Some Topics in Summability Theory

The chapter is divided into nine sections. In the first section, we prove a theorem, which gives an equivalent formulation of summability by weighted mean methods. A result of Hardy and a result of Móricz and Rhoades are particular cases of this theorem. We also prove another interesting result on weighted mean methods in this section. In Sect. 2, we introduce the \((M, \lambda _n)\) method (or Natarajan method) of summability and study its properties concerning its regularity, consistency and translativity. We also prove an inclusion theorem and an equivalence theorem. In Sect. 3, we study the connection between the Natarajan method and the Abel method. In Sect. 4, we establish an interesting product theorem involving the Euler and the Natarajan methods. Section 5 is devoted to a study of some more nice properties of the Natarajan method. In Sect. 6, a sequence space \(\Lambda _r\), \(r \ge 1\) being a fixed integer, is defined and a Steinhaus type theorem involving this space is proved. Section 7 is devoted to a study of the above sequence space in the context of sequences of zeros and ones. In Sect. 8, we study the core of a sequence. We prove an improvement of a result of Sherbakhoff, which incidentally gives rise to a very simple proof of Knopp’s core theorem. In the final Sect. 9, we present a study of the matrix class \((\ell , \ell )\).