Absolute Tauberian constants for H-J Hausdorff matrices

Abstract

Let C-k denote the Ces'aro matrix of order k>0, Sigma a(k) a series with partial sums s(n). Then, with C-n (k):= ((n+k)(n)) Sigma(n)(v=1) ((n-v+k)(n-v))a(v), Sherif [5] obtained estimates of the form Sigma vertical bar tau(n) - a(n)vertical bar <= K Sigma vertical bar Delta(na(n))vertical bar and Sigma vertical bar tau(n) - a(n)vertical bar <= K'Sigma(n)vertical bar Delta tau(n-1)], under the assumption that Sigma n vertical bar Delta tau(n-1)vertical bar is finite where Delta is the forward diference operator and tau(n) := c(n)(k) - c(n-l)(k). The constants K and k' he names absolute Tauberian constants. In a later paper [6] he obtained analogous results for regular Hausdorff matrices In this paper we obtain results similar to [6] for the H-J and E-J generalized Hausdorff matrices.