Moment multipliers for power series
Let \(\mu =\{\mu_ n\}_ 0^{\infty}\) be a Hausdorff moment sequence, \[ \mu_ n=\int^{1}_{0}t^ nd\chi (t),\quad \chi \in BV[0,1], \] and let \({\mathcal G}\) be the set of all sequences of quotients of positive Hausdorff moment sequences, \[ {\mathcal G}=\{\nu =\{\mu_ n^{(1)}/\mu_ n^{(2)}\}^{\infty}_{n=0}:\quad \mu_ n^{(i)}=\int^{1}_{0}t^ nd\chi^{(i)}(t)>0\quad for\quad all\quad n,\quad i=1,2\}. \] Now let \({\mathcal P}_ k\) denote the set of all power series \(P(\omega)=\sum^{\infty}_{k=0}a_ k\omega^ k\) with positive radius of convergence and let \(T^{\nu}P\) denote the operator on P defined by \[ T^{\nu}P=\sum^{\infty}_{k=0}\nu_ ka_ k\omega^ k,\quad \nu \in {\mathcal G}. \] Finally let \({\mathcal G}_ s\) denote the subset of \({\mathcal G}\) for which the Mittag-Leffler star \(M_{_ T\nu_ P}\) of \(T^{\nu}P\) is equal to the Mittag-Leffler star \(M_ P\) of P. (It is known that \(M_{_ T\nu_ P}\supset M_ P\), see \textit{E. Hille}, Analytic function theory. Vol. II (1962; Zbl 0102.294).) The author proves the following two theorems: Theorem 1. Given a sequence \(\{\phi_ n\}\) with \(0<\phi_ n<1\), \(\lim_{n\to \infty}\phi_ n=0\) and \(\lim_{n\to \infty}\phi_ n^{1/n}=1\), there exists a non- decreasing \(\chi\) such that the moments \(\mu_ n=\int^{1}_{0}t^ nd\chi (t)\) satisfy \(0<\mu_ n\leq \phi_ n\) and \(M_{_ T\nu^{\alpha}_ P}=M_ P\), \(P\in {\mathcal P}\), \(\alpha\) real where \(\nu^{\alpha}\) is the sequence of powers \(\{\nu_ n^{\alpha}\}^{\infty}_{n=0}\). Theorem 2. Given a nonnegative sequence \(\{a_ n\}\) with \(\limsup_{n\to \infty}a_ n^{1/n}=1\), and a sequence \(\{\epsilon_ n\}\), \(0<\epsilon_ n<1\), \(\lim_{n\to \infty}\epsilon_ n=0\), there exists \(\nu\in G\) and a subsequence \(\{n_ i\}\) such that \[ M_{_ T\nu^{\alpha}_ P}=M_ P,\quad P\in {\mathcal P},\quad \alpha \in {\mathbb{R}},\quad a_ n\nu_ n\leq 1,\quad n\leq n_ 0 \] and \(a_{n_ i}\nu_{n_ i}\geq e^{-\epsilon_{n_ i}}\), \(i=1,2,...\).
scientific article; zbMATH DE number 3923075