Tauberian theorems for \(J_ p\)-summability (Q909162)

Let \(p=(p_ 0,p_ 1,...)\) be a real sequence with \(p_ n\geq 0\). \(\forall n\geq 0\), \(P_ 0>0\), \(P_ n:=\sum^{n}_{k=0}p_ k\to \infty\), but \(p(x):=\sum^{\infty}_{k=0}p_ kx^ k<\infty\) for \(0<x<1\). A complex sequence \(s=(s_ 0,s_ 1,...)\) is said to be \(J_ p\)-summable to \(\xi\), we write \(s_ n\to \xi (J_ p)\), if \(\sigma (x):=(1/p(x))\sum^{\infty}_{k=0}p_ ks_ kx^ k\to \xi\) as \(x\to 1^-\) and \(\sum^{\infty}_{k=0}p_ ks_ kx^ k\) converges for \(0<x<1\). s is said to be \(M_ p\)-summable, we write \(s_ n\to \xi (M_ p)\), if \(\sigma_ n=(1/p_ n)\sum^{n}_{k=0}p_ ks_ k\to_{n}\xi.\) The authors prove a Tauberian theorem for \(J_ p\)- summable sequences as well as results of so called 0-Tauberian theorems for \(J_ p\)-methods and \(M_ p\)-methods. In the end, there are interesting discussions on these methods with various hypotheses on p.