The Dual Spaces of Sets of Difference Sequences of Order m and Matrix Transformations

Abstract

Let $p = {\left( {p_{k} } \right)}^{\infty }_{{k = 0}}$ be a bounded sequence of positive reals, m ∈ ℕ and u be s sequence of nonzero terms. If $x = {\left( {x_{k} } \right)}^{\infty }_{{k = 0}}$ is any sequence of complex numbers we write Δ^(m) x for the sequence of the m–th order differences of x and $\Delta ^{{{\left( m \right)}}}_{u} X = {\left\{ {x = {\left( x \right)}^{\infty }_{{k = 0}} :u\Delta ^{{{\left( m \right)}}} x \in X} \right\}}$ for any set X of sequences. We determine the α–, β– and γ–duals of the sets $\Delta ^{{{\left( m \right)}}}_{u} X$ for X = c ₀(p), c(p), ℓ∞(p) and characterize some matrix transformations between these spaces Δ^(m) X.