Modified means (Q916924)

Let \(t_ n=\sum^{\infty}_{k=1}a_{nk}s_ k\), and \(\phi_ n=t_ n-t_{n-1}\), \(t_{-1}=0\), \(\sum a_ n\in (A)\) if \(t\in C\) and \(\sum^{\infty}_{n=0}\phi_ n=s\). For a given sequence \((\lambda_ n)\) the author defines \(\tau_ n=\sum^{\infty}_{k=0}a_{nk}\lambda_ ka_ k\) and \(\psi_ n=\tau_ n/\lambda_ n=\sum^{\infty}_{k=0}b_{nk}a_ k,\) \((b_{nk})=(\lambda_ ka_{nk}/\lambda_ n)\); then \(\sum a_ n\) is said to be summable by modified A-means of weight \(\lambda\) or summable \((A',\lambda)\) to s if \(\sum \psi_ n=s\) and \(\sum a_ n\in (A',\lambda)\) if \(\psi\in \ell\). The method \((A)\sim (A',\lambda)\) if \(\sum \phi_ n\) converges (\(\Leftrightarrow\sum \psi_ n\) converges \(| A| \sim | A',\lambda |\) if \(\phi\in \ell \Leftrightarrow \psi \in \ell\). The author determines a class of sequences \(\lambda =(\lambda_ n)\) for which the summability method (A) and its modified method are equivalent. He also studies conditions on \(\lambda\) and p such that \(| N',p,\lambda | \Leftrightarrow | N,p|\).