
\Phi(0,x) = x, & x \in \mathbb R
\end{cases}
$$
where $f$ is function which is a non-...
- 109
3
votes
0
answers
73
views
"Almost" absolute continuity of derivative of BV function if ${\rm Tr}\,D_Sf = 0$
Let $f: \mathbb R^N \to \mathbb R^N$ be a $BV$ function.
Suppose that $\mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure: $\operatorname{div} f \ll \mathcal L^N$. This ...
- 839
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 $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...
user127612
3
votes
0
answers
177
views
Interesting stipulation about completely monotone functions
This question relates to a question I asked here. I thought of a well thought out generalization which appears to follow in the situations I've encountered it. I tried to generalize the answer ...
user78249
3
votes
0
answers
354
views
A Characterization of Closed Ideals in $C^{\infty}(\mathbb{R}^n)$
The space $C^{\infty}(\mathbb{R}^n)$ can be turned into a topological ring using the Whitney topology. Whitney's Spectral Theorem says that the closure of an ideal in this ring is the ideal of all ...
- 343
3
votes
0
answers
146
views
Variational Principle for a System of Differential Equations
I am studying a differential operator of the form $$ 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
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 ...
- 16.6k
3
votes
0
answers
188
views
Does the existence of an asymtpotic density imply the existence of a measure on infinite dimensional (path) space?
This question is related to the following question
Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
A couple of authors have observed that composing a ...
- 265
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:
$$A:=\left(
\begin{array}{cccccc}
1 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 &...
user136033
2
votes
0
answers
179
views
Analytic continuation of $\int_V (f(x_1,\cdots,x_n))^s dx_i$
Let $V$ be an $n$-dimensional simplex, let $f(\boldsymbol{x}) = f(x_1,\cdots,x_n)\in \mathbb{C}[x_1,\cdots,x_n]$ be a product of linear polynomials that is non-zero in interior of $V$. Also let $E(\...
- 528
2
votes
0
answers
102
views
Existence of unique-up-to-shift solution of a Volterra equation
Let $\Delta=\{(t,s):\ 0<s\leq t\leq1\}$, and suppose $k:\Delta\to\mathbb R$ and $f:(0,1]\to\mathbb R$ are continuous. Further assume that for every $t\in(0,1]$, the function $k(t,\cdot):(0,t]\to\...
- 537
2
votes
0
answers
97
views
On the second order analog of the upper 1-Lipschitz envelope of a function
Let $u: \mathbb R \to \mathbb R$ be a given function. Then we can consider its upper 1-Lip envelope
$$
\hat u(x) \doteq \inf\{g(x) \, \mid\, g \, \text{has Lipschitz constant 1 and}\, g(y) \geq u(y) \,...
- 355
2
votes
0
answers
81
views
Extension of a tangent vector field
Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...
- 434
2
votes
0
answers
138
views
Sufficient initial conditions for "non-local" PDE
I am studying a problem of the form $$i\, \partial_t \psi(t) = L \psi(t) + \int_0^t U(t-r) \psi(r) \, dr, \qquad \psi(0) = \psi_0,$$ where the evolution occurs in some Hilbert space, $L$ is a self-...
- 71
2
votes
0
answers
206
views
Failure of Calderón–Zygmund inequality at the endpoints
$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$I'd like to prove that the famous Calderón–Zygmund elliptic estimate $$ \norm{ \partial_{ij}u }_{L^p} \leq C \norm{\Delta u }_{L^...
- 457
2
votes
0
answers
136
views
Eigenfunction of $h\mapsto H(h')|_{[-1,1]}$?
Let $H$ be the Hilbert transform. Is there a continuous, even function $h:\mathbb{R}\to \mathbb{R}$ with support on $[-1,1]$ such that, for some $\lambda\in \mathbb{R}$,
$$H(h')(t) = \lambda h(t)$$
...
- 20.2k
2
votes
0
answers
201
views
Green function of a 2D exterior domain
Consider solutions of the laplace equation
\begin{equation}
\begin{split}
-\Delta u=f, \ \ u|_{\partial D}=0,
\end{split}
\end{equation}
where the domain $D\subset \mathbb{R}^2$. If $D$ is bounded ...
- 379
2
votes
0
answers
64
views
Scaling limit of ODE with double-well potential
Let us consider the ODE
$$
\frac{d}{dt}x_\epsilon(t) = -\frac{1}{\epsilon} W'(x_\epsilon(t))
$$
where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the ODE reads
$$...
- 839
2
votes
0
answers
170
views
Equivalence of implicit function theorem and Peano existence theorem in ODEs?
I was recently reading a book about the implicit function theorem (IFT): The implicit function theorem: history, theory, and applications, and before that I learned that Peano's existence theorem can ...
- 181
2
votes
0
answers
216
views
Fourier transform of Dirac delta distribution
Let $f,g$ be Schwartz functions on $\mathbb R^4$, we denote them as $\mathcal S(\mathbb R^4)$, one can then define the transform $V$ mapping $f,g$ to a Schwartz function $\mathcal S(\mathbb R^8)$
$$ V(...
- 73
2
votes
0
answers
117
views
Bounding integral expression with BV norm of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\...
user139844