Topological regular variation. II: The fundamental theorems
In its classical setting, regular variation deals with the asymptotic behaviour of \(h(tx)h(x)^{-1}\) for \(h:\mathbb R^+\rightarrow\mathbb R^+\), with \(t\) fixed and \(x\rightarrow \infty \). Fundamental theorems are the Uniform Convergence Theorem (UCT). The UCT states that under appropriate conditions, if \(\lim_{x\rightarrow \infty }h(tx)h(x)^{-1}\) exists for all \(t\in S\), then the limit exists for all \(t>0\) and the convergence holds locally uniformly in \(t>0\). In the present paper the authors consider functions \(h:X\rightarrow H\) and limits of the form \(\delta _{X}(t)=\lim_{\| x\| \rightarrow\infty }h(tx)h(x)^{-1}\), \(t\in S\), where \(X\) denotes a metric group, \(H\) is a group of self-homeomorphisms of \(X\) with finite norm, and \(S\subseteq T\subseteq H\). Under appropriate conditions on \(S,T\) and \(h\) the convergence holds for all \(t\in T\) and \(\delta_X(t)\) is a continuous homeomorphism from \(T\) to \(H\). A function \(h:X\rightarrow\mathbb R\) is \(\{\varphi_n\} \)-slowly varying if \(\lim_{n\rightarrow \infty }h(\varphi _{n}(t))-h(\varphi _{n}(t_{0}))=0\) , where \(\{ \varphi _n\} \) is a sequence of divergent self-homeomorphism of \(X\) and \(t_{0}\) is a ``distinguished point'' of \(X\). Using a bounded equivalence principle, the authors also prove a UCT for \(\{ \varphi _n\} \)-slowly varying functions.
scientific article; zbMATH DE number 5775720
scientific article