All Questions
Tagged with real-analysis fa.functional-analysis
1,447 questions
1
vote
1
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228
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Which norms on vectors can be consistently decomposed?
I need to know which permutation-invariant norms can be consistently decomposed in the sense that for any vector $v = (a,b,c)$ we have that
$$\|(a,b,c)\| = \|(\|(a,b)\|,c)\|.$$
More precisely, let $v ...
- 702
10
votes
1
answer
2k
views
Counting norms on an infinite dimensional vector space
It is known that whenever E is a finite dimensional real vector space, there is only one norm on E up to equivalence (actually one non discrete vector space topology).
Is it known what happens when E ...
- 101
2
votes
1
answer
336
views
Separability of $L^1$ in $L^2$ topology
In the space $L^1(0,1)$ take the topology generated by the $L^2$-balls
$$B^2_r(f)=\{g\in L^1(0,1):\; \|f-g\|_2<r\}.$$
Is $L^1(0,1)$ separable in this topology?
- 23
0
votes
1
answer
274
views
On Cantor sets every map is $C^{\infty}$ [closed]
For a fixed Cantor set $K\subset [0,1]$ and a continuous function $g:[0,1]\to \mathbb R.$ Is it always possible to find a $C^{\infty}$ map $f:[0,1]\to \mathbb R$ such that $g$ and $f$ coincide in $K?$
...
- 1,805
2
votes
1
answer
244
views
Growth rate of Lipschitz constants for derivatives of $C^\infty$ functions
Let $f\in C^\infty$ have bounded derivatives, i.e.
$$ \sup_{x\in\mathbb{R}}|f^{(p)}(x)| = B_p < \infty$$
for every $p\ge 1$.
I would like to find a proof or a counterexample for the following ...
- 392
7
votes
0
answers
549
views
Counter-example to the completeness of the Wasserstein metric
$\newcommand{\P}{\mathcal{P}}$
Let $(E,d)$ be a complete metric space, let $\P(E)$ be the set of all probability measures on $(E,\mathcal{B}(E))$. Let $W_d$ be the $1$-Wasserstein (Kantorovich) ...
- 931
12
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1
answer
191
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Spectra on different spaces
This is a method request: I am looking for techniques that allow me to investigate problems like this:
Let $T_1: \ell^1 \rightarrow \ell^1$ be a bounded operator with $\Re(\sigma(T_1)) \subset (-\...
- 305
1
vote
1
answer
262
views
Relationship between $f(t,x)$ as $t \to \infty$ and $f(t/\epsilon, x/\epsilon^2)$ as $\epsilon \to 0$ (periodic functions)
Let $f: (0,\infty)\times \mathbb {R} \to \mathbb{R}$ be $1$-periodic in the second variable and in $L^\infty((0,\infty)\times \mathbb{R}).$ If it is necessary, we can also assume $f$ to be continuous. ...
user89890
1
vote
0
answers
27
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How does the principal value affects to the limit here?
In Córdoba and Gancedo - Contour dynamics of incompressible 3-D fluids in a porous medium with different densities (page 4) I read that if
$$ v (x_1,x_2,x_3,t)=-\frac{\rho_2-\rho_1}{4\pi} \...
- 383
9
votes
3
answers
4k
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Is there a reference for compact imbedding theory of Hölder space?
This question is posted and unanswered from math.stackexchange.
Suppose $0 < \alpha < \beta$ and $\Omega$ is bounded. Then, the Hölder space $C^\beta(\Omega)$ is compactly imbedded to $C^\alpha(...
- 1,399
3
votes
0
answers
97
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Notions of $\beta$-Hölder smoothness when $\beta\in (1,2]$: are they equivalent?
I posted the following question on StackExchange a few months ago (https://math.stackexchange.com/questions/2898620/notions-of-beta-h%C3%B6lder-smoothness-when-beta-in-1-2-are-they-equivalent), but ...
- 31
9
votes
2
answers
553
views
Asymptotic behavior of Sturm-Liouville eigenvalues
I have two questions.
Consider the operator $Av = -v'' + a(x)v$ on $I = (0, L)$, with zero Dirichlet condition and $a \in C([0, L])$.
Let $(\lambda_n)$ denote the sequence of eigenvalues of $A$....
- 369
1
vote
1
answer
178
views
Bochner integrability within a subspace
Let $(H,||\cdot||_H)$ be a Banach space and $K$ a (not necessarily closed) subspace. Suppose that $K$ is a Banach space under another norm $||\cdot||_K$, which satisfies
$$||x||_H\leq ||x||_K$$
for ...
- 1,903
9
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0
answers
979
views
Strong convexity of the trace of the square root of a matrix function
Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
- 91
5
votes
1
answer
3k
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Equicontinuity and $L^2$ convergence imply uniform convergence
I'm currently working through an old Paper of Garsia, Rodemich and Rumsey (A Real Variable Lemma) and theres one thing i don't get. Suppose $(f_n)_{n\in\mathbb{N}}$ is a sequence of continuous real ...
- 61
7
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3
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4k
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Is a semicontinuous real function Borel measurable?
Let $f(x,u): [0,1]^2 \mapsto \mathbb{R}$ be a continuous
function.
[Q] Is $g(x) = \inf_{u\in [0,1]} f(x,u)$ always Borel measurable?
If not, can one find a counter-example?
Note that, for any $c$,
...
- 1,399
4
votes
2
answers
561
views
Reference request: concave/convex envelope
I'm seeking the references concerning on the regularity analysis of concave envelopes, i.e. given some measurable function $f:\mathbb R^d\to\mathbb R$ that is bounded from above ($d=1$ or $d\ge 1$), ...
- 1,835
8
votes
1
answer
242
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Does infinitesimal variance imply continuity?
Let $u:[0,1]\to\mathbb{R}^n$ be a bounded Borel function.
It is well-known that if, for any compact interval $I\subseteq [0,1]$,
$$ \int_I|u-u_I|^2\le C|I|^{1+\alpha} $$
for some $C,\alpha>0$ (here ...
- 3,146
3
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1
answer
1k
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Showing a singular integral operator takes Hölder continuous functions to Hölder continuous functions of the same order
I would like to show the following function is $\gamma$-Hölder continuous. Said function $F:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined by a singular integral operator of convolution type as ...
- 31
1
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1
answer
203
views
Why study the moment problem in one dimensional case( Hamburger moment problem)
I have been reading about moment problem and I have been curious about the following question.
What is the motivation for studying the Hamburger moment problem(one dimensional moment problem?
I ...
- 125
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1
answer
165
views
Is there an example of a one to one and onto mapping between these two spaces?
Let $\Omega$ be a convex open subset of $\mathbb{R}^d$ with a smooth boundary. Is there an example of a one to one and onto mapping of the form $$L^{d+1}(\Omega) \to W^{1,d+1}(\Omega)$$
- 698
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votes
3
answers
481
views
Quantum Mechanics and bilinear optimal control theory
I was wondering whether there are any rigorous results about the optimal controllability of Schrödinger operators.
So my question is something like this:
Let $i \partial_t \psi(x,t) = H_0(x)\psi(x,t)...
- 259
0
votes
0
answers
55
views
Smooth compactly supported function with good scaling with respect to the fractional Laplacian
Is there a smooth cut off function with compact support such that $\phi: \Omega \subset \mathbb R^N \to \mathbb R$, $\mathrm{supp} \phi \subset B_R(0) \subset \Omega$ and $$(-\Delta)^s \phi \le C R^{-...
user139845
1
vote
0
answers
170
views
non-analytic functions with arbitrary large derivatives [closed]
This may be a trivial question but I can't see it immediately.
Suppose $\{a_k\}$ is an increasing sequence of positive reals. Does there exist a smooth function $f \in C^{\infty}([0,1])$ such that $\...
- 4,143
1
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0
answers
211
views
Propagation of singularities and the Schrodinger equation
I always thought that the propagation of singularities theorem by Hörmander says (on $\mathbb R^n$ for a classical symbol $p(x,\xi)=\xi^2+V(x)$) that for a Schrödinger equation
$$(i \partial_t-p(x,D))...
- 11
1
vote
1
answer
334
views
Orthonormal basis and decay
Edit: I added smoothness, hoping to simplify the problem with this additional assumption.
Let me motivate this question first: In signal analysis it is often of interest to understand when a certain ...
- 501
2
votes
1
answer
800
views
Interpolation in Sobolev spaces
Let $H^s$, $0\leq s<\infty$ be the $L^2$ based Sobolev spaces such that
$$
\hat{f}(\xi)(1+|\xi|^2)^{s/2} \in L^2.
$$
Let $r_1,r_2,p_1,p_2>0$ be given parameters. Assume that a linear operator $...
- 843
4
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1
answer
860
views
Lebesgue's integrability condition in several variables
The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable
$f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost ...
- 21.8k
4
votes
0
answers
211
views
Inclusion of Hardy spaces
It is well-known that any convergence in $L^p$ for $p \in [1,\infty]$ implies convergence in $L^1_{\text{loc}}$ by Hölder's inequality.
It is also known that for $p>1$ it holds that $L^p(\mathbb R)...
3
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0
answers
53
views
Controlling a Schwartz kernel near the diagonal
Let $D$ be a first-order elliptic differential operator that is essentially self-adjoint on $L^2(\mathbb{R}^n)$. Consider the operator $(D+i)^q$ acting on $L^2(\mathbb{R}^n)$ with domain $C_c^\infty(\...
- 1,903
-1
votes
1
answer
227
views
Solving the integral identity $ \int_{a}^{b} f(x)dx = \int_{a}^{b} f(x)g(x)dx. $ [closed]
We know that 0 is the additive identity and 1 is the multiplicative identity. In the same spirit let us define the integral identity as follows.
Definition: Let $f(x)$ be integrable in $(a,b)$. If ...
- 5,470
9
votes
2
answers
2k
views
Mathematical equivalent to ladder operators?
A powerful method in theoretical physics are ladder operators. They are used in QM to solve problems like the harmonic oscillator and the hydrogen atom. The idea is to solve with their help the ...
user37929
1
vote
0
answers
126
views
identity involving spectral functions
Let $A$ be any compact operator and let $A^*$ denote its adjoint. Let $f$ be a spectral function. Then is the following true :
$$ A^* f(AA^*) = f(A^* A) A^*$$
- 519
0
votes
0
answers
63
views
Feller semigroups and fractional operators
Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
user60665
2
votes
1
answer
260
views
An elementary functional inequality
Let $g$ be a $C^1$ function with $g(0)=0$ and $g(t)>0$ for all $t>0$. I am surprised that for all such $g$ the following seems to hold
$\frac{\int_0^t(g'(s))^2ds}{g^2(t)}\geq \frac{1}{t}$ for ...
2
votes
2
answers
635
views
Continuous upper envelope of upper semicontinuous function
Let $u$ be a upper semicontinuous function on a compact set $K$ in $\mathbb R^d$. Define a space of continuous function dominating $u$ by
$$A = \{\phi \in C(K): \phi \ge u\}.$$
[Q.] Is the following ...
- 1,399
0
votes
1
answer
218
views
Heat semigroup dissipative
Consider the heat semigroup on $L^1(\mathbb{R}).$ I would like to know if the generator of this semigroup is dissipative in the sense of this definition.
On $L^2$ it would be completely trivial, but ...
- 167
1
vote
1
answer
192
views
Neumann-Poincare operator is in the Schatten class
Let $\Omega$ be a bounded domain in $\mathbb{R}^d$, $d\ge 3$. We define the Neumann-Poincare operator(or double layer potential) $K: L^2(\partial\Omega)\to L^2(\partial\Omega)$ by
$$(Kf)(x)=\int_{\...
- 171
4
votes
0
answers
349
views
Fractional integral inequality (Hardy-Littlewood-Sobolev)
I am investigating the following integral
\begin{equation}
I^*(x) = \int_{\mathbb{R}} \frac{f(y) \ln |y-x| }{|y - x|^{\mu}} \, dy
\end{equation}
where $f \in L_p(\mathbb{R})$, $ 1 < p < q <...
0
votes
1
answer
136
views
A question on existence of a Sobolev Hilbert space, where convergence implies uniform convergence [closed]
Is there a Sobolev Hilbert space $H^k(\Omega)$($\Omega$ open subset of $\mathbb{R}^m$, with a smooth boundary), for some $k \in \mathbb{N}$, such that, any sequence in the space $C^0(\bar{\Omega})\cap ...
- 698
8
votes
2
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785
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Is taking the product of signed measures weakly continuous?
For a Polish space $X$, let $C_b(X)$ denote the real Banach space of bounded continuous real-valued functions on $X$. Let $M(X)$ denote the space of all finite signed Borel measures on $X$, equipped ...
- 29.8k
1
vote
1
answer
518
views
Interpolation between Schatten classes
I was wondering if there is an analogue to the classical Riesz Thorin theorem for Schatten classes. I suppose the answer is yes, since Schatten classes are so similar to $\ell^p$ spaces for which the ...
- 305
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
$$\int_0^1 h(s) f(s) ds=0$$
$$...
- 501
1
vote
0
answers
78
views
Potential for a Monotone Operator
[Cross-posted from math.stackexchange]
I have a question about understanding the proof of Theorem 4.11 in the paper A Potential Theory for Monotone Multivalued Operators (accessible here). The ...
- 291
1
vote
1
answer
175
views
Stochastic operator on $\ell^1$ has dense range
Let $P:\ell^1(\mathbb{Z}^d) \rightarrow \ell^1(\mathbb{Z}^d)$
be given by
$$(Pz)(x)=\sum_{y \tilde \ x} \frac{1}{2d} z(y)$$
where the tilde indicates that $y$ is a neighboured vertex of $x.$
I ...
- 329
1
vote
2
answers
371
views
Weak convergence in vector-valued Hilbert space
Let $V$ be a separable Hilbert space and define $X=L^2(0,T;V)$. Then $u_m\to u$ weakly in $X$ means
for every $v\in X'=L^2(0,T;V')$
$$
\int_0^T\langle v(t),u_m(t)\rangle\ dt\to\int_0^T\langle v(...
user14319
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
4
votes
2
answers
4k
views
Embedding of $BV$ and $L^p$ spaces
An elementary question about Sobolev spaces:
Is there some explicit theorem about embedding relation between spaces $BV(\Omega)$ and $L^p(\Omega)$?
Formulated otherwise: is $BV$ a subset of $L^2$ (i....
- 53
4
votes
1
answer
1k
views
For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?
I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...
- 1,649
5
votes
1
answer
249
views
If $\mathcal R_j f\in L^1$ then $\widehat{\mathcal R_j f}=-i\frac{\xi_j}{|\xi|}\widehat{f}(\xi)$
For any $f\in L^1(\mathbb{R}^n)$ and $1\le j\le n$, recall that the Riesz transform $\mathcal{R}_jf\in L^{1,\infty}(\mathbb{R}^n)$ is given by
$$ \mathcal{R}_jf:=c_n\lim_{\epsilon\to 0}\left(\frac{x_j}...
- 3,146