In [1] S r ( Δ ) : = { x = ( x k ) : ( k r | Δ x k | ) k = 1 ∞ ∈ c 0 } for r ≥ 1 is studied. In this paper, we generalize this space to S r ( p , Δ ) for a sequence of strictly positive reals. We give a characterization of the matrix classes ( S r ( p , Δ ) , ℓ ∞ ) and ( S r ( p , Δ ) , ℓ 1 ) .