Some Minimax Results on Dense Sets and Dense Families of Functionals in C(K) and B(K)

Miscellaneous applications of certain minimax theorems. I

Here is one of the results of this paper (with the convention ${{1}\over {0}}=+\infty$): Let $X$ be a real Hilbert space and let $J:X\to {\bf R}$ be a $C^1$ functional, with compact derivative, such that $$α^*:=\max\left \{0,\limsup_{\|x\|\to +\infty}{{J(x)}\over {\|x\|^2}}\right \}<β^*:=\sup_{x\in X\setminus \{0\}}{{J(x)}\over {\|x\|^2}}<+\infty\ .$$ Then, for every $λ\in \left ]{{1}\over {2β^*}}, {{1}\over {2α^*}}\right [$ and for every convex set $C\subseteq X$ dense in $X$, there exists $\tilde y\in C$ such that the equation $$x=λJ'(x)+\tilde y$$ has at least three solutions, two of which …

Diagonalizations of dense families

We develop a unified framework for the study of properties involving diagonalizations of dense families in topological spaces. We provide complete classification of these properties. Our classification draws upon a large number of methods and constructions scattered in the literature, and on some novel results concerning the classical properties.

Estimation in Functional Regression for General Exponential Families

This paper studies a class of exponential family models whose canonical parameters are specified as linear functionals of an unknown infinite-dimensional slope function. The optimal minimax rates of convergence for slope function estimation are established. The estimators that achieve the optimal rates are constructed by constrained maximum likelihood estimation with parameters whose dimension grows with sample size. A change-of-measure argument, inspired by Le Cam's theory of asymptotic equivalence, is used to eliminate the bias caused by the non-linearity of exponential family models.

$Γ$-supercyclicity of families of translates in weighted $L^p$-spaces on locally compact groups

Let $ω$ be a weight function defined on a locally compact group $G$, $1\le p<+\infty$, $S\subset G$ and let us assume that for any $s\in S$, the left translation operator $T_s$ is continuous from the weighted $L^p$-space $L^p(G,ω)$ into itself. For a given set $Γ\subset\mathbb{C}$, a vector $f\in L^p(G,ω)$ is said to be $(Γ,S)$-dense if the set $\{ λT_sf:\, λ\in Γ, \,s\in S\}$ is dense in $L^p(G,ω)$. In this paper, we characterize the existence of $(Γ,S)$-dense vectors in $L^p(G,ω)$ in terms of the weight and the set $Γ$.

泛函分析中若干网族的代数通有性

Algebraic genericity of certain families of nets in Functional Analysis

本文从线性角度研究泛函分析中多类异常网的代数通有性，涵盖弱收敛无界网、不满足Banach-Steinhaus定理的网、弱稠密范数无界网及关联网发散的收敛级数，推广并改进了拓扑向量空间与赋范空间中的若干已有结果。

A holomorphic functional calculus for finite families of commuting semigroups

Let A be a commutative Banach algebra such that uA = {0} for u $\in$ A \ {0} which possesses dense principal ideals. The purpose of the paper is to give a general framework to define F (--$λ$1$Δ$T 1 ,. .. , --$λ$ k $Δ$T k) where F belongs to a natural class of holomorphic functions defined on suitable open subsets of C k containing the "Arveson spectrum" of (--$λ$1$Δ$T 1 ,. .. , --$λ$ k $Δ$T k), where $Δ$T 1 ,. .. , $Δ$T k are the infinitesimal generators of commuting one-parameter semigroups of multipliers on A belonging to one of the following classes (1) The class of strongly continous semi…