Special Summability Methods I

Abstract

This chapter introduces the Norlund method that is defined by the infinite matrix (ank) in certain conditions. It proves that any two regular Norlund methods (N, pn) and (N, qn) are consistent. It also proves an inclusion and an equivalence theorem for Norlund methods. Regular Norlund methods (N, pn), (N, qn) are equivalent if and only if ∑ |kn| < ∞ and ∑ |hn| < ∞. The chapter presents a limitation theorem on sequences that are summable and focuses on the Weighted Mean method. It introduces the Abel method and notes that the Abel method cannot be defined by an infinite matrix so that there are “nonmatrix summability methods”. An Abel method can also be regarded as a semicontinuous method. Poisson used this method in the summation of Fourier series and the method is sometimes attributed to Poisson.