Let l(infinity), c and c(0) be the Banach spaces of bounded convergent and null sequences x = (xk)(1)(infinity), respectively. Write Delta x = (x(k) - x(k+1))(1)(infinity) and Delta(2)x = (Delta x(k) - Delta x(k+1))(1)(infinity). In [Canad. Math. Bull. 24 (1981), 169-176] H. Kizmaz has introduced and studied the sequence spaces, E(Delta) = {x: Delta x is an element of E}, where E is an element of {c(0), c, l(infinity)}. Recently, [Doga Mat. 17 (1993), 18-24] Mikail Et defined the sets E(Delta(2)), {x: Delta(2)x is an element of E}. He obtained alpha-duals of these sets and characterized the matrix class (E, F(Delta(2))), where E, F is an element of {c(0), c, l(infinity)}. In this paper, we generalize these sets and define E(u; Delta(2)) = {x:u .Delta(2) x is an element of E}, where u = (u(k)) is another sequence such that u(k) not equal 0 (k = 1, 2,...). We obtain alpha- and beta-duals of these sets and further we characterize the matrix classes (E(u; Delta(2)), F) and (E, F(u; Delta(2))). (C) 1990 Academic Press, Inc.
