Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3,436 questions with no upvoted or accepted answers
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On the strong convergence of generators and the corresponding semigroups
Let $(E,\|\cdot\|)$ be a separable Hilbert space on $\mathbb{R}$. We consider a sequence of strongly continuous semigroup $\{T^{(n)}\}_{n=1}^\infty$ on $E$ [of course, for $n \in \mathbb{N}$ and $t>...
- 721
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749
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Notation for the space of eventually-zero sequences
An eventually-zero sequence is a real-valued sequence $(x_n)_{n=1}^\infty$ for which there exists an $N\in\mathbb{N}$ such that $x_n=0$ for each $n\geq N$. The space of eventually-zero sequences ...
- 153
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152
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Poisson Kernel and solution formula for fractional elliptic problem
$$
k (-\Delta)^s u + u = 0, \qquad x \in U, \\
u(x) = f(x), \qquad x \in \mathbb R^n \setminus U,
$$
with $f \in L^\infty(\mathbb R^n)$, $k>0$, and $(-\Delta)^s$ is the singular integral ...
- 839
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292
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Developing measure theory through $\delta$-rings
I am scratching my head trying to understand the notion of measurability in Fell's and Doran's book: Representations of (star)-algebras, locally compact groups, and Banach (star)-algebraic bundles.
...
- 155
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What is the topological characteristic of a separable metric space $X$ such that $|kX\setminus X|=\frak{c}$ for any completion $kX$ of $X$?
What is the topological characteristic of a separable metric space $X$ such that $|kX\setminus X|=\frak{c}$ for any completion $kX$ of $X$?
- 1,101
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Show that for all $\delta$, we have $\|u\|^2_{\Gamma}\le c_\delta(\|u\|^2_{\omega(\delta)}+\|u\|_{\omega(\delta)}\|\nabla u\|_{\omega(\delta)})$
I am reading the article On existence of weak solutions of the Navier-Stokes equations in regions with moving boundaries from Fujita and Kato and at some point they use an argument I have some trouble ...
- 452
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Question about Dirac operator
Let $D$ be a generalized Dirac operator on a complete Riemannian manifold. I'm a little confused to prove that there exists a constant $c>0$ such that
$$\|D\sigma\|^2\geq c^2\|\sigma\|^2$$
for $\...
- 563
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34
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Minimum eigenvalue of normal matrix with polynomial basis
For each $n\in \mathbb{N}\cup\{0\}$, let $x_n(t)=\frac{t^n}{n!}$ for all $t\in [0,1]$. As the functions $X_N=(x_0 ,\ldots, x_N)$ are linearly independent, the matrix
$
\int_0^1 X_N(t)^\top X_N(t)\,\...
- 503
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597
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What is $T T^*$ argument?
During my studying of many papers, some authors used what so-called $T T^*$ argument. I have no clue about this concept (or mathematical tool). Could you please enlighten me with some explanations or/...
- 159
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100
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$L_1$ convergence rates for multivariate kernel density estimation
Let $X$ be a random variable on $\mathbb R^d$ with probability density function $f$, and let $X_1,\ldots,X_n$ of $X$ be $n$ iid copies of $X$. Given a bandwidth parameter $h=h_n > 0$ and a kernel $...
- 6,853
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Uniformly continuous semigroups are analytic
Reposting from stackexchange.
I know that every analytic $C_0$-semigroup is differentiable and then every differentiable semigroup is norm continuous.
I want to know where uniform continuity fits in ...
- 131
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94
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Less strict holomorphy
Using the Grünwald-Letnikov definition of fractional derivative for complex-valued functions, can there be complex function $f(z)$ over all plane, that is not holomorphic but there is $r \in (0,1)$ s....
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The functional calculus on continuous functions
Let us consider $C[0,1]$ the space of continuous functions on the closed unit interval. For a given $x$ in $C[0,1]$, let us consider $A(sp(x))$, all analytic functions on a neighborhood of $sp(x)$.
...
- 4,058
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Extreme points of the unit ball of bounded operators on $L^p(\mathbb{R}_+)$
Is it known what the set of all extreme points of $B(L^p(\mathbb{R}_+))$ (bounded operators on $L^p(\mathbb{R}_+)$) is?
- 11
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Extreme case of K-interpolation
Suppose $X_0$ and $X_1$ are Banach spaces living in a larger Banach space
$X$. The $K$-functional is defined for each $f\in X_0+X_1$ and $t>0$ as
$$K(f,t,X_0,X_1)=\inf\{\|f_0\|_{X_0}+t\|f_1\|_{X_1}:...
- 319
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57
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How to show the solution map of NLS is not smooth?
Let $u(\delta, t)$ satisfy
$$iu_t +\Delta u+ |u|^{2k}u=0, \quad u(0)=\delta v_0$$
Note that the mapping:
$$\delta v_0\mapsto u(\delta, t)= S(t)(\delta v_0)-i\int_0^tS(t-\tau)(|u|^{2k}u)(\tau)d\tau $$
...
- 325
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Reference for unique classical solution to quasilinear uniformly parabolic PDEs
In this post, the author mentioned that "we know there is a unique classical solution (see the references below, for example)". I have tried to read the two references the author provided, ...
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Is it true that $\omega = \sum_{(k,l)\in I^2}\omega(p_k - p_l)$
Let $\{p_i\}_{i \in I}$ be a collection of projections in a $C^*$-algebra $A$ such that $\sum_{i \in I} p_i = 1$ in the strict topology (note here that $1$ is the unit of the multiplier algebra $M(A)$)...
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$H^s$ norm of dispersive semigroup
The Bourgain space is $X^{s,b} := X^{s,b}(\mathbb R \times \mathbb{T}^3)$ is the completion of $C^\infty (\mathbb R; H^s(\mathbb{T}^3))$ under the norm
$$\| u\|_{X^{s,b}}:= \|e^{- i t \triangle} u(t,x)...
- 159
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74
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Reference request: normal trace and the conormal derivative associated to the operator $Div (A \nabla)$ for a symmetric positive definite $A$
Let $A$ be a $3\times 3$ symmetric positive definite matrix. I am looking for a reference where I could find in which sense the normal trace $\gamma$ and conormal derivative $\gamma_n$ associated to ...
- 63
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Intersection of the kernel with the interpolation space
$\DeclareMathOperator\Ker{Ker}$Given two Banach spaces $X$ and $Y$ with a continuous inclusion $X\subset Y$, and another couple $X’ \subset Y’$ with the same properties. Take $f : Y \longrightarrow ...
- 11
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271
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Starlike sets in $\mathbb{C}^n$
Let $S$ be a bounded domain in $\mathbb{C}^n$. $S$ is called starlike about the point $x_0\in S$ if for every point of $S$, the segment of the straight line from the point to $x_0$ lies in $S$. If $S$ ...
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Properties of a kernel convolution $K'(x,y) = \int_X\int_X K_0(x,a)K(a,b)K_0(b,y)d\mu(a)d\mu(b)$ where $K$ and $K_0$ are kernels on $(X,\mu)$
Let $(X,\mu)$ be a probability measure space and $K:X \times X \to \mathbb R$ be a (psd) kernel on $X$. Let $K_0$ be another kernel on $X$ and defined a new kernel $\widetilde K$ on $X$ by
$$
\...
- 6,853
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A problem arising from Wiener-Levy theorem on the real line
Theorem (Wiener-Levy). Let $A(\mathbb{T})$ be the Fourier-algebra on the unit circle $\mathbb{T}$. Let $f$ be in $A(\mathbb{T})$ and suppose that $F$ is an analytic function on the range of $f$. Then $...
- 4,058
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Stability of Hajłasz-Sobolev class under post-composition
Informally: When is a Sobolev function, post-composed by a vector-valued function still Sobolev?
Assumptions/Setup
Let $(X,d_X,m_X)$ and $(Y,d_Y,m_Y)$ be complete and separable metric measure spaces; ...
- 5,405
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Is the universal representation an order isomorphism?
Let $A$ be a Banach *-algebra. By a *-representations of $A$, we mean a *-homomorphism $\pi:A\to B(H_\pi)$, where $B(H_\pi)$ is the space of all bounded linear maps on a Hilbert space $H_\pi$. Let $\...
- 4,058
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When are ellipsoids in a convex hull of a sequence with prescribed growth rate?
I am currently reading Dudley's 'Uniform Central Limit Theorems' and found two sections which together would have an interesting geometric interpretation for ellipses in Hilbert spaces. I would like ...
- 1,295
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Infinite matrices from $\ell^p$ to $\ell^{p/(p-1)}$ that are compact operators
I wanted to ask if my proof (sketch) of the following statement is correct. Namely, let $p>1$ and define $q= \frac{p}{p-1}$ we are given an operator $K : \ell^{p} \rightarrow \ell^{q}$ defined as $...
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Vector convolution?
I am working on a research problem which leads to the following optimization problem:
\begin{equation}
\hat{M} = \operatorname*{arg\,max}_M \Bigl\lVert\sum_{k=0}^{M-1} {\mathbf y}_k \exp\left(-j 2\pi ...
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143
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$\newcommand\v{\operatorname{vol}_d(C}$Compact subsets of $ℝ^d$ which maximize $\inf_{|v|\le1}\dfrac{\v\cap(𝜀v+C))}{\v)}$ for fixed $\v)$ and $𝜀>0$
Let $\operatorname{vol}_d$ be the volume measure on $\mathbb R^d$ and let $B_d$ be the unit-ball. For $\varepsilon \ge 0$ and a compact subset $C$ of $\mathbb R^d$ with $\operatorname{vol}_d(C)>0$, ...
- 6,853
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Closure of finite rank operators on $L^p$
It well-known that, an operator $T:H\to H$ on a Hilbert space, is compact if and only if T is limit of finite rank operators.
Besides this, the results by Per Enflo 1973 shows that this results is ...
- 1,101
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Small perturbation to a commuting family of hermitian matrices will hurt the nice properties?
Let $A_1, \dotsc A_N$ be a collection of finite Hermitian matrices that commute with one another and all have the matrix $2$-norm as $1$. Here $N$ is large but fixed.
Then, they are simultaneously ...
- 3,477
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Eigenvalues and eigenvectors of non-symmetric elliptic operators
We know that the operator $A=\Delta$ with domain $D(A)=\{u\in W^{2, 2}(\Omega): u=0 \ \ \text{on } \partial\Omega\}$ (say $\Omega$ is a bounded nice domain) has eigenvalues $\lambda_1>\lambda_2\ge \...
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Reference for smoothness of Nemytskii operator on fractional Sobolev spaces
Let $\varphi:\mathbb{R}\to\mathbb{R}$ be smooth and bounded (together with all of its derivatives). Define the operator
$$
\big(N_\varphi x\big)(t)=\varphi\big(x(t)\big)
$$
for $x\in H^s(T^d)$, the ...
- 93
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Intuition behind bound of second moment of Greens function by fractional moment
Consider the Hilbert space $ \mathcal{H} = l^2(\mathbb{Z}^d)$ for some dimension $d$ with basis given by the basisvectors $\{ \vert {x} \rangle \}_{x \in \mathbb{Z}^d} $.
Let $A$ be an either self-...
- 1,054
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Regularity of functions everywhere approximable by $n$-th degree polynomials
Let $(X, \lVert \cdot \rVert_X)$, $(Y, \lVert \cdot \rVert_Y)$ be two Banach spaces.
A function $P \colon X \to Y$ such that there exists $n \in \mathbb{N}$ such that for all $i \in \{ 0, \ldots, n \}$...
- 820
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96
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BV estimate for conservation law $u_t +( v(x)f(u))_x=0$
Let $u_0 \in BV(\mathbb R)$ and $f:\mathbb R \to \mathbb R$ be Lipschitz. Consider the Cauchy problem
$$
\begin{align*}
u_t +( v(x)f(u))_x&=0\\
u(0,\cdot) &= u_0
\end{align*}
$$
What is the ...
- 303
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99
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Estimate on integral with logarithmic weight
Let $f:\mathbb R^n \to \mathbb R$. What (minimal) assumptions are needed, in addition to $f \in L^1 \cap L^\infty$, to have
$$
\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} \, dx \, ...
- 839
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126
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Estimates for the Benjamin-Ono equation
Consider the Cauchy problem for the Benjamin-Ono equation
$$u_t + \frac{1}{2}(u^2)_x + \alpha \mathcal H(u_{xx}) - \beta u_{xx} = 0, \qquad t>0, \ x \in \mathbb R,$$
where $\mathcal H$ is the ...
- 839
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260
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Closure of smooth functions in Besov spaces
For real numbers $\alpha > \beta$, we know there is a continuous embedding of Besov spaces $B^\alpha_{\infty,\infty}\subset B^\beta_{\infty,\infty}$. We take the closure of the intersection $C^{\...
- 253
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239
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Expected value of the quotient times quotient of the expected values
I am looking for a reference -if there is any- about how to control the following expression:
$$\mathbb{E}\left[\frac{f(X)}{g(X)}\right]\cdot\frac{\mathbb{E}[g(X)]}{\mathbb{E}[f(X)]},$$
where $f$ and $...
- 11
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127
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Laplacian on the sphere and Moving Plane method
Consider the sphere $\mathbb{S}^n$ as a subset of $\mathbb{R}^{n+1}$, thus $\mathbb{S}^n=\{\omega\in \mathbb{R}^{n+1},\sum_{i=1}^{n+1}\omega_i^2=1\}.$
I am interested in studying positive solutions to ...
- 537
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95
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Existence of $C^{2, \alpha}$ solution to $a^{ij}(x,u,Du)D_{ij}u+b(x,u,Du)=0$ using the Leray–Schauder theorem in "Elliptic PDE" of Q. Han & F. Lin
In this part of the book "Elliptic PDE" of Qing Han & Fanghua Lin, the Leray–Schauder existence theorem is applied to prove the existence of $C^{2, \alpha}(\bar{\Omega})$ solution.
For $\...
- 809
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56
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Smooth approximation in Sobolev spaces for surfaces with boundary
Let $\mathbb{D}$ be the unit disk in $\mathbb{C}$ with closure $\overline{\mathbb{D}}$, and let $\varphi:\partial \mathbb{D}\to \partial \mathbb{D}$ be any continuous homeomorphism. Let $\mu$ be a ...
- 163
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133
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‘Linear’ intersection property of separable Banach spaces
Let $X$ be a separable Banach space. Denote $W(f,\varepsilon) = \{z\in X\colon \lvert\langle f,z\rangle\rvert < \varepsilon\}$ for some $f\in X^*$ .
Suppose that $U$ is an open set in $X$ such that ...
- 43
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49
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Knowledge on weighted integral operators?
There are tons of books and a huge literature on the properties of the following integral operator:
\begin{equation}
T(f) = \int_{\mathcal{X}} K(x,\cdot)f(x)dx,
\end{equation}
where $K(x,z)$ is, say, ...
- 11
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119
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product of two generalized functions
Let $f_n$ and $g_n$ two generalized functions such that :
the product $f_n g_n$ is well defined for all $n\in \mathbb{N}$, which means $WF(f_n)+WF(g_n)$ does not contain an element of the form $(x,0)$...
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74
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Regular inclusions: $\{b\in B:E(b^*b)=0\}$ is a two-sided ideal
From [Donsing-Pitts-2008, theorem 4.8]:
For $A\subseteq B$ a regular inclusion, with $A$ abelian, and $E:B\to A$ its unique conditional expectation it holds:
The left ideal
$$L(E):=\{b\in B:E(b^*b)=0\...
- 598
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53
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Systematic approach to Weierstrass factorization
If you want to calculate the Taylor expansion of a function, you only need to know the derivatives of the function at the point of expansion.
Is there a similar algorithmic approach that can be ...
- 369
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124
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Is it possible to define a Bochner integral for a $S'(\mathbb R^d)$-valued function?
I apologize in advance for the rather vague question.
While reading the book White noise distribution theory by H.H. Kuo, in particular the section 13.3 I came across the following statement (I'll ...
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