# Questions tagged [fa.functional-analysis]

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

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### Takesaki's duality in representation theory of $C^*$-algebras

In M.Takesaki's 1967 article titled A Duality in the Representation Theory of C-Algebras*, admissible operator fields are defined in order to generalize Gelfand transform to a non-abelian setting. ...
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### Dependence of functional integral on the function space

In physics, the following functional integral is considered \begin{gather} Z[J]= \int Df \exp(-\int d^dx( f\Box f+\lambda f^4 +Jf )) \end{gather} It is usually said that the integration is performed ...
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### Necessary conditions for convergence of convolution

In math.SE, I've asked a question about the convergence of convolution of two functions which have bilateral Laplace transform and also have disjoint Region Of Convergence (ROC) but the question didn'...
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### Series representation of functions

Let $H$ be a Hilbert space, consisting of functions $f:\mathbb{R} \to \mathbb{R}$. Let $$V = \left\{ f_J \in H: f_J= \sum_{j=1}^J c_j^{(J)} g_j, c_j^{(J)}\in\mathbb R, J\in \mathbb N \right\}$$ ...
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### Existence of the limit of periodic measures

Let $T: X \to X$ be a continuous map over a compact metric space. We say that a measure $\mu$ is $T$-invariant if $T_{\ast} \mu= \mu$. We denote by $M(X, T)$ the space of all $T$-invariant Borel ...
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### Prove if the fractional Laplacian of a function is bounded

Take $s\in (0, 1)$. I am trying to understand if $(-\Delta)^s (\log(1+x^2))$ is bounded, that is if there exists $R>0$ such that $|(-\Delta)^s (\log(1+x^2))|\le R$. Here $(-\Delta)^s$ is the ...
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### A question in functional analysis about selfadjoint operator [closed]

In Hilbert space $u$, Let $T_1$,$T_2$ is selfadjoint operator, if exit $c>0$ such that $cI\le T_1\le T_2$, prove $T_1$,$T_2$ have a bounded inverse operator and $c^{-1}I\ge T_1^{-1}\ge T_2^{-1}$. I ...
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### Sufficient condition such that the set of zeros of an analytic function $f:\mathbb{R}^n \to \mathbb{R}$ contains only isolated points

Consider a real- analytic function $f: \mathbb{R}^n \to \mathbb{R}$. We know that zeros of $f$, roughly speaking, live in the low dimensional manifolds. My question: Does a 'reasonable' sufficient ...
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### An integral inequality?

Let $v \in C^\infty(\mathbb R)$ such that $1 \ge v \ge 0$ and $\int_{\mathbb R} v \, dx = 1$. I want to show that if $$\int_{\mathbb R} v |v''|^2 \, dx < + \infty. \tag{\star}$$ then  \int_{\...
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### For any $\beta>0$, there is a constant $c>0$ such that $\left\|(1-\Delta)^{\frac{\beta}{2}} f\right\|_{\infty} \leq c\|f\|_{C_b^\beta}$

For any $n \in \mathbb{Z}^{+}$, let $C_b^n\left(\mathbb{R}^d\right)$ be the class of real functions $f$ on $\mathbb{R}^d$ with continuous derivatives $\left\{\nabla^i f\right\}_{0 \leq i \leq n}$ such ...
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