All Questions
Tagged with fa.functional-analysis reference-request
1,021 questions
3
votes
1
answer
246
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Stone-Weierstrass theorem for modules of non-self-adjoint subalgebras
In "Weierstrass-Stone, the Theorem" by Joao Prolla, there is a Stone-Weierstrass theorem for modules, stated as the following:
Let $\mathcal{A}$ be a subalegebra of $C(X, \mathbb{R})$ and $...
- 25.8k
3
votes
1
answer
141
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A function $V : \mathbb{R}^2 \to \mathbb{R}$ is a (logarithmic) potential
I'm looking for references given some sort of inverse problem in logarithmic potential theory. That is, given a function $V : \mathbb{R}^2 \to \mathbb{R}$, what is a sufficient (and perhaps necessary) ...
- 91.8k
0
votes
0
answers
122
views
Isolated points of the spectra of self-adjoint operators on Hilbert spaces
Let $T$ be a (everywhere defined) self-adjoint operator on a complex Hilbert space $\mathcal{H}$.
I am interested in results that give (non-trivial, possibly mild) sufficient conditions on $T$ to ...
3
votes
1
answer
109
views
Family of Pettis integrals functions "uniformly approximated" by sums
In this book (proof of $4.1.3.$ Lemma. exactly), one can find this passage, that I tried to rephrase here:
Let $f:I\times E\rightarrow E$ a Pettis integrable function, where $I:=[0,T]\subset \mathbb{...
- 95
9
votes
2
answers
611
views
When does $\left\Vert f(\mathbf{N}) - f(\mathbf{M})\right\Vert_{\mathrm{op}} \leq k\left\Vert \mathbf{N} - \mathbf{M}\right\Vert_{\mathrm{op}}$ hold?
Define the Frobenius norm of a matrix as $\left\Vert A \right\Vert_{\mathrm{F}}=\sqrt{\sum_{i,j} A_{ij}^2}$ and the operator norm as $\left\Vert A \right\Vert_{\mathrm{op}}=\sup_{x \not = 0} \frac{\...
- 63.9k
1
vote
1
answer
136
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Find this reference or an alternative where I can find this result
I need this reference, but I couldn't find it online as a PDF. Any help please?
J. Sun, X, Zhang, The fixed point theorem of convex-power condensing operator and applications to abstract semilinear ...
- 188k
6
votes
2
answers
1k
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Vector-Valued Stone-Weierstrass Theorem?
The standard statement of the Stone-Weierstrass theorem is:
Let $X$ be compact Hausdorff topological space, and $\mathcal{A}$ a subalgebra of the continuous functions from $X$ to $\mathbb{R}$ which ...
- 263
9
votes
1
answer
202
views
Literature request: Schatten class difference of semigroups
Let $\mathcal{H}$ be a Hilbert space and $A,B$ two operators on it (not necessarily self-adjoint) such that $A, A+B$ are generators of strongly continuous one parameter semigroups $e^{-tA},e^{-t(A+B)}$...
- 571
1
vote
0
answers
56
views
Moduli of continuity and Wasserstein differentiability of functions between measures
Let $X=\mathbb{R}^n$; I am also interested in the general case $X$ is a metric space but for simplicity let's focus on Euclidean space. Let $\mathcal{P}(X)$ denote the space of Borel probability ...
- 11
1
vote
0
answers
72
views
Multivarate "RKHS" Examples
I've been reading about RKHSs and Hilbert spaces of functions these days a bit these days and I haven't yet come across an example of a hilbert space $H$ whose elements are all functions $f:\mathbb{R}^...
- 5,405
4
votes
0
answers
146
views
When does an operator from $\ell_1$ to itself factor through $\ell_p$?
I would like to know whether a given operator from $\ell_1$ to itself, given by a matrix $A$, factors through $\ell_p$, for $p>1$.Does anyone know any references/results on this topic? I am ...
- 53
0
votes
1
answer
345
views
$L^p$ norm inequalities with respect to strongly-log-concave densities
Let $\pi(x)=\frac{e^{-f(x)}}{\int_{\mathbb{R}^d}e^{-f(u)}du}$ be a strongly-log-concave distribution, i.e., $f(x):\mathbb{R}^d\rightarrow R$ is an $m$-strongly convex function. Also, $f(x)$ has $L$-...
- 5,429
5
votes
0
answers
153
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Characterizing Besov spaces in terms of p-variation
For $s>1/p$ the Besov space $B_{p,q}^s([0,1])$ can be characterized in terms of the $p$-variation:
Let $p,q \in (1,\infty)$ and $s \in (0,1)$, $s>1/p$. A function $f:[0,1] \to \mathbb{R}$ is in ...
- 151
3
votes
0
answers
127
views
Rigorous stability analysis of infinite dimensional ODEs : How to bound the tails?
My question is about linear stability analysis of dynamical systems obtained by discretizing linear(ized) partial differential equations. Consider,
$\dot{x}=Ax$, where $x$ is the infinite dimensional ...
- 2,281
2
votes
1
answer
266
views
Approximate selection for finite-valued upper hemicontinuous/semicontinuous maps?
I'd like to know if there are any known-results on the existence of continuous approximation theorems for upper hemicontinuous (aka upper semicontinuous) maps $\phi: X\rightarrow Y$ which are finite ...
- 63.9k
1
vote
1
answer
387
views
$L^p$ compactness for a sequence of functions from compactness of product with cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 6,035
1
vote
1
answer
426
views
$L^p$ compactness for a sequence of functions from compactness of cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 2,670
8
votes
5
answers
545
views
Reference for : a Fréchet nuclear space is Montel
I'm looking for a reference to cite regarding the property presented in the title: "Closed and bounded sets of a nuclear Fréchet space are compact"
Thank you in advance for the help!
3
votes
0
answers
89
views
Error rate implying regularity
My question is a bit general/vague.
It is well known that the regularity of certain functions can be measured through the rate of decay of certain error quantity based on an approximation procedure (...
- 63.9k
0
votes
0
answers
113
views
Viewing limit as a map
Question: Let $X$ be a set of functions from $\mathbb{R}$ to itself. Consider the subset $X_0$ of the sequences $(f_n)_{n=1}^{\infty}\in X^{\mathbb{N}}$ for which
$$
f_{\infty}(x) = \lim\limits_{n \...
- 5,405
4
votes
3
answers
669
views
Gaussian bounds on Dirichlet heat kernel
Let $(M, g)$ be a compact Riemannian manifold and let $p(t, x , y)$ be the heat kernel of $M$. Then there exist constants $c, C > 0
$ such that $$\frac{c}{t^{n/2}}
e^{-\frac{1}{4t}d(x, y)^2} \leq ...
- 3,223
9
votes
1
answer
2k
views
Density of smooth functions on Hölder spaces
The following result is often cited without reference in the context of PDEs:
Let $\varOmega \subset\mathbb R^n$ be a bounded open set with smooth boundary. If $0<\beta<\alpha<1$ then $C^\...
- 42.8k
3
votes
1
answer
228
views
Reference request: Variational techniques for complex "iterated" Lagrangians
I am interested in solving variational problems of the form
$$
\min_u \int \Big\{L(x,y,u(x,y)) + \phi\Big(\int J(z,y,u(z,y))\,dz\Big)\Big\} p(x,y)\,dx\,dy.
$$
for some known, smooth functions $L,J,\...
CommunityBot
- 1
1
vote
0
answers
66
views
Outer-regular product of $\tau$-additive measures
Due to the deficiencies of the simple product measure defined on measurable rectangles, there have been many different constructions of product measures in more specialized circumstances.
Originally, ...
- 3,816
13
votes
2
answers
570
views
A conjecture of De Giorgi on weighted Sobolev spaces
Let $\mu$ be a probability measure on $\mathbb{R}^d$ which is absolutely continuous with respect to the Lebesgue measure with density $\rho$. Assume that, for all $t>0$,
\begin{align*}
\exp \left(...
- 778
9
votes
3
answers
2k
views
Real analyticity of solution of heat equation
Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient ...
- 1
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
138
views
A good reference for Bochner spaces
I am looking for some references on Bochner spaces containing basic stuff such as measurability, convergence and $L^p$ theory. I already have the Analysis in Banach Spaces: Volume I book which covers ...
- 597
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
1
answer
134
views
Equality in spectral inclusion theorem
I asked this question on Math SE but didn't receive any response.
Let $(T_t)$ be a $C_0$-semigroup on a Banach space $X$ with generator $A.$ If $\lambda_0\in \mathbb{C}$ is such that $e^{\lambda_0 t}$...
- 12.6k
3
votes
1
answer
409
views
Riesz transform of fractional operators
I am interested in Riesz transforms linked to the fractional Laplacian and other fractional operators. I have been hunting down in the literature to find related results but I have not been able to ...
- 778
6
votes
3
answers
852
views
Are nuclear operators closed under extensions?
Given $X_i, Y_i$ Banach spaces, $f_j, g_j, T_i$ bounded linear operators for $i=1,2,3$ and $j=1,2$. We have the following diagram
$\require{AMScd}$
\begin{CD}
0 @>>> X_1 @>f_1>> X_2 ...
- 4,461
4
votes
2
answers
691
views
Uniqueness of viscosity solutions of Hamilton-Jacobi equation
Consider the following Hamilton-Jacobi (HJ) equation:
$$u_t + H(\nabla u,x) = 0 \quad \text{ in } \mathbb{R}^n \times (0, T], $$ where $u:\mathbb{R}^n \times (0,T] \to \mathbb{R}$, and $H:\mathbb{R}^n ...
- 4,713
4
votes
1
answer
129
views
Meaningful generalization of viscosity solutions to higher order equations
Is there a meaningful generalization of the notion of viscosity solutions to third and fourth order equations?
- 5,380
4
votes
1
answer
370
views
Equivalence of viscosity and weak solutions for the Poisson equation
Suppose $\Omega$ is a bounded smooth domain in $\mathbb{R}^d$.
How does one prove that weak solutions are viscosity solutions and vice versa for the problem
$$
\begin{cases}
-\Delta u = f(x) & \...
- 5,380
0
votes
1
answer
178
views
Convergence in LB-spaces
Edit:
Let $X$ be a strict LB-space described by $\lim X_n$ and suppose that $\{x_n\}_{n \in \mathbb{N}}$ converges in $X$. I'm looking for a reference showing that $x_n$ must converge in some $X_N$.
- 16.4k
6
votes
2
answers
390
views
Another reference for higher order Fourier analysis
I am trying to read Tao's Higher order Fourier analysis but I would be very happy to find another book on the subject. I would like to learn something about the Gowers norm and about Roth's theorem (...
CommunityBot
- 1
9
votes
1
answer
429
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Is $\mathcal{S}(\mathbb{R}^n)$ a tame Fréchet space?
Hamilton's paper "The Inverse Function theorem of Nash and Moser" (1982, Bull. Amer. Math. Soc, vol. 7, n. 1, page $137$) proves that $C^{\infty}(M)$ is a tame Fréchet space when $M$ is a compact ...
- 16.4k
7
votes
1
answer
403
views
Why are we interested in operators that share a basis of eigenfunctions?
I hope this is an appropriate question for this forum. If not, I apologize. Before stating my question (which may be found at the end of this post), I will attempt to provide sufficient context.
I ...
- 606
1
vote
0
answers
109
views
Is this a positive definite kernel?
Under which conditions on the function :
\begin{array}{l|rcl}
K : & \mathbb R^+ & \longrightarrow & (0, 1)\\
&t & \longmapsto & K(t) \end{array}
is the symmetric ...
0
votes
0
answers
154
views
Use of this space of very rapidly decreasing continuous functions
Let $C_n$ denote the subspace of continuous function on $[0,\infty)$ supported on $[n,n+1]$. Denote the $\ell^p$-direct sum Banach space
$$
V_p
:=
\left\{
f \in C([0,\infty)):\,
\sum_{n=1}^{\infty} ...
- 5,405
0
votes
0
answers
132
views
Spectral Theorem for compact self-adjoint operators on real Hilbert spaces [duplicate]
Is the spectral theorem for self-adjoint compact operators on a Hilbert space also true if the Hilbert space is real (instead of complex)?
Wikipedia says this is true.
However, it seems to me that ...
- 63.9k
2
votes
0
answers
134
views
Stone–Weierstrass theorem for stronger topologies
The Stone–Weierstrass theorem gives an easy to check criterion on a (algebra) set of functions $D\subseteq C(X)$ which ensures that $D$ is dense.
Are any similar results for density in $C_b(X)$ ...
- 5,405
1
vote
0
answers
104
views
"Global" topologies between compact convergence and uniform convergence
Let $X$ and $Y$ be locally compact (but not compact), second countable, Hausdorff spaces with $Y$ metric. It is easy to see that the topology of compact convergence is weaker than the topology of ...
- 5,405
11
votes
4
answers
2k
views
Spectral theorem for unbounded self-adjoint operators on REAL Hilbert spaces
This question was posed on MathStackExchange but did not get an answer (even with a bounty).
In all books that I have checked the spectral theorem (every self-adjoint unbounded operator on a Hilbert ...
- 12.6k
1
vote
0
answers
102
views
Given a finite set of points, does there exist a linear function pass through a point and strictly below the other points for all the points?
I guess my question is a follow up question of this one: usul, Existence of a strictly convex function interpolating given gradients and values, version: 2019-04-13.
In usul's question, the answer ...
- 12.9k
3
votes
0
answers
188
views
Invariant subspaces of Markov operators
I am currently working on some kind of graph theoretic problem and the following question came up:
Suppose you have a Markov operator $T$ on $\ell^\infty$, that is a positive, bounded operator such ...
- 381
6
votes
1
answer
243
views
How to choose phase to give a desired Fourier transform
Cross posted from MSE.
I have a mathematical problem arising from a physics application, which I feel must have been solved before, but I don't know the terminology associated with it. I am looking ...
- 956
23
votes
5
answers
6k
views
Hahn-Banach without Choice
The standard proof of the Hahn-Banach theorem makes use of Zorn's lemma. I hear that, however, Hahn-Banach is strictly weaker than Choice. A quick search leads to many sources stating that Hahn-Banach ...
- 4,713
7
votes
1
answer
756
views
Taylor expansion with remainder on locally convex spaces
It is usual to introduce Fréchet and Gâteaux derivatives in Banach spaces. In this context, the familiar Taylor expansion with remainder is also at hand, as you can see on the picture below taken from ...
- 2,388