On relations between weighted mean and power series methods of summability
Let \(\{p_ n\}\) be a sequence of non-negative real numbers with \(p_ 0>0\) and \(P_ n=\sum^{n}_{k=0}p_ k\to +\infty\) (n\(\to \infty)\). Let \(p(x)=\sum^{\infty}_{n=0}p_ nx^ n<+\infty\) for \(0<x<1\). A sequence \(\{s_ n\}\) of real numbers is said to be \(M_ p\)-convergent \((s_ n\to s(M_ p))\) if \((1/P_ n)\sum^{n}_{k=0}p_ ks_ k\to s\) (n\(\to \infty)\) and \(J_ p\)-convergent \((s_ n\to s(J_ p))\) if \(\sum^{\infty}_{n=0}p_ ns_ nx^ n\) converges for all \(x\in (0,1)\) and \[ \lim_{x\to 1-}\frac{1}{p(x)}\sum^{\infty}_{n=0}p_ ns_ nx^ n=s. \] These two limitation methods are regular and \(s_ n\to s(M_ p)\) implies \(s_ n\to s(J_ p)\). In a paper of \textit{D. Borwein and \textit{A. Meir}} [(*) Math. Proc. Camb., Philos. Soc. 101, 283- 286 (1986; Zbl 0633.40013)] there is proved Theorem T. If \(s_ n\to s(J_ p)\), \(s_ n>-H\) \((n=0,1,...)\) and \((1)\quad \lim_{x\to 1- }p(x^ m)/p(x)=\lambda_ m>0\) for \(m=2\) and \(m=3\), then \(s_ n\to s(M_ p)\). It is proved in (*) that \((2)\quad \lim_{x\to 1-}p(x^ 2)/p(x)=\lambda\) does not imply (1) when \(0<\lambda <1\) but (1) and (2) are equivalent if \(\lambda =1\). The authors prove that Theorem T does not remains valid when (1) is replaced by (2) with \(0\leq \lambda <1\). Further for each \(\lambda\in [0,1)\) a function p(x) and a sequence \(\{s_ n\}\) of positive numbers are constructed such that p(x) satisfies (2) and \(a_ n\) is \(J_ p\)-convergent but not \(M_ p\)-convergent. Several further results related to Theorem T are proved here.
scientific article; zbMATH DE number 4175610
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