# Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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### Approximation of the product $(\bar{z} - a)^{-1} \cdot (z - b)^{-1}$

I would like to construct an approximation of the product \begin{equation} f(z) = \frac{1}{\overline{z}-a} \frac{1}{z-b}, \end{equation} where $a, b \in \mathbb{C}$, and $|{a}/{z}|, |{b}/{z}| <1$. ...
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### Action of Bochner integral of operator-valued functions on vectors

Consider a separable Hilbert space $\mathcal H$ and the bounded linear operators $B(\mathcal H)$. Consider a function $T: [0, \infty) \to B(\mathcal H)$, under what assumptions on $T(t)$ is it true ...
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### On $L^2$ spaces which have an orthogonal basis of characters (complex exponentials)

Suppose $\Omega \subset \mathbb{R}^n$. What conditions on $\Omega$ make it so there exists a countable set $\Lambda$ such that $\{e^{2\pi i\lambda t} \}_{\lambda \in \Lambda}$ form an orthogonal basis ...
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### Does the compactness of parameter of distribution function imply the compactness of the distribution (or probability measure) in Wasserstein space?

For a family of probability measures sharing the same form of distribution function $F(x; p)$ with different parameters (i.e., $p$'s), if the parameter falls in a compact subset of real line, can we ...
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Suppose that $\mu(dx) = \exp(-\psi(x)) \, \mathrm{dx}$ is a probability measure on $\mathbf{R}^d$. For suitable functions $g \geqslant 0$, define $$\text{Ent}(g) = \int \mu(dx) g(x) \log \left( \frac{... 0answers 142 views ### Weaker analogues of amenability for groups of piecewise projective homeomorphisms Let A be a subring of {\bf R} and let H(A) be the group defined/constructed in Monod's 2013 PNAS paper. Monod showed that provided A\neq {\bf Z}, H(A) is non-amenable. (The proof breaks down ... 0answers 78 views ### How can we define \chi_{\Omega}(A)? I was reading Spectrum and dynamics where Paolo Facchi discusses projection-valued measures and integrals. The discussion and constructions are all based on the fact that one can define characteristic ... 0answers 132 views ### Reference Aubin T.- Espaces de Sobolev sur les varietes Riemanniennes. Bull. Sc. Math. 100, (1976) 149-173 I have been searching without success for the reference: Aubin T.- Espaces de Sobolev sur les varietes Riemanniennes. Bull. Sc. Math. 100, (1976) 149-173 It is cited in many related works. In ... 1answer 82 views ### Limit of u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0 as \epsilon \to 0 Consider the initial-value problem associated to the PDE u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0. To prove that, as \epsilon \to 0, the weak solution ... 1answer 210 views ### What is the 'right' definition of zero measure subsets of Banach spaces? Question. There are several ways of defining a notion of a 'zero measure' subset of a Banach space X. Which one is the 'right' or failing that, the preferred notion? [See below for a more precise ... 1answer 284 views ### \|t\| = \sup_{\|z\| \le 1} \|\langle tz,z\rangle\| when t=t^* Let A be a C^*-algebra, E be a (right) Hilbert A-module and t \in \mathcal{L}_A(E) be an adjointable operator satisfying t=t^*. Is it true that$$\|t\| = \sup_{z \in E, \|z\| = 1} \|\...
Is there a fixed-point theorem that implies the following result? Let $F$ be a nonempty convex set of functions on a discrete group with values in $[0,1]$. Suppose $F$ is invariant with respect to ...