A note on \(\ell\)-to-\(\ell\) Nörlund method (Q801225)

\textit{K. Knopp} and \textit{G. G. Lorentz} [Arch. Math. 2, 10-16 (1949; Zbl 0041.184)] have proved the following basic result: The infinite matrix transformation \(A=(a_{nk})\) is \(\ell\)-to-\(\ell\) if and only if \(\sup_{k}\{\sum^{\infty}_{n=0}| a_{nk}| \}<M\), and \(\sum^{\infty}_{n=0}a_{nk}=1\) for each k. In this paper Nörlund methods of summability are studied as mappings from the sequence space \(\ell\) into itself. Some inclusion results for the Cauchy product of Nörlund methods are established. By an example it is shown that these inclusion results do not hold good for all Nörlund methods. At the end of this paper an open problem in the form of a conjecture is also presented.