Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3,435 questions with no upvoted or accepted answers
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$\ell_\infty$-norm covering number of RKHS ball $\{f\in\mathcal{H}: \|f\|_\mathcal{H} \leq R\}$
For any $\epsilon \in (0,1)$, let $N_\infty(\epsilon, \mathcal{H}, R)$ denote the $\epsilon$-covering number of the RKHS norm ball $\{f\in\mathcal{H}: \|f\|_\mathcal{H} \leq R\}$ with respect to the $\...
- 121
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51
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Conjugate of composition in Bochner spaces
Let $H$ be a separable Hilbert space (of non-zero dimension), let $(\Omega,\Sigma,\mu)$ be a finite measure space, and let $L^2(\mu;H)$ be the Bochner-space $\mu$-integrable $H$-valued functions. ...
2
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72
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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
2
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0
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52
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Reference Request: Dirichlet operators with singular coefficients
Let $d\geq 2$, $\delta \in (0,1)$ and let $\mathcal{L}_{d,\delta}$ be the second order differential operator defined by
\begin{align*}
\mathcal{L}_{d,\delta}(f)(x) = \Delta(f)(x)-\delta \|x\|^{\delta-...
- 778
2
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answers
66
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Continuity of weak solution of elliptic PDE
I am investigating the following standard elliptic PDE with mixed Dirichlet-Neumann boundary condition:
$-\Delta u=f$ on $\Omega$; $u=0$ on $\Gamma_D$; $\left<n,\nabla u=g\right>$ on $\Gamma_N$
...
- 21
2
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186
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Metric on space of Borel-measurable functions
Let $(X,d_X),(Y,d_Y)$ be metric spaces and $X$ is locally-compact and fix a Borel probability measure $\nu$ on $X$. For any Borel-measurable $f:X\rightarrow Y$, let $\mathcal{K}(f,\delta)$ be the set ...
2
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252
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A sequence of compact operators, weak convergence versus strong convergence
Let $(X,\mu)$ be a finite measure space.
We consider two sequences of bounded linear operators $\{T_n\}_{n=1}^\infty$ and $\{S_n\}_{n=1}^\infty$ on $L^2(X,\mu)$. We denote by $\mathcal{L}$ the space ...
- 721
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78
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How to compute the functional derivative of the following functional encountered in electrical impedance tomography?
Note:
I have raised this question in Mathematical stack exchange but it received no attention. That is why I proceed to here to ask this question. Please tell me if this is not appropriate, thank you....
- 161
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132
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When does a nonnegative $C^1$ function on $[a,b]$ have finitely many zeros?
It is well known that if $f:\mathbb R\to \mathbb R$ is real analytic then we have that either $f\equiv 0$ or $f$ has finitely many zeros on the interval $[a,b]$.
It is also known that you can have a ...
- 133
2
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206
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Helly-type theorem for infinite-dimensional spaces
Currently I'm reading a classical text Convex Sets by F. Valentine. While reasing it I'm trying to generalize results as much as I can. I'm a bit confused of authors use of a notions of Euclidean ...
- 519
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145
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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
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104
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Weak convergence rates for integral operators
Suppose $q=\sum_{i=1}^m\pi_i\delta_{x_i}$ is a discrete measure on $\mathbb{R}^n$ and let $q\ast \varphi_\epsilon$ denote the convolution of $q$ with some mollifier $\varphi_\epsilon$, so that $q\ast\...
- 75
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75
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Weighted translation semigroup
Given a strictly positive function $a:[0,1]\longrightarrow \mathbb{R}$, I have a question about the generation of a $C_0$-semigroup on $L^p([0,1])$ ($1\le p<\infty$) by the following maximal ...
- 299
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224
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Interpolation of embedded Hilbert spaces and intersection
I'm wondering under what hypothesis it is true a property like
$$[\mathcal{H}_1\cap X, \mathcal{H}_2\cap X]_{\theta}=\mathcal{H}_1\cap X\cap [\mathcal{H}_1, \mathcal{H_2}]_{\theta}$$
where $\mathcal{H}...
- 81
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Asking for results on critical points and similar properties of solutions of nonlinear Volterra integral equations - Physically coherent solutions
I have a system of nonlinear Volterra integral equations of form
$$x(t)=x_0+\int_0^t K(t,s)F(x(s))ds$$
and I am interested on the critical points of $x(t)$, I mean maximum, minimum, increasing and ...
- 123
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169
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Functions whose Fourier coefficients satisfy $ \sum_{k=1}^\infty |c_k| < 1 $?
Let $f:(0,1) \to \mathbb R$ be a function that can be written as $$f(x) = \sum_{k=1}^\infty c_k \phi_k(x),$$ where $\phi_k(x) = \cos(\pi k x)$. What is the minimal assumption required on $f$ to ...
- 839
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Decomposition of the Orlicz norm into sequential norm
I am bearing seeking for a sequential decomposition of the norm in Orlicz space.
Let me state what is known in the particular case of Lebesgue space $L^p(\Bbb R^d)$.
Given $u\in L^p(\Bbb R^d)$ let
$$n\...
- 1,101
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259
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Bochner integral in a Fréchet space
I have a Fréchet space $V$ whose topology is (if it helps) induced by a family $\mathcal{P}$ of norms - not just seminorms - and on this space I have a Borel probability measure $\nu$. Now, I would ...
- 661
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answers
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Compact embedding of the Sobolev space $H^m(\Omega)$ and $L^2(\Omega)$ from Rellich-Kondrachov theorem
From the Rellich-Kondrachov theorem we know that $H^m(\Omega)\hookrightarrow_c L^2(\Omega)$ when $\Omega$ is bounded of class $C^1$ and $m\geq 1$ is an integer. Also this is not true if $\Omega:=\...
- 657
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71
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Dominant singularity of a multivariate complex function and Cauchy integral formula
Suppose I have a function $f: \mathbb C^d \rightarrow \mathbb C$, that i know in close form, and from which i want to bound the taylor coefficients around 0. For that, I am using the cauchy integral ...
- 686
2
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350
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What is the explicit version of the Peter Weyl Theorem?
While the name "Peter-Weyl" is reserved for the compact group case, I prefer to talk in greater generality. Let $G$ be a unimodular type I topological group with a fixed Haar measure. The ...
- 2,081
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116
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How to prove surjectivity of biharmonic operator?
My question is regarding the Kirchhoff plate pde which is the so-called biharmonic operator.
Let $\Omega=[a,b]\times[a,b]\subset \mathbb{R}^2$, and $Z$ be the Hilbert space $Z=L^2[\Omega]$. The ...
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76
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wave equation with L^2 boundary data via spectral decomposition
It is classical that if $\Omega \subset \mathbb R^n$ is a bounded domain with smooth boundary, then the equation
\begin{equation}\label{pf2}
\begin{aligned}
\begin{cases}
\partial^2_{t}u- \Delta u=0\,\...
- 4,115
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answers
66
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Minimizing a certain norm of the identity operator on $\mathbb R^2$
$\newcommand\R{\mathbb R}\newcommand\Q{\mathcal Q}$For mutually orthogonal vectors unit vectors $a=[a_1,\dots,a_n]^T$ and $b=[b_1,\dots,b_n]^T$ in $\R^n=\R^{n\times1}$ (so that $n\ge2$) and for all $x=...
- 128k
2
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86
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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
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163
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Hilbert transform on weighted Sobolev spaces
Let $\mathscr H\,f$ denote the Hilbert transform of a function $f \in L^2(\mathbb R)$. We know that $\mathscr H$ is an isometry on $L^2(\mathbb R)$, but I want to know to what is the mapping ...
- 4,115
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164
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What are (the different aspects of) harmonic analysis good for?
Let $G$ be a locally compact group. To the best of my understanding, harmonic analysis has three legs that all work perfectly in the case that $G$ is in addition compact and abelian, but have ...
- 2,081
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142
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Radon-Nikodým-like theorem for Radon measures
Let $(E,d)$ be a metric space, $\mu$ be a nonnegative Radon$^1$ measure on $\mathcal B(E)$ and $\nu$ be a finite (signed) Radon measure on $\mathcal B(E)$.
I'm searching for a Radon-Nikodým-like ...
- 167
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382
views
Poincaré inequality holds on Riemannian manifolds (min max principle)
In YuChang Xia's book "Eigenvalues on Riemannian Manifolds" Page 4 equation (1.16）Poincaré inequality:
I want to know which manifold(s)/function(s) can make the inequality hold. What if we ...
- 71
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50
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A question about strong slopes (nonsmooth analysis)
Context. I'm reading the manuscrip "Nonlinear Error Bounds via a Change of Function" by Dominique Azé and Jean-Noël Corvellec (J Optim Theory Appl 2016), and I'm having a hard time ...
- 6,853
2
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113
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$W^{1,p}-$regularity on the boundary for solution of Laplace equation with Robin boundary condition
I came across with the attached paper and here is the part that I try to understand.
If the non-tangential maximal function of $\nabla u$, i.e $(\nabla u)^*$, belongs in $L^p(\partial \Omega)$, then ...
- 227
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129
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Is there any nontrivial measure on $L^p$ or Sobolev spaces?
One may find a counting measure or trivial probability measure on $L^p$. However, is there any meaningful measure constructed on $L^p$, so that one may try to find a size of subsets of $L^p$? The same ...
2
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308
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Which domains have a Poincare-Wirtinger inequality? Which don't?
A Poincare-Wirtinger inequality holds over a domain $\Omega \subseteq \mathbb R^n$ with exponentnt $1 \leq p \leq \infty$ if there exists $C(p,\Omega) > 0$ such that
$\| u - \operatorname{avg}(u) \|...
- 5,327
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answers
323
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Continuous injective functions with dense image
Let $X$ be the set of continuous, injective functions from $\mathbb{R}^n$ to $\mathbb{R}^n$ with dense image; and equip $X$ with the (relative) compact-open topology. What is known about this space?
...
- 5,405
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115
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Showing that for measurable $\Omega \subseteq \mathbb{R}^n$, $L^1(\Omega; C_0(\mathbb{R}^n))$ is separable
Here we're integrating "Banach-valued" functions $u: \Omega \rightarrow C_0(\mathbb{R}^n))$ , and by $u \in L^1(\Omega; C_0(\mathbb{R}^n))$ I mean that
$$\int_{x \in \Omega} \| u(x) \|_{\...
- 21
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Lidskii's theorem for non trace-class operators
Lidskii's theorem states that for a trace-class operators, the trace is actually equal to the sum of eigenvalues.
Now, there are operators with finite trace that are not trace-class. Of course, in ...
- 536
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42
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Generalized Hardy operator and Lorentz gamma spaces
I would like to find an inequality which would 'place' the generalized Hardy operator $\int_0^th(y)dy\int_y^tk^*(s)ds$ in between two Lorentz gamma spaces.
Any literature or ideas would be greatly ...
- 109
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275
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Extension of Lipschitz functions that preserve the Frobenius norm of the Jacobian
Let $n,m\ge 1$ be integers and let $f:E\to R^m$ be $L$-Lipschitz for some subset $E\subset R^n$.
Kirszbraun's theorem, https://en.wikipedia.org/wiki/Kirszbraun_theorem, states that there exists ...
- 1,724
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99
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Lower bound on iterated matrix application
Let $n \in \mathbb Z^2$ such that the non self-adjoint weighted Laplacian is
$$(\Delta u)(n)=u(n_1+1,n_2)-u(n_1-1,n_2) + i( u(n_1,n_2+1)- u(n_1,n_2-1))$$
the adjoint operator is then
$$(\Delta^* u)(n)=...
- 192
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110
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A strong duality for convex functional optimization that admits Lipschitz continuity constraints?
Problem Statement
I am looking for formal proof---hopefully textbook material---of two items:
an analogue to Slater's condition [1] that obtains strong duality for optimization of convex functionals; ...
2
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395
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A question in Sobolev spaces involving time
Let $X$ be a Banach space, we understand $L^1(0, T, X)$ is the space of strongly measurable functions from $[0, T]$ valued in $X$, that is integrable. Assume ${\bf u}\in L^1(0, T, X)$, we say ${\bf v}\...
- 637
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395
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On strong resolvent convergence of unbounded operators
Given a sequence of unbounded self-adjoint operators $\Delta_N$ defined on some fixed domain of the Hilbert space $L^2(\mathbb{R})$ converging to i*identity in the sense that: for all $\phi\in\...
- 63
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347
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Can quotient space be isomorphically isometric to some closed subspace of original one?
Suppose $\mathcal{B}$ is a Banach space which is not isomorphic to a Hilbert space. What condition should I pose on $\mathcal{B}$ such that for every closed subspace $\mathcal{M}$, the quotient space $...
- 53
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votes
0
answers
89
views
Prove integral inequality for divergence-free vector fields
Let $u$ be a divergence-free vector field $u:\mathbb R^n \to \mathbb R^ n$. Does the following inequality hold?
$$\Big( \int_{\mathbb R^n} |u|^2 dx\Big)^2 \le C\Big(\int_{\mathbb R^n} |u|^2|x|^2 dx \...
- 839
2
votes
0
answers
137
views
Conditions on the inequality with a gauge norm
Let $\Phi(x)=\int_0^x \phi(y)\,dy$, $x \in \mathbb{R}_+$, be an N-function, and let $u$ be locally inferable on $\mathbb{R}_+$. Consider the gauge norm
$$
\rho_{\Phi,u}(f)=\inf\{\lambda>0: \int_{\...
- 343
2
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0
answers
126
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Mixed partial derivatives of planar functions converging to delta distribution
Given a sequence $(f_k)_{k\in\mathbb{N}}\subset C^2(\mathbb{R}^2)$ of strictly positive functions $f_k\equiv f_k(x,y)$ with $\|f_k(x,\cdot)\|_{L^1}=1$ for all $x\in\mathbb{R}$ and such that for each $...
- 51
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answers
87
views
Estimate in vanishing viscosity for the difference $\Vert u^\epsilon - u^\eta \Vert_{L^2(\mathbb R^N)} $
Consider the following advection-diffusion equation
$$
\begin{cases}
u^\epsilon_t + f(u^\epsilon)_x = \epsilon \Delta u^\epsilon\\
u^\epsilon(0,\cdot) = u_0,
\end{cases}
$$
How can one prove an ...
- 839
2
votes
0
answers
131
views
Does a spectral theorem exist for linear operator pencils?
I was wondering if a version of the spectral theorem (the projection valued measure case) holds for linear pencils of the form
$$
A-\lambda B
$$
where $A,B$ are self-adjoint on some Hilbert space $\...
- 223
2
votes
0
answers
120
views
Hilbert transform on a Besov space
Consider the usual Hilbert transform of periodic functions
$$H(f) = \frac{1}{2\pi}P.V.\int_{-\pi}^{\pi}\cot(\frac{x-y}{2})f(y)dy.$$
We know $H$ does not map $L^\infty$ continuously to $L^\infty$. Now ...
- 903
2
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0
answers
176
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Uniqueness for measure valued ode
Morning! Basically I'm working on a mean field scaling for some measure valued process (valued on $M_F(\mathbb{N})$). The limit turns up as a (deterministic) solution to a measure valued ODE. Let's ...
