All Questions
Tagged with real-analysis fa.functional-analysis
1,447 questions
2
votes
1
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265
views
characterization of normality by selection theorem
The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...
- 4,713
1
vote
1
answer
162
views
Does there exist a class of real-valued upper semicontinuos functions on $X$ such that $\mathcal{F}$ is countable?
Ian Morris quoted the following:
For any upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a nonempty topological space $X$ there exists a nonempty set $\mathcal{F}\...
- 623
4
votes
1
answer
1k
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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 $f(x)=e^{-|x|}$, ...
CommunityBot
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15
votes
3
answers
2k
views
Asymptotic expansion of $\sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$
I've been trying to find an asymptotic expansion of the following series
$$C(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$$
and
$$L(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{...
CommunityBot
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0
votes
0
answers
308
views
Invertible operator
We consider the operator $$T=I + {{{\partial ^2}} \over {\partial {x^2}}}:{H^2}(0,L) \cap H_0^1(0,L) \to {L^2}(0,L)$$
We hope to prove that $T$ is invertible if and only if $L = n\pi $.
and for this ...
- 6,259
18
votes
1
answer
3k
views
How bad can the second derivative of a convex function be?
One can easily construct an example of a measurable function $f:(a,b)\to \mathbb{R}$ which satisfies the following property:
$$\label{p}\tag{P}
f\notin L^1(I),\ \mbox{for each interval}\ I\subset (a,...
2
votes
0
answers
73
views
A question on groupoids and measurable fields of Hilbert spaces
Suppose that we have the following data:
$ \mathcal{G} $ is a locally compact Hausdorff groupoid, with its source and
range maps denoted by $ s $ and $ r $ respectively.
$ (\lambda^{x})_{x \in \...
- 1,396
11
votes
3
answers
3k
views
Dual space of $L^2(\mathbb{R},L^1(0,1))$?
I was wondering what the dual space of $L^2(\mathbb{R},L^1(0,1))$ is? (equipped with Lebesgue measures)
Formally, one would suspect that it is just $L^2(\mathbb{R},L^{\infty}(0,1))$. But this may be a ...
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1
vote
1
answer
130
views
Resolvent difference of absolute values!
Let $T$ be a bounded operator. Then, the operators $\left\lvert T \right\rvert:=\sqrt{T^*T}$ and $\left\lvert T^* \right\rvert:=\sqrt{TT^*}$ are well-defined.
Is there a way to write
$$(\left\lvert ...
CommunityBot
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3
votes
3
answers
2k
views
Determining if a set is a Basis for l^2
For each $ n\ge 1$ Define the vectors $e_n = (e_{nk})$ where $ k\ge 1$ and $ e_{nk} = \frac{1}{k^n}$
Is this set a basis for $l^2$?
Thanks,
- 462
2
votes
1
answer
102
views
Evolution equation invariance of sets
Let $A: D(A) \subset X \rightarrow X$ be a generator of a $C_0-$semigroup and $Z$ be a bounded operator on $X$, then the evolution equation for $u \in C([0,T], \mathbb{R})$
$$\varphi'(t) = A \varphi(t)...
- 16.4k
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
$$ \mathcal{R}_jf:=c_n\lim_{\epsilon\to 0}\left(\frac{x_j}...
- 3,146
12
votes
1
answer
927
views
On an Inequality of Lars Hörmander
Let $P(z)$ be a non-null complex polynomial in $\nu$ variables $z=(z_1,\dots,z_n)$ of degree $\mu$:
\begin{equation}
P(z)=\sum_{|\alpha| \leq \mu} c_{\alpha} z^{\alpha},
\end{equation}
where as usual ...
CommunityBot
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5
votes
2
answers
359
views
Proof without distributions
I was wondering whether there is a way to show this identity
$$\pi \int_{\mathbb{R}^3} \frac{f(x)}{|x|} dx = \int_{\mathbb{R}^3} \frac{\widehat{f(x)}}{|x|^2} dx $$ without using distributions for $f ...
- 23k
2
votes
1
answer
250
views
Density in the Space of absolutely convergent Fourier series
It is possible to approximate a function $f$ on $[0,2\pi]$ by a continuous function whose derivative is zero almost everywhere (as can be seen here : https://math.stackexchange.com/questions/67334/...
CommunityBot
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1
vote
1
answer
91
views
Design measure, which cannot be factorized as a product of measures
Let $\mathcal{S}_x$ and $\mathcal{S}_y$ be a finite discrete sets, such that
$$
0 < |\mathcal{S}_x| < \infty, \qquad 0 < |\mathcal{S}_y| < \infty, \qquad \mathcal{S}_x \cap \mathcal{S}_y =...
- 24.8k
2
votes
0
answers
226
views
degree theory argument in elliptic pde; apparent contradiction
i have a question regarding a degree theory argument and an apparent contradiction. Let me point out that I am a complete novice with degree theory and really i am just pushing some symbols with no ...
- 1,385
1
vote
0
answers
180
views
Implicit function theorem for operators
Let $P: (-a,a) \rightarrow \Psi_h^0(\mathbb{R}),$ be a pseudodifferential operator in Weyl quantization with $(-a,a) \ni z \mapsto P(z)$ depending smoothly on this parameter $z$. Note that this ...
CommunityBot
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3
votes
0
answers
280
views
Helmholtz-Hodge decomposition
I have a question regarding a decomposition of a vector field. So fix $ 1<p<\infty$ and let $ \Omega$ denote a smooth bounded domain in $ R^N$. Now let $ F $ denote a smooth vector field $F:\...
- 1,385
7
votes
0
answers
187
views
distance distributions on a hypersphere?
Fix a real number $0\leq t\leq 1$ and an integer $n>1$. Let
$\mathbb{S}^{n-1}\subset\mathbb{R}^n$ denote the unit hypersphere. Define
$$d_N(n;t):=\max\sum_{i<j}\Vert P_i-P_j\Vert_2^t$$
where ...
- 43.2k
1
vote
3
answers
653
views
When does $f^{-1}=\frac{1}{f}$ with $f$ a function mapping $\mathbb{R}^{*}$ to $\mathbb{R}$?
In mathematics, an inverse function is a function that "reverses" another function: if the function $f$ applied to an input $x$ gives a result of $y$, then applying its inverse function $g$ to $y$ ...
- 4,713
3
votes
1
answer
195
views
Boundedness of different Fourier transforms
Let $f: \mathbb{R}^n \rightarrow \mathbb{C}$ be in $L^2\cap L^1,$ then the Fourier transform is in $L^2 \cap L^\infty.$
Does this imply that we can take common norms in the sense that we can estimate ...
- 33
1
vote
1
answer
287
views
Interpolation between $L^1$ and $L^2$ spaces
I was wondering whether the following interpolation between $L^1$ and $L^2$ spaces is true:
Let $f \in \mathbb{R}^n$ be such that
$$ \alpha_1:= \int_{\mathbb{R}} \left\lVert f(x_1,\cdot,....\cdot) \...
- 7,249
2
votes
0
answers
194
views
A question regarding mollifiers on Sobolev spaces on closed manifolds
Let $M$ be a closed Riemannian manifold and denote by $H^s(M), \, s\in \mathbb{R} $ the standard Sobolev spaces on $M$ defined using powers of $1+\triangle$. Let $J_n: \mathcal{D}'(M)\rightarrow \...
- 505
5
votes
1
answer
1k
views
Trace-norm of integral operator
Let me start by saying that I do appreciate any insight on this. So also if you have a partial result, please share it as a comment or answer.
This is somewhat unrelated to what I normally do, so I ...
- 24.2k
0
votes
1
answer
104
views
Operator identity for convergent series
Let $T_i$ and $S_i$ be a sequence of bounded operators such that
$$ \sum_{k,i,j=0}^{\infty} S_j^* T_i^* T_i S_k$$ converges unconditionally in operator norm on some Hilbert space. The limit is then ...
- 23
1
vote
1
answer
124
views
On a weaker condition of summability for Fourier series
The Wiener algebra $W:=W(\mathbb{T}^n)$ on the torus is defined as the algebra of all continuous fonctions $f$ on $\mathbb{T}^n$ such that $(\widehat f(k))_{k\in \mathbb{Z}^n} \in \ell^1(\mathbb{Z}^n)$...
- 109k
2
votes
0
answers
115
views
Does this Sobolev-space like construction have a name?
Take $\Omega \subset \mathbb{R}^n$ arbitrary then define as $X$ the closure of $C^1(\Omega) \cap W^{1,1}(\Omega)$ w.r.t. the norm $f \mapsto \left\lVert f \right\rVert_{\infty} + \left\lVert \nabla f \...
- 305
3
votes
0
answers
177
views
Interesting stipulation about completely monotone functions
This question relates to a question I asked here. I thought of a well thought out generalization which appears to follow in the situations I've encountered it. I tried to generalize the answer ...
CommunityBot
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0
votes
0
answers
58
views
in search of convergent daughter sequences
Let $\{f_n\}\subset L^1(\Omega,\mu)$, where $\mu$ is the Lebesgue measure, and $\Vert f_n\Vert_1\leq M$ and $\Vert Df_n\Vert_{1/2}\leq C$ uniformly in $n$.
Question. Is there a subsequence $\{f_{...
- 43.2k
1
vote
1
answer
518
views
Interpolation between Schatten classes
I was wondering if there is an analogue to the classical Riesz Thorin theorem for Schatten classes. I suppose the answer is yes, since Schatten classes are so similar to $\ell^p$ spaces for which the ...
2
votes
0
answers
341
views
Trace class operators convergent series
On wikipedia it is mentioned that if we are on some (separable) Hilbert space $H$ and there is an ONB $(e_n)$ such that any compact operator $K$ can be written as
$$ K = \sum_{n,m =0}^{\infty} K_{n,m}...
- 305
0
votes
1
answer
150
views
Solutions to Schrödinger equation parameter dependence
This is somewhat unrelated to what I normally do in mathematics, which is why it may be obvious to some of you, but I was puzzled by this:
If we look for classical solutions on $[0,1]$ to
$$-y''(x) =...
- 305
5
votes
1
answer
101
views
Does minimum of an analytic map restricted to analytic curves implies minimum?
Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be an analytic function such that its restriction to any arbitrary analytic curve $\gamma$ passing through the origin $0\in \mathbb{R}^n$ attains a local ...
- 60.5k
4
votes
0
answers
298
views
Operator topologies
Let $L(H)$ be the space of bounded operators on some Hilbert space.
We can endow this space with the operator norm topology, the strong operator topology (SOT) and the weak operator topology (WOT).
...
- 41
3
votes
0
answers
235
views
Is this "differentiation map" uniquely determined by these properties?
Let $A$ be the set of all real-valued functions having their domain a subset of $\Bbb R$ which are at least differentiable on an open set, and for $f \in A$, let $U_f$ be the largest open set on which ...
CommunityBot
- 1
3
votes
1
answer
167
views
Recovering residue using local real information
Let $f(z)$ be defined by a Laurent series at z = 0 with real coefficients. In particular, $f(x) \in \mathbb{R}$ for $x \in \mathbb R$.
Compute the residue of $f(z)$ at z = 0 using just the ...
- 91.8k
9
votes
1
answer
2k
views
Alternative proof of a theorem of Riesz
My question is not research level, but I have not received any feedback on Mathstack; so I am posting it here. I am aware of the traditional proof of the Riesz Theorem that relates linear functionals ...
- 3,816
1
vote
1
answer
211
views
Representation of Hilbert transform by a singular integral
Hilbert transform defines as follow:
$$ H: L^2(\mathbb R) \to L^2(\mathbb R) $$
$$ H(f)= \mathcal{F}^{-1}[{F(\gamma) \mathrm{sign}(\gamma)]}$$
Where $F(\gamma)= \mathcal{F}(f) (\gamma)= \...
- 43.2k
-3
votes
1
answer
451
views
Exponential decay of kernel
Let $A: \ell^2 \rightarrow \ell^2$ be a bounded operator given by
\begin{equation}
(Au)(\alpha) = \sum_{\beta}A(\alpha,\beta)u(\beta)
\end{equation}
where $\left|A(\alpha,\beta) \right|\le Ce^{-|\...
- 79.9k
2
votes
0
answers
139
views
Existence of solution of a variational inequality
Let $K\subseteq \mathbb{R} ^n$ be closed and convex, and let $F:K \to \mathbb R^n $ be a continuous function. If for every $x,y \in K$ we have $$(x-y)^T(F(x)-F(y))\ge \alpha ||x-y||^2 \, ;\quad \...
- 24.2k
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 $...
CommunityBot
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4
votes
1
answer
264
views
Density of the max set of a non-differentiable function
For $f : [0;1] \to \mathbb{R}$, let $M_f := \{x \in [0;1] \mid f(x)$ is a local strict maximum of $f\}$. It is easy to see that for any $f$, $M_f$ is at most countable. It is also easy to see that ...
- 4,679
1
vote
1
answer
114
views
Reference request: regularity of functionals on the space of probability measures
Let $\mathcal M=\mathcal M(\mathbb R^d)$ be the space of finite measures on $\mathbb R^d$, and $\mathcal P=\mathcal P(\mathbb R^d)\subset\mathcal M$ be the space of probability measures. Let $F:\...
CommunityBot
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2
votes
0
answers
92
views
Estimating the size of a subset of $\mathbb{R}^N$
This concrete geometric question has arisen out of the problem of counting arithmetic functions with a particular property. The details of the relationship between the counting procedure and this ...
- 2,480
1
vote
1
answer
392
views
Integral kernel smooth
Assume that $Tf(x):=\int_{\mathbb{R}^n} K(x,y)f(y) dy$ is an operator such that $T \in L( H^{-k}, H^k)$ is continuous for any $k$, where $H^k$ is the $k-th$ order Sobolev space on $\mathbb{R}^n$.
...
- 16.2k
5
votes
2
answers
1k
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Derivatives of $C^{\infty}$ non analytic function
Question: Given $f\in C^{\infty}$ which is not analytic on a bounded domain $\Omega \subseteq \mathbb{R}$. What can we say about the sequence $\lbrace f^{(m)} \rbrace _{m=1}^{\infty} $? Specifically - ...
CommunityBot
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2
votes
1
answer
497
views
Are the partial derivatives of a function increasing in both variables measurable?
Let $f$ be a function from $[0,1]\times[0,1]$ to $\mathbb{R}$ that is nondecreasing in both variables, i.e. $f(x_1,y_1)\le f(x_2,y_2)$ whenever
$x_1\le x_2$ and $y_1\le y_2$. It is known that the ...
- 3,816
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?
- 446
0
votes
1
answer
302
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Approximation of a $C^{\infty}_c$ function with tensor products of a constant tensor rank
I asked the following question a few days ago:
Approximation of a $C^{\infty}_c$ function by tensor products
However, I then realised that I actually need a stronger result in my proof.
As in the ...
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