# Questions tagged [fa.functional-analysis]

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

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### Characterizing Besov spaces in terms of p-variation

For $s>1/p$ the Besov space $B_{p,q}^s([0,1])$ can be characterized in terms of the $p$-variation: Let $p,q \in (1,\infty)$ and $s \in (0,1)$, $s>1/p$. A function $f:[0,1] \to \mathbb{R}$ is in ...
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### Continuity of operator exponential on subspace

Let $H$ be a separable Hilbert space with overcomplete basis $(e_t)_{t \in (0,\infty)}$ such that $\langle e_t, e_s \rangle = e^{-\vert t-s \vert}.$ We can then define the projection $P_t$ which is ...
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### Fréchet derivative of evaluation-like functional (multivariate)

I'm fairly new to functional calculus but and posting here since the question seems more appropriate than for MSE. When coming across this post I could not help but wonder the following. Let $H$ be ...
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### Conditions on triangle inequality for integral kernel

Consider $\int_RK(x,y)f(y)dy$, where $K(x,y) \in M_+(R^2)$. Let $L(t,s)$ be an iterated rearrangement of $K$. Let also $$A(t,v)=\int_0^{1/v}L(1/t,s)ds,$$ which is decreasing with $v$ and ...
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### A functional integral inequality

Suppose $f:I=(0,1)\to \mathbb R$ is a continuous function that satisfies $$\int_I f(t) e^{at}\,dt \geq 0\quad \text{for all a \in \mathbb R}.$$ Does it follow that $f\geq 0$ on $I$?
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### parametric model with parameters described as gaussian processes

Let's assume that I have some data $y_{t_i}$ (i = 0, 1, ..., N) and a model $\hat{y}(a(t), b(t))$, where the parameters of my model (a, b) evolve with time t in a stochastic manner. I am wondering if ...
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### $C^j$-topology considered by Greene and Krantz

My question is about the $C^j$ topology used by Greene and Krantz in their paper "Deformations of Complex Structures, Estimates for the $\bar{\partial}$-equation, and stability of the Bergman ...
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### discontinuous functions on the Sobolev borderline

The Sobolev embedding theorem implies that every function of class $W^{k,p}$ on a reasonable $n$-dimensional domain is continuous if $kp > n$. Cases with $kp=n$ are known as "borderline" ...
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### Analytic approximations of smooth vector fields

Let $M$ be the set of smooth divergence-free vector fields $u$ on $\mathbb{R}^3$ with $$|\partial_x^{\alpha} u(x)| \leq C_{\alpha K}(1+|x|)^{-K}$$ on $\mathbb{R}^3$ for any $\alpha,K$. Further, we ...
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### Closed convex hull in infinite dimensions vs. continuous convex combinations

tl;dr: When is the closed convex hull of a set $K$ equal to the set of "continuous" convex combinations of $K$? I am essentially asking for the most general, infinite-dimensional analogue of ...
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### $L^p$ compactness for a sequence of functions from compactness of cut-off

Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions \{f_{n}\psi_m(...
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### An example of a certain continuous and strictly convex function

Let $X$ be a locally convex topological vector space. I am looking for an example of a function $f: X \times X \to [0,\infty]$ with the following properties: (1) For all $x,y \in X$, $f(x,y) = 0$ if ...
Let $\mathcal{H}$ be a Hilbert space and $A,B$ two operators on it (not necessarily self-adjoint) such that $A, A+B$ are generators of strongly continuous one parameter semigroups $e^{-tA},e^{-t(A+B)}$...
Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The ...