Some remarks on summability factors

Abstract

Bosanquet [2] showed that a necessary and sufficient condition for Σ k = 1 ∞ x k y k \Sigma _{k = 1}^\infty {x_k}{y_k} to be Cesàro summable of order n n ( n n is a nonnegative integer) whenever σ k n ( y ) = o ( k ) \sigma _k^n(y) = o(k) where σ k n ( y ) \sigma _k^n(y) is the k k th Cesàro mean of y y of order n n is that Σ k = 1 ∞ k n + 1 | Δ n + 1 x k | > ∞ \Sigma _{k = 1}^\infty {k^{n + 1}}|{\Delta ^{n + 1}}{x_k}| > \infty and lim k → 0 k x k = 0 {\lim _{k \to 0}}k{x_k} = 0 . The main result of this paper is to show that a necessary and sufficient condition for Σ k = 1 ∞ x k y k \Sigma _{k = 1}^\infty {x_k}{y_k} to be Cesàro summable of order n n ( n n is a nonnegative integer) whenever Σ k = 1 ∞ k n + 1 | Δ n + 1 x k | > ∞ \Sigma _{k = 1}^\infty {k^{n + 1}}|{\Delta ^{n + 1}}{x_k}| > \infty and lim k → ∞ k x k = 0 {\lim _{k \to \infty }}k{x_k} = 0 is that σ k n ( y ) = o ( k ) \sigma _k^n(y) = o(k) .