All Questions
Tagged with fa.functional-analysis reference-request
1,021 questions
25
votes
3
answers
13k
views
Fourier transform of the unit sphere
The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula
$$
\int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu ...
- 14k
11
votes
1
answer
486
views
Resources for divergent / asymptotic series
This series is divergent; therefore, we may be able to do something with it. -- Oliver Heaviside
[Edit (1/14/21) from the answer by Count Iblis to a recent MO-Q on math vids: An enthusiastic intro is ...
- 10.5k
3
votes
0
answers
182
views
Parabolic regularization for the Navier-Stokes equations
I'm looking for some references about a result on the Navier-Stokes equations which seems to be folklore but for which I didn't manage to find a proof. The setting is the following :
Let $Q=\mathbb{R}^...
- 161
8
votes
0
answers
360
views
The many theories of integration
Diclaimer: In what follows, I will be loose in the usage of terminology since the very nature of the question is of a similar flavour.
In the mathematics literature, one can find a zoo of theories of ...
- 183
4
votes
0
answers
95
views
When the Jacobian of unstable measure converges
Let $T:X \to X$ be a hyperbolic map on the compact metric space $X$. Hyperbolicity means that $T$ has local stable and unstable sets with uniform exponential bounds, which satisfy a local product ...
- 1,043
1
vote
0
answers
36
views
Existence and uniqueness for fractional parabolic equation with transport term
Let us consider the problem
\begin{equation}
\begin{cases}
u_t+(-\Delta)^{\sigma}u+\mathrm{div}(a(t,x)u) = 0 & \text{in } \mathbb{R}^n \times [0, T) \\
u(x,0)=u_0(x) & \text{in } \...
- 99
3
votes
0
answers
111
views
Infinite ordered products (reference request)
While writing arXiv:1510.05757v2, I found myself proving some basic facts about products of Banach algebra elements over an infinite totally ordered set. (Statements and proofs are in Appendix C.) The ...
- 3,065
4
votes
0
answers
65
views
A standard name of a strongly extremal point of a convex set
I need to name somehow points $x$ of a bounded convex set $C$ in a Banach space $X$ such that the set $$\{x^*\in X^*:x^*(x)=\max x^*[C]\}$$ of support functionals at $x$ has non-empty interior in the ...
- 41.8k
1
vote
0
answers
280
views
On measurability of certain group actions on spaces of bounded measurable functions
Let $\mathcal{H}$ be a separable Hilbert space equipped with a strongly continuous unitary representation of a locally compact group $G$. Denote by $\mathcal{L}^{\infty}(H)$ the space of the bounded ...
- 505
4
votes
0
answers
147
views
Continuous extension preserving modulus of continuity
Let $X$ be a (non-empty) compact subset of $D(0,M):=\left\{x\in \mathbb{R}^n:\, \|x\|\leq M\right\}$, and let $f:X\rightarrow Y$ be uniformly continuous; for some metric space $Y$. Are there any ...
- 63.9k
3
votes
1
answer
178
views
Banach Mazur distance between the cube and the cross-polytope in the dimensions for which a Hadamard matrix exists
The Banach-Mazur distance between two centrally symmetric convex bodies $K,L\in\mathbb{R}^n$ can be defined as
$$ d(K,L) = \inf \{ r : \exists T\colon \mathbb{R}^n \to \mathbb{R}^n \text{ linear such ...
- 183
6
votes
1
answer
573
views
English translation of Schwartz's papers on vector-valued distributions
I am interested in systematically studying the theory of vector-valued distributions. The original two papers due to Laurent Schwartz entitled Théorie des distributions à valeurs vectorielles. I & ...
- 188k
0
votes
0
answers
104
views
Rigorous energy estimate for advection-diffusion equation
Let $a \in L^q([0,T];L^p(\mathbb R^N))$ with $2/q + N/p \le 1$ and
$q \in [2,\infty), p \in (N,\infty) \text{ if } N \ge 2$
$q \in [2,4], p \in [2,\infty] \text{ if } N = 1$
and consider the ...
- 61
1
vote
1
answer
195
views
Existence and regularity for fractional elliptic problem with gradient term: $ (-\Delta)^s u + v\cdot \nabla u = 0$ with $v \in \dot H^s$
Let us consider the problem
$$ (-\Delta)^s u + v\cdot \nabla u = 0 \quad \text{ in } \mathbb R^n, $$
where $s \in (0,1)$, $(-\Delta)^s$ is the fractional Laplace operator and
$v:\mathbb R^n \to \...
- 99
3
votes
1
answer
146
views
Translation of a paper by Salvatore Pincherle
Anyone know of an English translation of the oft-cited paper "Funktionaloperationen und Gleichungen" by Salvatore Pincherle, Encyklopädie der Mathematischen Wissenschaften mit Einschluss ...
- 3,065
9
votes
4
answers
905
views
Defining the value of a distribution at a point
Let $\omega\in D'(\mathbb R^n)$ be a distribution and $p\in \mathbb R^n$. If there is an open set $U\subset \mathbb R^n$ containing $p$ such that $\omega|_U$ is given by a continuous function $f\in C(...
- 6,367
3
votes
1
answer
1k
views
Friedrichs mollifiers and Sobolev spaces
$\renewcommand{\epsilon}{\varepsilon}$The following is from John Roe's book Elliptic operators, topology and asymptotic methods. $S$ is a vector bundle on a compact manifold $M$, but I think for my ...
- 1,942
1
vote
0
answers
66
views
Well-posedness of hyperbolic system with constant coefficients in finite domains
I'm studying the PDE
$$
\frac{\partial u}{\partial t} + A_x\frac{\partial u}{\partial x} + A_y\frac{\partial u}{\partial y} + A_z\frac{\partial u}{\partial z} = 0
$$
with $A_x, A_y, A_z$ being ...
CommunityBot
- 1
2
votes
0
answers
145
views
Are Weyl sequences polynomially bounded?
Look at the Hilbert space $l^2( \mathbb{Z}) $ and let $A$ be a translation invariant band operator. I.e. if $\{ e_n \}_{n \in \mathbb Z} $ is the standard basis for $l^2( \mathbb{Z}) $ then it holds ...
- 1,054
4
votes
1
answer
312
views
Literature on the product of two distributions satisfying the Hörmander condition
I am currently studying some basic questions concerning the product $uv\in \mathscr{D}'(\mathbb R^n)$ of two Schwartz distributions $u,v\in \mathscr{D}'(\mathbb R^n)$ satisfying the Hörmander ...
- 1,942
2
votes
0
answers
72
views
Product of Besov and Lorentz functions
Let us fix $n\in\mathbb{N}^+$ and $p,q\in [1,\infty)$. Given $r_1,r_2,r_3\in[1,\infty)$, I would like to understand whether we have the bound
$$
\|fg\|_{L^{q,r_3}(\mathbb{R}^n)}\lesssim \|f\|_{B^{n/p}...
- 282
5
votes
0
answers
61
views
Minimizers of variational problems with less symmetry
In am interested in understanding the so-called Hartree ground states, namely, minimizers of variational problems of the form
$$\inf_{\phi\in H^1,~\|\phi\|_2=1}\left\{\int|\nabla\phi(x)|^2dx
-\int|\...
- 891
3
votes
0
answers
104
views
Regular Lagrangian flows on a domain of $\mathbb{R}^d$ with a boundary
I'm looking for some references about the theory of regular Lagrangian flows on a smooth domain $\Omega$ of $\mathbb{R}^d$ (say a smooth bounded open set of $\mathbb{R}^d$ or a half space).
Here, ...
- 161
1
vote
1
answer
111
views
Sum of positive self-adjoint operator and an imaginary "potential": literature request
To keep things simple, let us consider the following: $L$ is a positive, unbounded S.A. operator on $L_2(\mathbb{R},f(x))$, where $f(x)$ is a Gaussian. Assume that we know the spectrum and ...
- 188k
5
votes
1
answer
297
views
A scaled fractional ''Sobolev inequality''
Does a fractional interpolation inequality similar to $$
\int_{B_R(0)} |u| dx \le C R^{2} \sqrt{\log(2R)} \Big( \int_{\mathbb R^2}\int_{\mathbb R^2} \frac{|u(x)-u(y)|^2}{|x-y|^{2+2s}} dxdy + \int_{B_1(...
- 61.9k
4
votes
0
answers
159
views
Is there a name for this geometric property of metric spaces?
My research has lead me to metric spaces $(M, \rho)$ which have the following geometric property:
Suppose $x, y \in M$ and $r, s > 0$ such that
$(x, r) \neq (y, s)$,
$B[y; s] \subseteq B[x; r]$,
$...
- 191
4
votes
0
answers
176
views
If $u \in W^{\alpha,p}(0,T;X)$ for $\alpha \in (0,1)$, then $f(u) \in W^{\alpha,q}(0,T;Y)$ for some good $f:X \to Y$
Let $u \in W^{\alpha,p}(0,T;X)$ for some reflexive Banach space $X$ (you can also take a Hilbert space if it helps) for $\alpha \in (0,1)$ and $p \geq 2$. I am fine with both the Sobolev-Slobodeckij ...
- 51
2
votes
0
answers
169
views
Functions whose Fourier coefficients satisfy $ \sum_{k=1}^\infty |c_k| < 1 $?
Let $f:(0,1) \to \mathbb R$ be a function that can be written as $$f(x) = \sum_{k=1}^\infty c_k \phi_k(x),$$ where $\phi_k(x) = \cos(\pi k x)$. What is the minimal assumption required on $f$ to ...
- 839
1
vote
0
answers
42
views
On the boundary integral of Neumann eigenfunctions
Let $v$ be an eigenfunction corresponding to the first nonzero Neumann Laplacian eigenvalue on a domain $\Omega \subset \mathbb{R}^2$. By definition, we know that $\int_{\Omega} v \, dx=0$. If $\Omega$...
- 1,350
3
votes
1
answer
94
views
Applications of coupled Volterra-Hammerstein in Banach space
I'm looking to study the existence solutions of the following coupled equation:
\begin{equation}
\left\{\begin{matrix}
x(t)&=&\int_{0}^{t} K\big(t, s\big) f\big(s, x(s),y(s)\big) d s, \quad t \...
- 188k
4
votes
1
answer
256
views
Regularity of Nemitskii maps on Sobolev spaces
Let $\Omega\subset \mathbb R^N$ be a bounded smooth domain, and $f\colon\Omega\times \mathbb R\to \mathbb R$ be a smooth function (let's say $C^2$).
Let $X=W^{1,p}(\Omega)$ with $p>1$ be the ...
- 63.9k
0
votes
1
answer
106
views
Existence of uniform approximator that also approximates derivative
Let $S$ be a subset of $C^1([0, 1], \mathbb{R})$. It is a well-known fact that given a function $f\in C^1([0, 1], \mathbb{R})$ and a sequence $\{f_n\}\subset C^1([0,1], \mathbb{R})$ such that $f_n\to ...
CommunityBot
- 1
7
votes
0
answers
351
views
Fractional Laplacian and chain rule
For the classical Laplacian, we have
$$\Delta (h(u)) = h'\Delta u + h''(u)|\nabla u|^2$$
for smooth functions $h$ and $u$.
Does a similar chain rule hold (up to a reminder term) also for the ...
- 161
3
votes
1
answer
951
views
Specific criterion for the sum of two closed sets to be closed
Let $Y$ and $Z$ be two closed subspaces of a Banach space $X$ with $Y\cap Z=\{0\}$.
I know that $Y+Z$ is a closed subspace of $X$ $\iff \exists \alpha > 0:\quad \lVert y\rVert \le \alpha\lVert y+z\...
- 12.9k
1
vote
0
answers
123
views
Derivatives of measures of bounded variation on intervals
Investigating an abstract Cauchy problem on the space of measures with bounded variation I came up with the following space:
Let $\operatorname{BV}[a,b]$ the space of all functions $f:[0, 1] \to \...
- 301
3
votes
3
answers
455
views
$\ell^1$ functor as left adjoint to unit ball functor
In a comment to this answer
https://mathoverflow.net/a/38755/1106
Yemon Choi notes that "The $\ell^1$ functor is the free Banach space functor, left adjoint to the forgetful unit ball functor&...
- 118k
3
votes
2
answers
782
views
Relation between optimal transport cost and difference between topological invariants?
I was working on some mathematics of Wasserstein GAN and found out a seemingly interesting research problem but I am not quite sure whether it has already been studied in some recent literature of ...
- 2,274
3
votes
1
answer
688
views
Positive definite kernels involving the $\min$ function
I am interested in the positive kernels of the form $k(x,y) = \min\{a(x,y), b(x,y)\}$ (assuming $k(x,y) = k(y,x)$). Some examples including $\min\{x,y\}$ and $\min\{f(x)g(y), f(y)g(x)\}$, but are ...
- 4,713
1
vote
1
answer
196
views
Holomorphic semigroups on $L^1$ spaces
Let $E$ be a locally compact metric space and $\mu$ a non-negative Radon measure on $E$ (we also assume that the support is $E$).
I am concerned with holomorphic semigroups on $L^1(E,\mu)$. In ...
- 12.6k
1
vote
0
answers
62
views
Properties of the Fourier Transform of Countably Supported Functions on $[0,1)$
Identifying $\mathbb{R}/\mathbb{Z}$ with the interval $\left[0,1\right)$, let $C_{\textrm{coun}}\left(\mathbb{R}/\mathbb{Z}\right)$ denote the set of all functions $f:\mathbb{R}/\mathbb{Z}\rightarrow\...
- 1,284
1
vote
1
answer
217
views
A result on the convergence of the vanishing viscosity approximation to the viscosity solution of an IVP for Hamilton-Jacobi equation
In [CL84], Crandall and Lions considered an initial value problem for an Hamilton-Jacobi equation and proved [CL84, Theorem 5.1] a result on the convergence of the vanishing viscosity approximation to ...
CommunityBot
- 1
2
votes
0
answers
86
views
Eigenvalues of the operator $A = -v'' + B(x) v$
How can I prove that for the eigenvalues of the operator $$A := -v'' + B(x) v$$ on $(0,L)$ with zero Dirichlet boundary condition it holds that
$$
\left| \lambda_n - \frac{\pi^2n^2}{L^2}\right| \le ||...
- 217
3
votes
1
answer
190
views
Laplace eigenfunction on a polygonal domain symmetric about an axis
Consider a polygon $\Omega \subseteq \mathbb{R}^2$, and let us consider the usual Laplacian operator $\Delta = \partial_x^2 + \partial_y^2$, with Dirichlet boundary conditions. My question comes from ...
- 24.2k
1
vote
0
answers
74
views
Dimension dependence: boundedness result of the fractional Riesz integral
I am looking for the best known constant in the boundedness result of the fractional Riesz integral. In particular, I am interested in the dependence on the dimension $d$ and on the parameter $\alpha&...
- 63.9k
1
vote
0
answers
42
views
Energy estimate for $\theta_t + H(\theta)_x = 0$ in $t>0, x >0$?
Consider the IBVP for $$\theta_t + H(\theta)_x = 0, \qquad t>0, \ x>0$$ with $$H(\theta) = \frac{1}{\pi} \text{pv}\int_{0}^\infty \frac{\theta(y)}{y-x} dy$$
with Dirichlet boundary conditions. ...
- 161
1
vote
0
answers
122
views
Metric transforms that preserve $\ell^1$ embeddability
Consider a function $f$ from reals to reals such that $f$, when applied to pairwise Manhattan distances between $n$ points, always results in a set of Manhattan distances.
Work by Schoenberg and ...
- 63.9k
3
votes
3
answers
562
views
Did anyone ever introduce an "oscillating unity"?
I wonder whether anyone ever tried to introduce an extension of real numbers by adding an element $\nu$ which would signify the behavior of the function $(-1)^x$ as $x$ goes to infinity?
In other ...
- 41
0
votes
0
answers
44
views
Solving nonlinear equations involving expectations
Let $X$ be a random variable and $g(x,y)$ be a function of two variables. Consider the equation
$$
\mathbb{E}_Xg(X,y) = 0
$$
Are there any specialized techniques for solving such equations (...
- 11
1
vote
0
answers
74
views
Good source for Jordan Fréchet algebras
Is there any good source for Jordan Fréchet (or more generally, Jordan locally convex) algebras?
I'm looking for something on the level similar to the level of the book "Banach and Locally Convex ...
- 63.9k
0
votes
1
answer
86
views
Kolmogorov entropy of a subset of $L^1$
How can we estimate the Kolmogorov $\epsilon$-entropy
$$H_\epsilon (A,L^1(\mathbb R))$$
where
$
A = \{f:\mathbb R \to [0,K] \text{ s.t. $f \in L^1$ and has total variation $TV(f) \le M$}\}
$?
- 128k