Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
9,353
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A question about Stroock's notes on the Weyl lemma
On p.4 of these notes, D. Stroock gives a quick and efficient construction of the Markov transition functions of a certain diffusion. The idea of his construction (on page 4) is to 'freeze' the ...
- 1,451
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74
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Very weak solution to parabolic PDE (pointwise a.e. in time with time derivative on test function)
Consider the parabolic PDE
$$u' + Au = 0$$
as an equality in $L^2(0,T;V^*)$ for some Hilbert space $V$ with $A\colon L^2(0,T;V) \to L^2(0,T;V^*)$ a coercive, bounded linear operator. Here $u'$ is the ...
- 107
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310
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Sequence of Hilbert Schmidt operators
Consider the Banach space $\mathcal K=S_2(H)$ of Hilbert Schmidt operators on a Hilbert space $H$. I am looking for an example of two pairs of sequences $\{T^{(i)}\},\{\tilde T^{(j)}\}$ and $\{S^{(i)}\...
- 243
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399
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Smallest eigenvalue for large kernel matrix
I am interested in the the asymptotics of the minimum eigenvalue $\lambda_n^n$ of a class of kernel matrix $P = [ K(x_i - x_j) ]_{i,j}$, with $x_i$ equally spaced in the unit cube of $\mathbb{R}^d$.
...
- 151
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84
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A question on the Dieudonné property
Recall that a Banach space $X$ is said to have the Dieudonné property if for every Banach space $Y$, an operator $T:X\rightarrow Y$ that transforms weakly Cauchy sequences into weakly convergent ...
- 3,333
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81
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Embedding random variables in infinite-dimensional spaces
Let $H$ be a reproducing kernel Hilbert space of functions $f:E\to F$ with kernel $k$. A point in $E$ may be embedded into $H$ via the canonical embedding $x\mapsto k(x,\cdot)$. Similarly, a random ...
- 680
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128
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Topology on function spaces for pointwise convergence
Suppose I have some collection of maps $T_\lambda: C^\infty(\Omega)\rightarrow C^\infty(\Omega)$ which are linear and parameterized by a parameter $\lambda >0$. (Perhaps more generally take $C^\...
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925
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Dunford-Pettis theorem
Let $\mathcal{F}$ be a bounded set in $L^{1}(\Omega)$. Then $\mathcal{F}$ has compact closure in the weak topology $\sigma(L^{1},L^{\infty})$ if and only if $\mathcal{F}$ is equi-integrable, that is,
\...
- 43
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273
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Strong data-processing inequality ? Upper bound on a certain modified total-variation metric
Let $\mathcal X=(\mathcal X,d)$ be a Polish space equipped with the Borel sigma-algebra. Let $p\ge 1$ and $P_1,P_2$ be probability distributions on $\mathcal X$ such that $\max_{k=1,2}\int d(x,x_0)^...
- 6,726
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59
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A determinantal mixture of probability densities
I came up with this operation after playing with determinantal point processes:
Given two probability densities $f,g$ defined on some measurable space with reference measure $\mu$, set
$$
f\star g(x)...
- 2,095
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27
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Approximation of multipliers by multipliers of a smaller set 2
This question is a refinement of my previous question.
Let $X$ be a compact metric space, and let $B$ be a bounded Banach Disk in $C(X)$ such that for every $x\in X$ there is $f\in B$ with $f(x)\ne 0$...
- 5,385
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How does the principal value affects to the limit here?
In Córdoba and Gancedo - Contour dynamics of incompressible 3-D fluids in a porous medium with different densities (page 4) I read that if
$$ v (x_1,x_2,x_3,t)=-\frac{\rho_2-\rho_1}{4\pi} \...
- 373
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Gluing together dense subset of Projective Limit in $Ban_1$
Let $(X_n,\pi_n^{m})$ be a countable projective system in the category Ban$_1$ of Banach spaces and short linear maps (is (continuous) linear constructions). Then (co)-completeness of this category ...
- 4,969
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638
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Reciprocal expansion of modified Bessel function
I am reading Sherstyukov and Sumin - Reciprocal expansion of modified Bessel function in simple fractions and obtaining general summation relationships containing its zeros. The authors say they are ...
- 111
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87
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Explanation for the energy method used here
I am reading a paper where the authors prove
$$ \frac{d}{dt} \Vert f \Vert_{H^4} (t)\leq C\Vert f \Vert^5_{H^4}(t) $$
Where $f=f(x,t)$ and $H^4$ stands for the usual Sobolev space. Using Gronwall's ...
- 373
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Do we have $M\hat{\otimes}_A N\cong M\otimes_A N$ if $M$ is a finitely generated projective $A$-module over a nuclear Frechet algebra $A$?
Let $A$ be a nuclear Frechet algebra with unit. Let $M$ be a right Frechet $A$-module and $N$ be a left Frechet $A$-module. Both $M$ and $N$ are assumed to be non-degenerate. We can define the ...
- 8,707
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291
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Continuity of the Legendre transform of a Lipschitz function
Let $p > 1$, $f$ defined on $L^p(\mathbb{R}^n,\mathbb{R})$, locally Lipschitz and concave, with $f(x) = 0$ when $x \geq 0$ a.e. We define, with $q$ dual to $p$, for any $y\in L^q$ , $g(y) := \sup_{...
- 31
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A different kind of weighted Hardy space
Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$, let $\mathcal{A}\left(\mathbb{D}\right)$ denote the vector space of all complex-valued functions which are holomorphic on $\mathbb{D}$, and ...
- 1,266
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79
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Minimization of a smooth integral functional over a closed convex set
Let $(E,\mathcal E,\mu)$ be a probability space, $I$ be a finite nonempty set, $\gamma:(E\times I)^2\to[0,\infty)$ be measurable, $$F_1(g,w):=\sum_{i\in I}\int\mu({\rm d}x)w_i(x)g(x)\sum_{j\in I}\int\...
- 161
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162
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Solution to Heat Equation By Projection
Let $X\subseteq C(\mathbb{R}\times [0,\infty))$ be the collection of all solutions to the IV heat equation
$$
\partial_t u(t,x) = \partial^2 u(t,x) \qquad u(0,x)=p(0,x),
$$
for some fixed $p\in C^2(\...
- 4,969
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85
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Gaussian width and restricted isometry
It is known that, for an $m$ dimensional space and an $n\times m$ dimensional random matrix $U$ whose entries are iid Gaussian, then $\|I-(1/n)U^TU\|$ is bounded by $\sqrt{m/n}$ when $n>m$.
If a ...
- 11
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101
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Weak convergence to a Gaussian measure in coarser topology induced by a covariance operator
I'm currently studying Gaussian measures on Hilbert spaces and would like to find conditions under which convergence to a Gaussian measure with respect to a coarser topology induced by a covariance ...
- 617
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79
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One dimensional periodic travelling waves to some pde
Travelling wave equation on one dimension to Gross Pitaeavkii equation is $$ \phi '' +ic\phi'+\phi (1-|\phi|^2)=0\qquad (1) $$ where $c\in (0,\sqrt{2})$ and $ \phi$ is a complex valued function. I am ...
- 373
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45
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How can we show that a solution depends on more than one variable? [closed]
I have obtained theoretically a solution to a nonlinear Schrödinger-type equation in dimension two. I also proved that is not constant. Now, I wonder if it depends on two variables and not only in one,...
- 373
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110
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Trace and second-order inverse trace on space with Gibbs measure
Consider $(t, x)\in [0,T]\times (\mathbb{R}^d,d\mu)$, where the measure $d\mu(x)=K^{-1}\exp(-U(x))dx$ is a reasonable Gibbs measure (it satisfies a Poincaré or log-Sobolev inequality. One can, for ...
- 73
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101
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Extension of a compactly supported pseudo-differential operator
Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ and $P \in \Psi^{m}(\Omega)$ a compactly supported pseudo-differential operator, that is, the kernel of $P$, is compact. Is it true that $P$ extends ...
- 481
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86
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Uniform continuity of sequence of semigroups
Let $T(t)$, $t\in [0,\tau]$, be a $C_0$ semigroup on an Banach space $X$. Also, let $T_n(t)$ be a sequence of semigroups that satisfies for all $x\in X$
$$\lim_{n\to \infty}\sup_{t\in [0,\tau]}\|T_n(...
- 395
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148
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Lyapounov's inequality for Orlicz norms
When a sequence $f \in \ell_1$, there is a very simple bound on its $\ell_q$-norms given by $\|f\|_q^q \leq \|f\|_1 \cdot \|f\|_\infty^{q-1}$.
This inequality is a special (or rather limit) case of ...
- 4,342
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114
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Banach space isometric to its dual
Let $X$ be a real or complex Banach space linearly isometric to its Banach dual $X^\star$. Is it true that $X$ is reflexive?
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160
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Little help showing strong convergence in $H^1$
Let $u_n $ be a sequence in $H^1 (\Omega,\mathbb{C})$ where $\Omega \subset \mathbb{R} ^N$ is bounded. Assume that we know that all the functions $u_n$ are smooth and $\Vert u_n \Vert _{C^m} \leq K(m,...
- 373
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109
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Algebra structure on Haagerup tensor product of operator spaces
Let $A$ and $B$ be operator spaces. Is there any algebra structure on Haagerup tensor product of operator spaces such that the Haagerup tensor product becomes Banach Algebra?
Any references or ideas?
- 1,085
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110
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Connection between traces in Cameron-Martin spaces of two Gaussian measures
Let $\mu$ and $\nu$ be two Gaussian measures defined on a common separable Banach space $B$. Denote their two Cameron-Martin spaces by $H(\mu)$ and $H(\nu)$, respectively.
Let $T: B \to B^{\ast}$ be a ...
- 95
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96
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Minimize $\langle(1-\kappa)^{-1}f,f\rangle$ for a parameter-dependent integral operator $\kappa$
I've got a contractive self-adjoint linear integral operator $\kappa$ of the form $$(\kappa g)(x):=g(x)+\int\lambda({\rm d}y)k(x,y)(g(y)-g(x))\;\;\;\text{for }g\in L^2(\mu),$$ where $k$ depends on the ...
- 161
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125
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Almost every where divergent Fourier series
Does there exist any continuous function $f:[\pi,\pi]\to \mathbb{C}$ whose Fourier series $\sum \hat{f}(n)e^{int}$ is almost every where divergent?
- 3,982
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84
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A Hölder version of the Johnson-Lindenstrauss Lemma on essentially bounded functions
Does there exist a Hölder (not necessarily linear) projection from $L^{\infty}(\mathbb{R}^d)$ to any finite-dimensional linear subspace? This is known when $L^{\infty}(\mathbb{R}^d)$ is replaced by a ...
- 4,969
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148
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Existence of continuous integral kernel
Let $(X_i,\mathcal{F}_i,\mu_i)$, $i\in \{1,2\}$, be a $\sigma$-finite measure space and $E_i$ an ideal of measurable functions on $X_i$ with full carrier (for example $E_i=L^p(X_i,\mu_i)$).
A ...
- 31
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153
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Gradient formula for Clarke's generalized gradient on a general Banach space
In Theorem 10.27 of the book Functional Analysis, Calculus of Variations and Optimal Control, there is the following gradient formula:
($\operatorname{co}$ deotes the convex hull).
Is there an ...
- 161
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80
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Measurability of a generalized point spectrum
Assume that $ T:H\oplus H\rightarrow H\oplus H$ is a unitary linear operator on the double sum of a separable Hilbert space $H$ with itself.
Let us call a pair $(\lambda, \mu)\in\mathbb{C}\oplus\...
- 621
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72
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Finding the solutions of the inequality $\|xb-a\|<1$
Let $H$ be a Hilbert space and consider bounded operators $a$ and $b$ on $H$.
For given operators $a$ and $b$, I am looking a way to get all solutions (bdd operators $x$) of the inequality $\|xb-a\|...
- 3,982
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133
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orthonormal basis of ${H^2} \cap H_0^1$
we consider the following eigenvalue problem for the Laplacian
$$ - \Delta w\left( x \right) = \lambda w\left( x \right),\,x \in \left( {0,1} \right),\,w\left( 0 \right) = w\left( 1 \right) = 0.$$
By ...
- 183
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Is the integral functional $I(x) = \int_{0}^{T} \Lambda (t , x(t), \dot{x} (t)) \; dt $ lipschitz around $x_0$?
Let the function $\Lambda : [0,T] \times \mathbb{R^n} \times \mathbb{R^n} \to \mathbb R$ be continuously differentiable. Assume the integral functional $I(x) = \int_{0}^{T} \Lambda (t , x(t), \dot{x} (...
- 359
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75
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Existence of a `right' sequence
Suppose that $f_n: \mathbb{R} \times [0,\infty) \to \mathbb{R}$ is a sequence of bounded variation functions such that there exists $C>0$: $|f_n| < C$ for every $n$ and, as $n \to \infty$, for $...
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164
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A consequence of De Giorgi oscillation lemma
The following lemma is true (see DeGiorgi oscillation lemma)
Let $u$ be a subsolution of $$\mathrm{div}(A(x)\nabla u) = 0,$$
where $A$ is bounded, measurable and uniformly elliptic ($C^{-1}\...
- 819
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70
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Domain of definition of a hamiltonian with delta(contact) potential
I am having a hard time making sense of the so-called "delta function potential well" in quantum theory. The Hamiltonian operator is defined as (with $\mathscr D_H\subset \mathscr H=L^2(\mathbb R)$)
$$...
- 111
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109
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Commutative subalgebras of $B(H)$ whose all automorphisms are in the form of unitary conjugation
Let $H$ be a complex Hilbert space.
Is there a compact Hausdorff space $X$ such that $C(X)$ is embeded in $B(H)$ and for every homeomorphism $\alpha$ of $X$ there exist a unitary operator $u\in B(H)$ ...
- 312
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0
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52
views
Asymptotically periodic potentials
Who came up with the idea of solving elliptic equations with periodic potentials and from there solving elliptic equations with asymptotically periodic potentials?
- 67
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52
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Spectrum of a $1$-parameter family of symmetric linear operators
I am working with certain submanifolds of symmetric spaces and, using a construction in Terng-Thorbergson, we ended up in the following Hilbert space problem:
Let $H$ be a (real) Hilbert Space and $...
- 163
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138
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Spectrum of a differential operator on $L^2(0, \infty)$
Let $A:H^n(0, \infty) \subset L^2(0, \infty) \to L^2(0, \infty)$ be the differential operator defined by
$$Af:= \sum_{j=0}^na_j f^{(j)}$$
for all $f \in H^n(0, \infty),$ where $a_j \in \mathbb{C}$ and ...
- 229
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244
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Why does this PDE have a solution?
Let $\Sigma$ be a compact surface and denote by $\nu$ the unit conormal for $\partial \Sigma$. Let
$$E = \left\{ \phi \in C^{2,\alpha}(\Sigma) : \int_{\Sigma} \phi \, \mathrm{d}A = 0 \right\} $$
and
$$...
- 2,085
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0
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183
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What imaginations of Lebesgue spaces or other Banach spaces do people intuitively share?
At several occasions I heared people discussing about the „colors“ of Lebesgue spaces $L^p$: $L^2$ is red, $L^1$ is white, $L^\infty$ is black, and the other $L^p$ are blue or violett. Of course this ...
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