Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Almost convergence of double sequences and strong regularity of summability matrices

F. Móricza1 and B. E. Rhoadesa2

a1 University of Szeged, Bolyai Institute, Aradi vertanuk tere 1, 6720 Szeged, Hungary

a2 Indiana University, Department of Mathematics, Bloomington, Indiana 47405, U.S.A.

A double sequence x = {xjk: j, k = 0, 1, …} of real numbers is called almost convergent to a limit s if

S0305004100065464_eqnU001

that is, the average value of {xjk} taken over any rectangle {(j, k): mjm + p − 1, nkn + q − 1} tends to s as both p and q tend to ∞, and this convergence is uniform in m and n. The notion of almost convergence for single sequences was introduced by Lorentz [1].

(Received December 16 1986)

(Revised December 15 1987)